3.384 \(\int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx\)
Optimal. Leaf size=254 \[ -\frac {(a (A-B)-b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 (a B+A b)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(a (A+B)+b (A-B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a (A+B)+b (A-B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)} \]
[Out]
1/2*(a*(A-B)-b*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/2*(a*(A-B)-b*(A+B))*arctan(1+2^(1/2)*tan
(d*x+c)^(1/2))/d*2^(1/2)+1/4*(b*(A-B)+a*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-1/4*(b*(A-B
)+a*(A+B))*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)+2*(A*a-B*b)/d/tan(d*x+c)^(1/2)-2/5*a*A/d/tan(d*
x+c)^(5/2)-2/3*(A*b+B*a)/d/tan(d*x+c)^(3/2)
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Rubi [A] time = 0.26, antiderivative size = 254, normalized size of antiderivative = 1.00,
number of steps used = 13, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used
= {3591, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {(a (A-B)-b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}-\frac {2 (a B+A b)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(a (A+B)+b (A-B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a (A+B)+b (A-B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]
[Out]
-(((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a*(A - B) - b*(A + B))*Arc
Tan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]])/(2*Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])
/(2*Sqrt[2]*d) - (2*a*A)/(5*d*Tan[c + d*x]^(5/2)) - (2*(A*b + a*B))/(3*d*Tan[c + d*x]^(3/2)) + (2*(a*A - b*B))
/(d*Sqrt[Tan[c + d*x]])
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 1168
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]
Rule 3529
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
Rule 3534
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]
Rule 3591
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2
+ b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*
c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\tan ^{\frac {7}{2}}(c+d x)} \, dx &=-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\int \frac {A b+a B-(a A-b B) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx\\ &=-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\int \frac {-a A+b B-(A b+a B) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\int \frac {-A b-a B+(a A-b B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx\\ &=-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {-A b-a B+(a A-b B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}-\frac {(b (A-B)+a (A+B)) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {(a (A-B)-b (A+B)) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(b (A-B)+a (A+B)) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(b (A-B)+a (A+B)) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {(a (A-B)-b (A+B)) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}\\ &=\frac {(b (A-B)+a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}+\frac {(a (A-B)-b (A+B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}\\ &=-\frac {(a (A-B)-b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(b (A-B)+a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a A}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b+a B)}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 (a A-b B)}{d \sqrt {\tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 198, normalized size = 0.78 \[ -\frac {30 \sqrt {2} (a (A-B)-b (A+B)) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )+\frac {40 (a B+A b)}{\tan ^{\frac {3}{2}}(c+d x)}-\frac {120 (a A-b B)}{\sqrt {\tan (c+d x)}}-15 \sqrt {2} (a (A+B)+b (A-B)) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )+\frac {24 a A}{\tan ^{\frac {5}{2}}(c+d x)}}{60 d} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2),x]
[Out]
-1/60*(30*Sqrt[2]*(a*(A - B) - b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Ta
n[c + d*x]]]) - 15*Sqrt[2]*(b*(A - B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1
+ Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (24*a*A)/Tan[c + d*x]^(5/2) + (40*(A*b + a*B))/Tan[c + d*x]^(
3/2) - (120*(a*A - b*B))/Sqrt[Tan[c + d*x]])/d
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fricas [B] time = 20.00, size = 14358, normalized size = 56.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c))*(A+B*tan(d*x+c))/tan(d*x+c)^(7/2),x, algorithm="fricas")
[Out]
-1/60*(60*sqrt(2)*(d^5*cos(d*x + c)^4 - 2*d^5*cos(d*x + c)^2 + d^5)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4
+ 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt((
(A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2
*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 +
(A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^
2*B^2 + B^4)*b^4)/d^4)^(3/4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2
+ B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4)*arctan(-(((A^8 + 2*A^6*B^2 - 2*A
^2*B^6 - B^8)*a^8 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^7*b + 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^
6*b^2 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^5*b^3 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^3*b^
5 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^6 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a*b^7 - (A^8 + 2
*A^6*B^2 - 2*A^2*B^6 - B^8)*b^8)*d^4*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (
A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2
*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) - sqrt(2)*((B*a + A*b)*d^7*s
qrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(
((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3
)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + ((A^3 + A*B^2)*a^3 - (A^2*B + B^3)*a^2*b + (A^3 + A*B^2)*a*b^2 -
(A^2*B + B^3)*b^3)*d^5*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^
4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*
sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A
^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a
*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt((((A^6 - A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8*(A^5*B - A*B^5)*a^5*b - (A
^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^2*b^4 + 8*(A^5*B - A*B^5
)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^
4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) + sqrt(2)*(((A^5 - 2*A^3*B^2 + A*B^4)*a^5 - (9*A^4
*B - 10*A^2*B^3 + B^5)*a^4*b - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^3*b^2 + 2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^2*b^3
+ (A^5 - 10*A^3*B^2 + 9*A*B^4)*a*b^4 - (A^4*B - 2*A^2*B^3 + B^5)*b^5)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) + ((A^6*B - A^4*B^3 - A^2*
B^5 + B^7)*a^7 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a^6*b - (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^5*b
^2 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^4*b^3 - (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^3*b^4 - (A^7
- 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^2*b^5 + (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a*b^6 + (A^7 - A^5*B^2 -
A^3*B^4 + A*B^6)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 +
(A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B
- A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))
*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*
A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^7*b +
16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^5*b^3 - 2*(A^8 - 16*A^6*B
^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^5 + 16*(A^6*B^2 + 2*
A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2*A^4*B^4 + B^8)*b^8)*sin(d*
x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B
^4)*b^4)/d^4)^(3/4) + sqrt(2)*(((A^4*B - B^5)*a^5 + (A^5 - 4*A^3*B^2 - 5*A*B^4)*a^4*b - 4*(A^4*B + A^2*B^3)*a^
3*b^2 - 4*(A^3*B^2 + A*B^4)*a^2*b^3 - (5*A^4*B + 4*A^2*B^3 - B^5)*a*b^4 - (A^5 - A*B^4)*b^5)*d^7*sqrt(((A^4 +
2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^
2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A
^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + ((A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*a^7 - (5*A^6*B + 9*A^4*B^3 + 3*A^2*B^5 -
B^7)*a^6*b + (A^7 + 5*A^5*B^2 + 7*A^3*B^4 + 3*A*B^6)*a^5*b^2 - (9*A^6*B + 17*A^4*B^3 + 7*A^2*B^5 - B^7)*a^4*b^
3 - (A^7 - 7*A^5*B^2 - 17*A^3*B^4 - 9*A*B^6)*a^3*b^4 - (3*A^6*B + 7*A^4*B^3 + 5*A^2*B^5 + B^7)*a^2*b^5 - (A^7
- 3*A^5*B^2 - 9*A^3*B^4 - 5*A*B^6)*a*b^6 + (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*b^7)*d^5*sqrt(((A^4 - 2*A^2*B^2 +
B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*
A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2
*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A
^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^
3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt(sin(d*
x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^
4)*b^4)/d^4)^(3/4))/((A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*a^12 - 8*(A^11*B
+ 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^11*b + 2*(A^12 + 10*A^10*B^2 + 31*A^8*B^4 + 44*A^6
*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^10*b^2 - 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9
- A*B^11)*a^9*b^3 - (A^12 - 62*A^10*B^2 - 257*A^8*B^4 - 388*A^6*B^6 - 257*A^4*B^8 - 62*A^2*B^10 + B^12)*a^8*b^
4 - 16*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^7*b^5 - 4*(A^12 - 22*A^10*B^2 - 97*
A^8*B^4 - 148*A^6*B^6 - 97*A^4*B^8 - 22*A^2*B^10 + B^12)*a^6*b^6 + 16*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*
B^7 - 3*A^3*B^9 - A*B^11)*a^5*b^7 - (A^12 - 62*A^10*B^2 - 257*A^8*B^4 - 388*A^6*B^6 - 257*A^4*B^8 - 62*A^2*B^1
0 + B^12)*a^4*b^8 + 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^3*b^9 + 2*(A^12 + 1
0*A^10*B^2 + 31*A^8*B^4 + 44*A^6*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^2*b^10 + 8*(A^11*B + 3*A^9*B^3 + 2*A
^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^11 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B
^10 + B^12)*b^12)) + 60*sqrt(2)*(d^5*cos(d*x + c)^4 - 2*d^5*cos(d*x + c)^2 + d^5)*sqrt(((A^4 + 2*A^2*B^2 + B^4
)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a
*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d
^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B -
A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2
+ (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^
4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4)*arctan((((A^8 + 2*
A^6*B^2 - 2*A^2*B^6 - B^8)*a^8 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^7*b + 2*(A^8 + 2*A^6*B^2 - 2*A^2*
B^6 - B^8)*a^6*b^2 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^5*b^3 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 +
A*B^7)*a^3*b^5 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^6 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a*b
^7 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^8)*d^4*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4
)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*
(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + sqrt(2)*((B*a
+ A*b)*d^7*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^
4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(
A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + ((A^3 + A*B^2)*a^3 - (A^2*B + B^3)*a^2*b + (A^3 + A
*B^2)*a*b^2 - (A^2*B + B^3)*b^3)*d^5*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10
*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B
^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b
^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3
*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt((((A^6 - A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8*(A^5*B - A*B^
5)*a^5*b - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^2*b^4 + 8*(
A^5*B - A*B^5)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2
*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - sqrt(2)*(((A^5 - 2*A^3*B^2 + A*B^4)
*a^5 - (9*A^4*B - 10*A^2*B^3 + B^5)*a^4*b - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^3*b^2 + 2*(5*A^4*B - 14*A^2*B^3 +
B^5)*a^2*b^3 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a*b^4 - (A^4*B - 2*A^2*B^3 + B^5)*b^5)*d^3*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) + ((A^6*B - A
^4*B^3 - A^2*B^5 + B^7)*a^7 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a^6*b - (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5
+ B^7)*a^5*b^2 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^4*b^3 - (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^
3*b^4 - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^2*b^5 + (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a*b^6 + (A^7
- A^5*B^2 - A^3*B^4 + A*B^6)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^
4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a
^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2
+ B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^
2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*
B^7)*a^7*b + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^5*b^3 - 2*(A
^8 - 16*A^6*B^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^5 + 16*
(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2*A^4*B^4 + B^8
)*b^8)*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 +
2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4) - sqrt(2)*(((A^4*B - B^5)*a^5 + (A^5 - 4*A^3*B^2 - 5*A*B^4)*a^4*b - 4*(A^4*B
+ A^2*B^3)*a^3*b^2 - 4*(A^3*B^2 + A*B^4)*a^2*b^3 - (5*A^4*B + 4*A^2*B^3 - B^5)*a*b^4 - (A^5 - A*B^4)*b^5)*d^7*
sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt
(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^
3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + ((A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*a^7 - (5*A^6*B + 9*A^4*B^3 +
3*A^2*B^5 - B^7)*a^6*b + (A^7 + 5*A^5*B^2 + 7*A^3*B^4 + 3*A*B^6)*a^5*b^2 - (9*A^6*B + 17*A^4*B^3 + 7*A^2*B^5
- B^7)*a^4*b^3 - (A^7 - 7*A^5*B^2 - 17*A^3*B^4 - 9*A*B^6)*a^3*b^4 - (3*A^6*B + 7*A^4*B^3 + 5*A^2*B^5 + B^7)*a^
2*b^5 - (A^7 - 3*A^5*B^2 - 9*A^3*B^4 - 5*A*B^6)*a*b^6 + (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*b^7)*d^5*sqrt(((A^4
- 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^
3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 +
(A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*
B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)
)*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2
*A^2*B^2 + B^4)*b^4)/d^4)^(3/4))/((A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*a^12
- 8*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^11*b + 2*(A^12 + 10*A^10*B^2 + 31*A^8
*B^4 + 44*A^6*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^10*b^2 - 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7
- 3*A^3*B^9 - A*B^11)*a^9*b^3 - (A^12 - 62*A^10*B^2 - 257*A^8*B^4 - 388*A^6*B^6 - 257*A^4*B^8 - 62*A^2*B^10 +
B^12)*a^8*b^4 - 16*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^7*b^5 - 4*(A^12 - 22*A
^10*B^2 - 97*A^8*B^4 - 148*A^6*B^6 - 97*A^4*B^8 - 22*A^2*B^10 + B^12)*a^6*b^6 + 16*(A^11*B + 3*A^9*B^3 + 2*A^7
*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^5*b^7 - (A^12 - 62*A^10*B^2 - 257*A^8*B^4 - 388*A^6*B^6 - 257*A^4*B^8
- 62*A^2*B^10 + B^12)*a^4*b^8 + 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^3*b^9
+ 2*(A^12 + 10*A^10*B^2 + 31*A^8*B^4 + 44*A^6*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^2*b^10 + 8*(A^11*B + 3*
A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^11 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*
B^8 + 2*A^2*B^10 + B^12)*b^12)) + 15*sqrt(2)*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2
+ (A^4 + 2*A^2*B^2 + B^4)*b^4)*d*cos(d*x + c)^4 - 2*((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*
a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d*cos(d*x + c)^2 + ((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 +
B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d + 2*((A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3*cos(d*x + c)^4 -
2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3*cos(d*x + c)^2 + (A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3)*sqrt((
(A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^
4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*
b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^
2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^
2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2
*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4)*log((((A^6 - A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8*(A
^5*B - A*B^5)*a^5*b - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^
2*b^4 + 8*(A^5*B - A*B^5)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) + sqrt(2)*(((A^5 - 2*A^3*B
^2 + A*B^4)*a^5 - (9*A^4*B - 10*A^2*B^3 + B^5)*a^4*b - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^3*b^2 + 2*(5*A^4*B - 1
4*A^2*B^3 + B^5)*a^2*b^3 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a*b^4 - (A^4*B - 2*A^2*B^3 + B^5)*b^5)*d^3*sqrt(((A^4
+ 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) +
((A^6*B - A^4*B^3 - A^2*B^5 + B^7)*a^7 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a^6*b - (9*A^6*B - 17*A^4*B^3 -
25*A^2*B^5 + B^7)*a^5*b^2 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^4*b^3 - (A^6*B - 17*A^4*B^3 - 17*A^2*B^
5 + B^7)*a^3*b^4 - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^2*b^5 + (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a
*b^6 + (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A
^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 +
2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B
^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4
- 2*A^2*B^2 + B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 +
B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3 - A
^3*B^5 - A*B^7)*a^7*b + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^5
*b^3 - 2*(A^8 - 16*A^6*B^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^
3*b^5 + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2*A^
4*B^4 + B^8)*b^8)*sin(d*x + c))/cos(d*x + c)) - 15*sqrt(2)*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2
+ B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d*cos(d*x + c)^4 - 2*((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A
^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d*cos(d*x + c)^2 + ((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4
+ 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d + 2*((A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3*cos
(d*x + c)^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3*cos(d*x + c)^2 + (A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*
b)*d^3)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d
^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(
A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2
+ (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2
*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4)*log((((A^6 - A^4*B^2 - A^2*B^4 + B
^6)*a^6 - 8*(A^5*B - A*B^5)*a^5*b - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2
*B^4 + B^6)*a^2*b^4 + 8*(A^5*B - A*B^5)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^
2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - sqrt(2)*((
(A^5 - 2*A^3*B^2 + A*B^4)*a^5 - (9*A^4*B - 10*A^2*B^3 + B^5)*a^4*b - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^3*b^2 +
2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^2*b^3 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a*b^4 - (A^4*B - 2*A^2*B^3 + B^5)*b^5)*d
^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*c
os(d*x + c) + ((A^6*B - A^4*B^3 - A^2*B^5 + B^7)*a^7 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a^6*b - (9*A^6*B
- 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^5*b^2 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^4*b^3 - (A^6*B - 17*A^4*B
^3 - 17*A^2*B^5 + B^7)*a^3*b^4 - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^2*b^5 + (9*A^6*B - A^4*B^3 - 9*A^
2*B^5 + B^7)*a*b^6 + (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4
+ 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 - 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^
2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/(
(A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)
*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4
+ 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B
+ A^5*B^3 - A^3*B^5 - A*B^7)*a^7*b + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^
5 - A*B^7)*a^5*b^3 - 2*(A^8 - 16*A^6*B^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B
^5 - A*B^7)*a^3*b^5 + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7
+ (A^8 - 2*A^4*B^4 + B^8)*b^8)*sin(d*x + c))/cos(d*x + c)) - 8*(5*((A^4*B + 2*A^2*B^3 + B^5)*a^5 + (A^5 + 2*A
^3*B^2 + A*B^4)*a^4*b + 2*(A^4*B + 2*A^2*B^3 + B^5)*a^3*b^2 + 2*(A^5 + 2*A^3*B^2 + A*B^4)*a^2*b^3 + (A^4*B + 2
*A^2*B^3 + B^5)*a*b^4 + (A^5 + 2*A^3*B^2 + A*B^4)*b^5)*cos(d*x + c)^4 - 5*((A^4*B + 2*A^2*B^3 + B^5)*a^5 + (A^
5 + 2*A^3*B^2 + A*B^4)*a^4*b + 2*(A^4*B + 2*A^2*B^3 + B^5)*a^3*b^2 + 2*(A^5 + 2*A^3*B^2 + A*B^4)*a^2*b^3 + (A^
4*B + 2*A^2*B^3 + B^5)*a*b^4 + (A^5 + 2*A^3*B^2 + A*B^4)*b^5)*cos(d*x + c)^2 - 3*((6*(A^5 + 2*A^3*B^2 + A*B^4)
*a^5 - 5*(A^4*B + 2*A^2*B^3 + B^5)*a^4*b + 12*(A^5 + 2*A^3*B^2 + A*B^4)*a^3*b^2 - 10*(A^4*B + 2*A^2*B^3 + B^5)
*a^2*b^3 + 6*(A^5 + 2*A^3*B^2 + A*B^4)*a*b^4 - 5*(A^4*B + 2*A^2*B^3 + B^5)*b^5)*cos(d*x + c)^3 - 5*((A^5 + 2*A
^3*B^2 + A*B^4)*a^5 - (A^4*B + 2*A^2*B^3 + B^5)*a^4*b + 2*(A^5 + 2*A^3*B^2 + A*B^4)*a^3*b^2 - 2*(A^4*B + 2*A^2
*B^3 + B^5)*a^2*b^3 + (A^5 + 2*A^3*B^2 + A*B^4)*a*b^4 - (A^4*B + 2*A^2*B^3 + B^5)*b^5)*cos(d*x + c))*sin(d*x +
c))*sqrt(sin(d*x + c)/cos(d*x + c)))/(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4
+ 2*A^2*B^2 + B^4)*b^4)*d*cos(d*x + c)^4 - 2*((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2
+ (A^4 + 2*A^2*B^2 + B^4)*b^4)*d*cos(d*x + c)^2 + ((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^
2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}{\tan \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c))*(A+B*tan(d*x+c))/tan(d*x+c)^(7/2),x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)/tan(d*x + c)^(7/2), x)
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maple [B] time = 0.11, size = 497, normalized size = 1.96 \[ -\frac {2 A b}{3 d \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a B}{3 d \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a A}{d \sqrt {\tan \left (d x +c \right )}}-\frac {2 B b}{d \sqrt {\tan \left (d x +c \right )}}-\frac {2 a A}{5 d \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d}-\frac {A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d}-\frac {A \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b}{4 d}-\frac {a B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d}-\frac {a B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d}-\frac {a B \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d}+\frac {a A \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d}+\frac {a A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d}+\frac {a A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d}-\frac {B \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b}{4 d}-\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d}-\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*tan(d*x+c))*(A+B*tan(d*x+c))/tan(d*x+c)^(7/2),x)
[Out]
-2/3/d/tan(d*x+c)^(3/2)*A*b-2/3/d/tan(d*x+c)^(3/2)*a*B+2*a*A/d/tan(d*x+c)^(1/2)-2/d/tan(d*x+c)^(1/2)*B*b-2/5*a
*A/d/tan(d*x+c)^(5/2)-1/2/d*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b-1/2/d*A*2^(1/2)*arctan(-1+2^(1/2)*t
an(d*x+c)^(1/2))*b-1/4/d*A*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c)))*b-1/2/d*a*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))-1/2/d*a*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)
^(1/2))-1/4/d*a*B*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+
1/4/d*a*A*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+1/2/d*a*
A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+1/2/d*a*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-1/4/d*B*2^(
1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*b-1/2/d*B*2^(1/2)*arc
tan(1+2^(1/2)*tan(d*x+c)^(1/2))*b-1/2/d*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b
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maxima [A] time = 0.45, size = 211, normalized size = 0.83 \[ \frac {30 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 30 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \frac {8 \, {\left (15 \, {\left (A a - B b\right )} \tan \left (d x + c\right )^{2} - 3 \, A a - 5 \, {\left (B a + A b\right )} \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c))*(A+B*tan(d*x+c))/tan(d*x+c)^(7/2),x, algorithm="maxima")
[Out]
1/60*(30*sqrt(2)*((A - B)*a - (A + B)*b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 30*sqrt(2)*((A
- B)*a - (A + B)*b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - 15*sqrt(2)*((A + B)*a + (A - B)*b
)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + 15*sqrt(2)*((A + B)*a + (A - B)*b)*log(-sqrt(2)*sqrt(ta
n(d*x + c)) + tan(d*x + c) + 1) + 8*(15*(A*a - B*b)*tan(d*x + c)^2 - 3*A*a - 5*(B*a + A*b)*tan(d*x + c))/tan(d
*x + c)^(5/2))/d
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mupad [B] time = 12.50, size = 1473, normalized size = 5.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*tan(c + d*x))*(a + b*tan(c + d*x)))/tan(c + d*x)^(7/2),x)
[Out]
2*atanh((32*A^2*a^2*d^3*tan(c + d*x)^(1/2)*((2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2)/(4*d^4) + (A
^2*a*b)/(2*d^2))^(1/2))/(16*A*b*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2) - 16*A^3*a^3*d^2 + 16*A^
3*a*b^2*d^2) - (32*A^2*b^2*d^3*tan(c + d*x)^(1/2)*((2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2)/(4*d^
4) + (A^2*a*b)/(2*d^2))^(1/2))/(16*A*b*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2) - 16*A^3*a^3*d^2
+ 16*A^3*a*b^2*d^2))*((2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2)/(4*d^4) + (A^2*a*b)/(2*d^2))^(1/2)
- 2*atanh((32*A^2*a^2*d^3*tan(c + d*x)^(1/2)*((A^2*a*b)/(2*d^2) - (2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*
d^4)^(1/2)/(4*d^4))^(1/2))/(16*A*b*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2) + 16*A^3*a^3*d^2 - 16
*A^3*a*b^2*d^2) - (32*A^2*b^2*d^3*tan(c + d*x)^(1/2)*((A^2*a*b)/(2*d^2) - (2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A
^4*a^4*d^4)^(1/2)/(4*d^4))^(1/2))/(16*A*b*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2) + 16*A^3*a^3*d
^2 - 16*A^3*a*b^2*d^2))*((A^2*a*b)/(2*d^2) - (2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2)/(4*d^4))^(1
/2) + 2*atanh((32*B^2*a^2*d^3*tan(c + d*x)^(1/2)*(- (2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2)/(4*d
^4) - (B^2*a*b)/(2*d^2))^(1/2))/(16*B*a*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2) - 16*B^3*b^3*d^2
+ 16*B^3*a^2*b*d^2) - (32*B^2*b^2*d^3*tan(c + d*x)^(1/2)*(- (2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(
1/2)/(4*d^4) - (B^2*a*b)/(2*d^2))^(1/2))/(16*B*a*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2) - 16*B^
3*b^3*d^2 + 16*B^3*a^2*b*d^2))*(- (2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2)/(4*d^4) - (B^2*a*b)/(2
*d^2))^(1/2) - 2*atanh((32*B^2*a^2*d^3*tan(c + d*x)^(1/2)*((2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/
2)/(4*d^4) - (B^2*a*b)/(2*d^2))^(1/2))/(16*B*a*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2) + 16*B^3*
b^3*d^2 - 16*B^3*a^2*b*d^2) - (32*B^2*b^2*d^3*tan(c + d*x)^(1/2)*((2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d
^4)^(1/2)/(4*d^4) - (B^2*a*b)/(2*d^2))^(1/2))/(16*B*a*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2) +
16*B^3*b^3*d^2 - 16*B^3*a^2*b*d^2))*((2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1/2)/(4*d^4) - (B^2*a*b)
/(2*d^2))^(1/2) - ((2*A*a)/5 + (2*A*b*tan(c + d*x))/3 - 2*A*a*tan(c + d*x)^2)/(d*tan(c + d*x)^(5/2)) - ((2*B*a
)/3 + 2*B*b*tan(c + d*x))/(d*tan(c + d*x)^(3/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )}{\tan ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(d*x+c))*(A+B*tan(d*x+c))/tan(d*x+c)**(7/2),x)
[Out]
Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))/tan(c + d*x)**(7/2), x)
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