3.379 \(\int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=254 \[ \frac {(a (A+B)+b (A-B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a (A+B)+b (A-B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 (a B+A b) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {(a (A-B)-b (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d} \]

[Out]

-1/2*(b*(A-B)+a*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*(b*(A-B)+a*(A+B))*arctan(1+2^(1/2)*ta
n(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a*(A-B)-b*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-1/4*(a*(A-
B)-b*(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)*tan(d*x+c)^(1/2)/d+2/3*(A*b+B*a)*t
an(d*x+c)^(3/2)/d+2/5*b*B*tan(d*x+c)^(5/2)/d

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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 = {3592, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {2 (a B+A b) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {(a (A+B)+b (A-B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a (A+B)+b (A-B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {(a (A-B)-b (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]

[Out]

((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*ArcTan
[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] +
Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2
*Sqrt[2]*d) + (2*(a*A - b*B)*Sqrt[Tan[c + d*x]])/d + (2*(A*b + a*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*b*B*Tan[c +
d*x]^(5/2))/(5*d)

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 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

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 3592

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(B*d*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e
+ f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b
*c - a*d, 0] &&  !LeQ[m, -1]

Rubi steps

\begin {align*} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \tan ^{\frac {3}{2}}(c+d x) (a A-b B+(A b+a B) \tan (c+d x)) \, dx\\ &=\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \sqrt {\tan (c+d x)} (-A b-a B+(a A-b B) \tan (c+d x)) \, dx\\ &=\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \frac {-a A+b B-(A b+a B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx\\ &=\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 \operatorname {Subst}\left (\int \frac {-a A+b B+(-A b-a B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 (a A-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\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-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {(b (A-B)+a (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)) \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 {\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 {\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)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (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-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {(b (A-B)+a (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 {(b (A-B)+a (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 {(b (A-B)+a (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(b (A-B)+a (A+B)) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a (A-B)-b (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a (A-B)-b (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-b B) \sqrt {\tan (c+d x)}}{d}+\frac {2 (A b+a B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b B \tan ^{\frac {5}{2}}(c+d x)}{5 d}\\ \end {align*}

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Mathematica [C]  time = 1.05, size = 134, normalized size = 0.53 \[ \frac {15 \sqrt [4]{-1} (a-i b) (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (5 (a B+A b) \tan (c+d x)+15 (a A-b B)+3 b B \tan ^2(c+d x)\right )+15 \sqrt [4]{-1} (a+i b) (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{15 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]

[Out]

(15*(-1)^(1/4)*(a - I*b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 15*(-1)^(1/4)*(a + I*b)*(A + I*B)*A
rcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(15*(a*A - b*B) + 5*(A*b + a*B)*Tan[c + d*x] + 3*
b*B*Tan[c + d*x]^2))/(15*d)

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fricas [B]  time = 22.45, size = 13541, normalized size = 53.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(3/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/60*(60*sqrt(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)*((A*a - B*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^2*B + B^3)*a^3 + (A^3 + A*B^2)*a^2*b + (A^2*B + B^3)*a*b^2 + (A^3 + A*B^2)*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^4*B - 2*A^2*B^3 + B^5)*a^5 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14
*A^2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^2*b^3 + (9*A^4*B - 10*A^2*B^3 + B^5)*a*b^4 + (A^5 -
2*A^3*B^2 + A*B^4)*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^7 - A^5*B^2 - A^3*B^4 + A*B^6)*a^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^
5 + B^7)*a^6*b - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^5*b^2 + (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a
^4*b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*b^4 + (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 +
(A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*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^5 - A*B^4)*
a^5 - (5*A^4*B + 4*A^2*B^3 - B^5)*a^4*b + 4*(A^3*B^2 + A*B^4)*a^3*b^2 - 4*(A^4*B + A^2*B^3)*a^2*b^3 - (A^5 - 4
*A^3*B^2 - 5*A*B^4)*a*b^4 + (A^4*B - B^5)*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^6*B + A^
4*B^3 - A^2*B^5 - B^7)*a^7 + (A^7 - 3*A^5*B^2 - 9*A^3*B^4 - 5*A*B^6)*a^6*b - (3*A^6*B + 7*A^4*B^3 + 5*A^2*B^5
+ B^7)*a^5*b^2 + (A^7 - 7*A^5*B^2 - 17*A^3*B^4 - 9*A*B^6)*a^4*b^3 - (9*A^6*B + 17*A^4*B^3 + 7*A^2*B^5 - B^7)*a
^3*b^4 - (A^7 + 5*A^5*B^2 + 7*A^3*B^4 + 3*A*B^6)*a^2*b^5 - (5*A^6*B + 9*A^4*B^3 + 3*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^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^1
1 + (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))*cos(d*x + c)^2 + 60*sqrt(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)*((A*a - B*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^2*B + B^3)*a^3
+ (A^3 + A*B^2)*a^2*b + (A^2*B + B^3)*a*b^2 + (A^3 + A*B^2)*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 - 1
0*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) - sq
rt(2)*(((A^4*B - 2*A^2*B^3 + B^5)*a^5 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^
3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^2*b^3 + (9*A^4*B - 10*A^2*B^3 + B^5)*a*b^4 + (A^5 - 2*A^3*B^2 + A*B^4
)*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^7 - A^5*B^2 - A^3*B^4 + A*B^6)*a^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a^6*b - (
A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^5*b^2 + (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^4*b^3 - (A^7 - 17
*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*b^4 + (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 + (A^7 - 9*A^5*B^2
- A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*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^5 - A*B^4)*a^5 - (5*A^4*B + 4
*A^2*B^3 - B^5)*a^4*b + 4*(A^3*B^2 + A*B^4)*a^3*b^2 - 4*(A^4*B + A^2*B^3)*a^2*b^3 - (A^5 - 4*A^3*B^2 - 5*A*B^4
)*a*b^4 + (A^4*B - B^5)*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^6*B + A^4*B^3 - A^2*B^5 -
B^7)*a^7 + (A^7 - 3*A^5*B^2 - 9*A^3*B^4 - 5*A*B^6)*a^6*b - (3*A^6*B + 7*A^4*B^3 + 5*A^2*B^5 + B^7)*a^5*b^2 + (
A^7 - 7*A^5*B^2 - 17*A^3*B^4 - 9*A*B^6)*a^4*b^3 - (9*A^6*B + 17*A^4*B^3 + 7*A^2*B^5 - B^7)*a^3*b^4 - (A^7 + 5*
A^5*B^2 + 7*A^3*B^4 + 3*A*B^6)*a^2*b^5 - (5*A^6*B + 9*A^4*B^3 + 3*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^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*sq
rt(((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))*cos(d*x + c)^2 + 15*sqrt(2)*(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)*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*cos(d*x + c)^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 + 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^4*B - 2*A^2*B^3 + B^5)*a^5 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a^4*b - 2
*(5*A^4*B - 14*A^2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^2*b^3 + (9*A^4*B - 10*A^2*B^3 + B^5)*
a*b^4 + (A^5 - 2*A^3*B^2 + A*B^4)*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^7 - A^5*B^2 - A^3*B^4 + A*B^6)*a^7 - (9*A^6*B - A^4*
B^3 - 9*A^2*B^5 + B^7)*a^6*b - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^5*b^2 + (A^6*B - 17*A^4*B^3 - 17*A^
2*B^5 + B^7)*a^4*b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*b^4 + (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 +
B^7)*a^2*b^5 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*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)*(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)*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*cos(d*x + c)^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 + 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^4*B - 2*A^2*B^3 + B^5)*a^5 + (A^5 - 10*A^3*B^2
+ 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^2*b^3 + (9*A^4*B
- 10*A^2*B^3 + B^5)*a*b^4 + (A^5 - 2*A^3*B^2 + A*B^4)*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^7 - A^5*B^2 - A^3*B^4 + A*B^6)*a
^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a^6*b - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^5*b^2 + (A^6*B
- 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^4*b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*b^4 + (9*A^6*B - 17*A^4
*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B - A^4*B^3 - A^2*B^5 +
B^7)*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*(3*(A^4*B + 2*A^2*B^3 + B^5)*a^4*b + 6*(A^4*B + 2*A^2*B^3 + B^5)*a^2*b^3 + 3*(A^4*B + 2*A^2*B^3
+ B^5)*b^5 + 3*(5*(A^5 + 2*A^3*B^2 + A*B^4)*a^5 - 6*(A^4*B + 2*A^2*B^3 + B^5)*a^4*b + 10*(A^5 + 2*A^3*B^2 + A*
B^4)*a^3*b^2 - 12*(A^4*B + 2*A^2*B^3 + B^5)*a^2*b^3 + 5*(A^5 + 2*A^3*B^2 + A*B^4)*a*b^4 - 6*(A^4*B + 2*A^2*B^3
+ B^5)*b^5)*cos(d*x + c)^2 + 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)*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)^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(3/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.10, size = 497, normalized size = 1.96 \[ \frac {2 b B \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5 d}+\frac {2 A \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) b}{3 d}+\frac {2 B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) a}{3 d}+\frac {2 A \left (\sqrt {\tan }\left (d x +c \right )\right ) a}{d}-\frac {2 b B \left (\sqrt {\tan }\left (d x +c \right )\right )}{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 \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 ) a}{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 {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}\, \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 {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 \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 {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 ) a}{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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(3/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x)

[Out]

2/5*b*B*tan(d*x+c)^(5/2)/d+2/3/d*A*tan(d*x+c)^(3/2)*b+2/3/d*B*tan(d*x+c)^(3/2)*a+2/d*A*tan(d*x+c)^(1/2)*a-2*b*
B*tan(d*x+c)^(1/2)/d-1/2/d*a*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-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)))*a-1/2/d*a*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+
c)^(1/2))+1/2/d*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b+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)*arctan(1+2^(1/2)*tan(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*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)*tan(d*x+c)^(1/2))*b-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)))*a-1/2/d*a*B*2^(1/2)*ar
ctan(-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))

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maxima [A]  time = 0.67, size = 210, normalized size = 0.83 \[ \frac {24 \, B b \tan \left (d x + c\right )^{\frac {5}{2}} - 30 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 30 \, \sqrt {2} {\left ({\left (A + B\right )} a + {\left (A - B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left ({\left (A - B\right )} a - {\left (A + B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 40 \, {\left (B a + A b\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + 120 \, {\left (A a - B b\right )} \sqrt {\tan \left (d x + c\right )}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(3/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(24*B*b*tan(d*x + c)^(5/2) - 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*sq
rt(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(tan(d*x + c)) + tan(d*x + c) + 1) + 40*(B*a + A*b)*tan(d*x + c)^(3/2) + 120*(A*a - B
*b)*sqrt(tan(d*x + c)))/d

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mupad [B]  time = 11.62, size = 1492, normalized size = 5.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^(3/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x)),x)

[Out]

atan((A^2*a^2*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)*32i)/((16*A^3*b^3)/d + (16*A*a*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2))/d^3 - (16*A
^3*a^2*b)/d) - (A^2*b^2*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)*32i)/((16*A^3*b^3)/d + (16*A*a*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2))/d
^3 - (16*A^3*a^2*b)/d))*((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)*2i - atan((A^2*a^2*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)*32i)/((16*A*a*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2))/d^3 - (16*A^3*b^3
)/d + (16*A^3*a^2*b)/d) - (A^2*b^2*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)*32i)/((16*A*a*(2*A^4*a^2*b^2*d^4 - A^4*b^4*d^4 - A^4*a^4*d^4)^(1/2))/d^3 -
(16*A^3*b^3)/d + (16*A^3*a^2*b)/d))*(- (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)*2i - atan((B^2*a^2*tan(c + d*x)^(1/2)*((B^2*a*b)/(2*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))^(1/2)*32i)/((16*B^3*a^3)/d + (16*B*b*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4
*d^4)^(1/2))/d^3 - (16*B^3*a*b^2)/d) - (B^2*b^2*tan(c + d*x)^(1/2)*((B^2*a*b)/(2*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))^(1/2)*32i)/((16*B^3*a^3)/d + (16*B*b*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4
- B^4*a^4*d^4)^(1/2))/d^3 - (16*B^3*a*b^2)/d))*((B^2*a*b)/(2*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))^(1/2)*2i + atan((B^2*a^2*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)*32i)/((16*B*b*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a^4*d^4)^(1
/2))/d^3 - (16*B^3*a^3)/d + (16*B^3*a*b^2)/d) - (B^2*b^2*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)*32i)/((16*B*b*(2*B^4*a^2*b^2*d^4 - B^4*b^4*d^4 - B^4*a
^4*d^4)^(1/2))/d^3 - (16*B^3*a^3)/d + (16*B^3*a*b^2)/d))*((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)*2i + (2*A*a*tan(c + d*x)^(1/2))/d + (2*A*b*tan(c + d*x)^(3/2))/(3*d) + (2
*B*a*tan(c + d*x)^(3/2))/(3*d) - (2*B*b*tan(c + d*x)^(1/2))/d + (2*B*b*tan(c + d*x)^(5/2))/(5*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(3/2)*(a+b*tan(d*x+c))*(A+B*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))*tan(c + d*x)**(3/2), x)

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