3.583 \(\int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx\)

Optimal. Leaf size=271 \[ -\frac {(a-b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} d \left (a^2+b^2\right )}-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d} \]

[Out]

2*a^(7/2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(5/2)/(a^2+b^2)/d+1/2*(a-b)*arctan(-1+2^(1/2)*tan(d*x+c)^
(1/2))/(a^2+b^2)/d*2^(1/2)+1/2*(a-b)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/4*(a+b)*ln(1-2^(
1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)+1/4*(a+b)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2
+b^2)/d*2^(1/2)-2*a*tan(d*x+c)^(1/2)/b^2/d+2/3*tan(d*x+c)^(3/2)/b/d

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Rubi [A]  time = 0.57, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3566, 3647, 3654, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {2 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} d \left (a^2+b^2\right )}-\frac {(a-b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x]),x]

[Out]

-(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt
[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(
a^2 + b^2)*d) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a +
b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - (2*a*Sqrt[Tan[c + d*x]])/(
b^2*d) + (2*Tan[c + d*x]^(3/2))/(3*b*d)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 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 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^m*(c + d
*Tan[e + f*x])^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3654

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + a^2*C)/(a^2 + b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^
2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {2 \int \frac {\sqrt {\tan (c+d x)} \left (-\frac {3 a}{2}-\frac {3}{2} b \tan (c+d x)-\frac {3}{2} a \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b}\\ &=-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {4 \int \frac {\frac {3 a^2}{4}+\frac {3}{4} \left (a^2-b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 b^2}\\ &=-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {4 \int \frac {\frac {3 a b^2}{4}-\frac {3}{4} b^3 \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )}+\frac {a^4 \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {8 \operatorname {Subst}\left (\int \frac {\frac {3 a b^2}{4}-\frac {3 b^3 x^2}{4}}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 b^2 \left (a^2+b^2\right ) d}+\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac {\left (2 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {2 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {(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} \left (a^2+b^2\right ) d}-\frac {(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} \left (a^2+b^2\right ) d}\\ &=\frac {2 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}\\ &=-\frac {(a-b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a \sqrt {\tan (c+d x)}}{b^2 d}+\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d}\\ \end {align*}

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Mathematica [A]  time = 0.87, size = 222, normalized size = 0.82 \[ \frac {-\frac {6 \sqrt {2} b^2 (a-b) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{a^2+b^2}-\frac {3 \sqrt {2} b^2 (a+b) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{a^2+b^2}+\frac {24 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right )}-24 a \sqrt {\tan (c+d x)}+8 b \tan ^{\frac {3}{2}}(c+d x)}{12 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x]),x]

[Out]

((-6*Sqrt[2]*(a - b)*b^2*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a
^2 + b^2) + (24*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)) - (3*Sqrt[2]*b^2*(
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]
]))/(a^2 + b^2) - 24*a*Sqrt[Tan[c + d*x]] + 8*b*Tan[c + d*x]^(3/2))/(12*b^2*d)

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fricas [B]  time = 11.32, size = 7356, normalized size = 27.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

[1/12*(12*sqrt(2)*(a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d^5*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b
^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/(
(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2
*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*
d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*s
qrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*
b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2
*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2
*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4
)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^
3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x +
c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 +
2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((a^11 + 3*a^9*b^2 + 2*a^7*b^4 - 2*a^5*b^6 - 3*a^3*b^8 - a*b^10)*d^7*sq
rt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b
^4)*d^4)) + (a^8*b + 2*a^6*b^3 - 2*a^2*b^7 - b^9)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b
^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*
a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4
))^(3/4))/(a^4 - 2*a^2*b^2 + b^4))*cos(d*x + c) + 12*sqrt(2)*(a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d^5*sqrt(
(a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2
*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*
a^2*b^2 + b^4)*d^4))^(3/4)*arctan(-((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^9 + 4*a^
7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*
b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^4 -
2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5
*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b
^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^6*b - a^4*b^3 - a^2*b
^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c))*s
qrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2
*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 +
b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^11 + 3*a^9*b^2 + 2*a^7*
b^4 - 2*a^5*b^6 - 3*a^3*b^8 - a*b^10)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b
^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^8*b + 2*a^6*b^3 - 2*a^2*b^7 - b^9)*d^5*sqrt((a^4 -
2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*
b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/c
os(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4))*cos(d*x + c) + 6*a^3*sqrt(-a/b)
*cos(d*x + c)*log(-(6*a*b*cos(d*x + c)*sin(d*x + c) - (a^2 - b^2)*cos(d*x + c)^2 - b^2 + 4*(a*b*cos(d*x + c)^2
- b^2*cos(d*x + c)*sin(d*x + c))*sqrt(-a/b)*sqrt(sin(d*x + c)/cos(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c)
+ (a^2 - b^2)*cos(d*x + c)^2 + b^2)) + 3*sqrt(2)*(2*(a^3*b^3 + a*b^5)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4
))*cos(d*x + c) + (a^2*b^2 + b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*
d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*lo
g(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^6*b -
a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*
cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*
d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4
- 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - 3*sqrt(2)*(2*(a^3*b^3 + a*b^5)*d^3*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4))*cos(d*x + c) + (a^2*b^2 + b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3
+ a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)
)^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*
((a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 +
a*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b
^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1
/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - 8*(3*(a^3 + a*b^2)*cos(d*x + c) - (a^2*b + b^3)*si
n(d*x + c))*sqrt(sin(d*x + c)/cos(d*x + c)))/((a^2*b^2 + b^4)*d*cos(d*x + c)), 1/12*(12*sqrt(2)*(a^6*b^2 + 3*a
^4*b^4 + 3*a^2*b^6 + b^8)*d^5*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2
*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4
*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*
sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)) + sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^6*b + 3*a^4*b^3
+ 3*a^2*b^5 + b^7)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sq
rt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*
a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) +
sqrt(2)*((a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*
a^3*b^2 + a*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)
*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + s
qrt(2)*((a^11 + 3*a^9*b^2 + 2*a^7*b^4 - 2*a^5*b^6 - 3*a^3*b^8 - a*b^10)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^8*b + 2*a^6*b^3 -
2*a^2*b^7 - b^9)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqr
t((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a
^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4
))*cos(d*x + c) + 12*sqrt(2)*(a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d^5*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*
b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b
^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arcta
n(-((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2
*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 +
a*b^8)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 +
2*a^2*b^2 + b^4)*d^4)) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*
b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt
(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/(
(a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a
^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(
a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x +
c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))
*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^11 + 3*a^9*b^2 + 2*a^7*b^4 - 2*a^5*b^6 - 3*a^3*b^8 - a*
b^10)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2
*a^2*b^2 + b^4)*d^4)) + (a^8*b + 2*a^6*b^3 - 2*a^2*b^7 - b^9)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b
^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(
1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^
2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4))*cos(d*x + c) + 24*a^3*sqrt(a/b)*arctan(b*sqrt(a/b)*sqrt(sin(d*x
+ c)/cos(d*x + c))/a)*cos(d*x + c) + 3*sqrt(2)*(2*(a^3*b^3 + a*b^5)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))
*cos(d*x + c) + (a^2*b^2 + b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^
2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(
((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^6*b - a^
4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*co
s(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^
4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 -
2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - 3*sqrt(2)*(2*(a^3*b^3 + a*b^5)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4))*cos(d*x + c) + (a^2*b^2 + b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 +
a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^
(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((
a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a
*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4
) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - 8*(3*(a^3 + a*b^2)*cos(d*x + c) - (a^2*b + b^3)*sin(
d*x + c))*sqrt(sin(d*x + c)/cos(d*x + c)))/((a^2*b^2 + b^4)*d*cos(d*x + c))]

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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)^(7/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.30, size = 334, normalized size = 1.23 \[ \frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3 b d}-\frac {2 a \left (\sqrt {\tan }\left (d x +c \right )\right )}{b^{2} d}+\frac {2 a^{4} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right )}{d \,b^{2} \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {a \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d \left (a^{2}+b^{2}\right )}+\frac {a \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )}+\frac {a \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {b \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d \left (a^{2}+b^{2}\right )}-\frac {b \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {b \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.48, size = 202, normalized size = 0.75 \[ \frac {\frac {24 \, a^{4} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{2} + b^{2}} + \frac {8 \, {\left (b \tan \left (d x + c\right )^{\frac {3}{2}} - 3 \, a \sqrt {\tan \left (d x + c\right )}\right )}}{b^{2}}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(24*a^4*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2*b^2 + b^4)*sqrt(a*b)) + 3*(2*sqrt(2)*(a - b)*arctan(
1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*
x + c)))) + sqrt(2)*(a + b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(a + b)*log(-sqrt(2)*
sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^2 + b^2) + 8*(b*tan(d*x + c)^(3/2) - 3*a*sqrt(tan(d*x + c)))/b^2)/d

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mupad [B]  time = 6.18, size = 4927, normalized size = 18.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^(7/2)/(a + b*tan(c + d*x)),x)

[Out]

(2*tan(c + d*x)^(3/2))/(3*b*d) - atan((((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((1/(b^2*d^2*1i - a
^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) - (16*tan(c
+ d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4
- 16*a^6*b^6*d^4))/(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + (32*tan(c + d*x)^(1/2)*(14*
a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4)))/2 - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^
6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5)))/2 - (32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4))*(1/
(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i)/2 - ((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((1/(
b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^
5) + (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 1
6*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - (32*tan(c + d
*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4)))/2 - (32*(4*a^9*d^2 + a*b^
8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5)))/2 + (32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/
(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i)/2)/(((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2)
)^(1/2)*(((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5
*d^4))/(b^3*d^5) - (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^12*d^4 + 16*a^
2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 +
(32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4)))/2 - (32*(4*
a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5)))/2 - (32*tan(c + d*x)^(1/2)*(
2*a^8 + b^8))/(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + ((((1/(b^2*d^2*1i - a^2*d^2*1i +
2*a*b*d^2))^(1/2)*(((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 +
12*a^6*b^5*d^4))/(b^3*d^5) + (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^12*
d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))
^(1/2))/2 - (32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4)))/
2 - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5)))/2 + (32*tan(c + d
*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - (64*(a^6*b - a^4*b^3)
)/(b^3*d^5)))*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*1i - atan(((((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*
d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) - (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*
b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*
2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4))
*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 -
16*a^7*b^2*d^2))/(b^3*d^5))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*tan(c + d*x)^(1/2)*(2*a^8 +
b^8))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i - (((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^
4 + 12*a^6*b^5*d^4))/(b^3*d^5) + (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^
12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i
)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4))*(
1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 1
6*a^7*b^2*d^2))/(b^3*d^5))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*a^8 + b
^8))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i)/((((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4
+ 12*a^6*b^5*d^4))/(b^3*d^5) - (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^12
*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i))
)^(1/2) + (32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4))*(1i
/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*
a^7*b^2*d^2))/(b^3*d^5))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*tan(c + d*x)^(1/2)*(2*a^8 + b^8
))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (((((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*
a^6*b^5*d^4))/(b^3*d^5) + (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^12*d^4
+ 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/
2) - (32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4))*(1i/(4*(
b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b
^2*d^2))/(b^3*d^5))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b
^3*d^4))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (64*(a^6*b - a^4*b^3))/(b^3*d^5)))*(1i/(4*(b^2*d^2
- a^2*d^2 + a*b*d^2*2i)))^(1/2)*2i - (2*a*tan(c + d*x)^(1/2))/(b^2*d) - (atan(((((32*tan(c + d*x)^(1/2)*(2*a^8
+ b^8))/(b^3*d^4) + (((32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5)
- (((32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4) + ((-a^7*b
^5)^(1/2)*((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) - (32*tan(c + d*x)^(1/2)*(-a^7*b^
5)^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^8*d^5*(a^2 + b^2))))/(b^5*d*(a^
2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2)*1i)/
(b^5*d*(a^2 + b^2)) + (((32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4) - (((32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b
^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5) + (((32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 -
4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4) - ((-a^7*b^5)^(1/2)*((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*
b^5*d^4))/(b^3*d^5) + (32*tan(c + d*x)^(1/2)*(-a^7*b^5)^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4
- 16*a^6*b^6*d^4))/(b^8*d^5*(a^2 + b^2))))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2)))*(-a^7*b
^5)^(1/2))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2)*1i)/(b^5*d*(a^2 + b^2)))/((64*(a^6*b - a^4*b^3))/(b^3*d^5) +
(((32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4) + (((32*(4*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^
2 - 16*a^7*b^2*d^2))/(b^3*d^5) - (((32*tan(c + d*x)^(1/2)*(14*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5
*b^5*d^2))/(b^3*d^4) + ((-a^7*b^5)^(1/2)*((32*(12*a^2*b^9*d^4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) -
(32*tan(c + d*x)^(1/2)*(-a^7*b^5)^(1/2)*(16*b^12*d^4 + 16*a^2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^
8*d^5*(a^2 + b^2))))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2
+ b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2)) - (((32*tan(c + d*x)^(1/2)*(2*a^8 + b^8))/(b^3*d^4) - (((32*(4
*a^9*d^2 + a*b^8*d^2 + a^3*b^6*d^2 + 16*a^5*b^4*d^2 - 16*a^7*b^2*d^2))/(b^3*d^5) + (((32*tan(c + d*x)^(1/2)*(1
4*a*b^9*d^2 + 16*a^9*b*d^2 - 4*a^3*b^7*d^2 - 2*a^5*b^5*d^2))/(b^3*d^4) - ((-a^7*b^5)^(1/2)*((32*(12*a^2*b^9*d^
4 + 24*a^4*b^7*d^4 + 12*a^6*b^5*d^4))/(b^3*d^5) + (32*tan(c + d*x)^(1/2)*(-a^7*b^5)^(1/2)*(16*b^12*d^4 + 16*a^
2*b^10*d^4 - 16*a^4*b^8*d^4 - 16*a^6*b^6*d^4))/(b^8*d^5*(a^2 + b^2))))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/
(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2)))*(-a^7*b^5)^(1/2))/(b^5*d*(a^2 + b^2))))*(-a^7*b^5)
^(1/2)*2i)/(b^5*d*(a^2 + b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{\frac {7}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(7/2)/(a+b*tan(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**(7/2)/(a + b*tan(c + d*x)), x)

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