3.843 \(\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=343 \[ \frac{7 c^{9/2} (-7 B+2 i A) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{a^{5/2} f}+\frac{7 c^4 (-7 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 c^3 (-7 B+2 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 c^2 (-7 B+2 i A) (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 c (-7 B+2 i A) (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \]

[Out]

(7*((2*I)*A - 7*B)*c^(9/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/
(a^(5/2)*f) + (7*((2*I)*A - 7*B)*c^4*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*a^3*f) + (7*((2
*I)*A - 7*B)*c^3*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(6*a^3*f) + (14*((2*I)*A - 7*B)*c^2*
(c - I*c*Tan[e + f*x])^(5/2))/(15*a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) - (2*((2*I)*A - 7*B)*c*(c - I*c*Tan[e + f*
x])^(7/2))/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(9/2))/(5*f*(a + I*a*Tan[
e + f*x])^(5/2))

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Rubi [A]  time = 0.380989, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3588, 78, 47, 50, 63, 217, 203} \[ \frac{7 c^{9/2} (-7 B+2 i A) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{a^{5/2} f}+\frac{7 c^4 (-7 B+2 i A) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 c^3 (-7 B+2 i A) \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 c^2 (-7 B+2 i A) (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 c (-7 B+2 i A) (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^(5/2),x]

[Out]

(7*((2*I)*A - 7*B)*c^(9/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/
(a^(5/2)*f) + (7*((2*I)*A - 7*B)*c^4*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*a^3*f) + (7*((2
*I)*A - 7*B)*c^3*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(6*a^3*f) + (14*((2*I)*A - 7*B)*c^2*
(c - I*c*Tan[e + f*x])^(5/2))/(15*a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) - (2*((2*I)*A - 7*B)*c*(c - I*c*Tan[e + f*
x])^(7/2))/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(9/2))/(5*f*(a + I*a*Tan[
e + f*x])^(5/2))

Rule 3588

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)^{7/2}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}-\frac{((2 A+7 i B) c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{7/2}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{\left (7 (2 A+7 i B) c^2\right ) \operatorname{Subst}\left (\int \frac{(c-i c x)^{5/2}}{(a+i a x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 a f}\\ &=\frac{14 (2 i A-7 B) c^2 (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}-\frac{\left (7 (2 A+7 i B) c^3\right ) \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{\sqrt{a+i a x}} \, dx,x,\tan (e+f x)\right )}{3 a^2 f}\\ &=\frac{7 (2 i A-7 B) c^3 \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 (2 i A-7 B) c^2 (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}-\frac{\left (7 (2 A+7 i B) c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{\sqrt{a+i a x}} \, dx,x,\tan (e+f x)\right )}{2 a^2 f}\\ &=\frac{7 (2 i A-7 B) c^4 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 (2 i A-7 B) c^3 \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 (2 i A-7 B) c^2 (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}-\frac{\left (7 (2 A+7 i B) c^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 a^2 f}\\ &=\frac{7 (2 i A-7 B) c^4 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 (2 i A-7 B) c^3 \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 (2 i A-7 B) c^2 (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{\left (7 (2 i A-7 B) c^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c-\frac{c x^2}{a}}} \, dx,x,\sqrt{a+i a \tan (e+f x)}\right )}{a^3 f}\\ &=\frac{7 (2 i A-7 B) c^4 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 (2 i A-7 B) c^3 \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 (2 i A-7 B) c^2 (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{\left (7 (2 i A-7 B) c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{c x^2}{a}} \, dx,x,\frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c-i c \tan (e+f x)}}\right )}{a^3 f}\\ &=\frac{7 (2 i A-7 B) c^{9/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{a^{5/2} f}+\frac{7 (2 i A-7 B) c^4 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 a^3 f}+\frac{7 (2 i A-7 B) c^3 \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^3 f}+\frac{14 (2 i A-7 B) c^2 (c-i c \tan (e+f x))^{5/2}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{2 (2 i A-7 B) c (c-i c \tan (e+f x))^{7/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{5 f (a+i a \tan (e+f x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 15.1775, size = 247, normalized size = 0.72 \[ -\frac{\sqrt{2} c^4 e^{-4 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left (105 (7 B-2 i A) e^{5 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^2 \tan ^{-1}\left (e^{i (e+f x)}\right )-2 i A \left (-8 e^{2 i (e+f x)}+56 e^{4 i (e+f x)}+175 e^{6 i (e+f x)}+105 e^{8 i (e+f x)}+6\right )+B \left (-56 e^{2 i (e+f x)}+392 e^{4 i (e+f x)}+1225 e^{6 i (e+f x)}+735 e^{8 i (e+f x)}+12\right )\right )}{15 a^2 f \left (1+e^{2 i (e+f x)}\right )^2 \sqrt{a+i a \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^(5/2),x]

[Out]

-(Sqrt[2]*c^4*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*((-2*I)*A*(6 - 8*E^((2*I)*(e + f*x)) + 56*E^((4*I)*(e + f*x))
+ 175*E^((6*I)*(e + f*x)) + 105*E^((8*I)*(e + f*x))) + B*(12 - 56*E^((2*I)*(e + f*x)) + 392*E^((4*I)*(e + f*x)
) + 1225*E^((6*I)*(e + f*x)) + 735*E^((8*I)*(e + f*x))) + 105*((-2*I)*A + 7*B)*E^((5*I)*(e + f*x))*(1 + E^((2*
I)*(e + f*x)))^2*ArcTan[E^(I*(e + f*x))]))/(15*a^2*E^((4*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^2*f*Sqrt[a +
I*a*Tan[e + f*x]])

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Maple [B]  time = 0.125, size = 899, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(9/2)/(a+I*a*tan(f*x+e))^(5/2),x)

[Out]

1/30/f*(-c*(-1+I*tan(f*x+e)))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*c^4/a^3*(-3881*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)
*(a*c)^(1/2)*tan(f*x+e)+4410*I*B*ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*tan
(f*x+e)^2*a*c+2014*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^3+840*I*A*ln((a*c*tan(f*x+e)+(a*c*(
1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)^3*a*c-210*A*ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^
2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)^4*a*c-735*I*B*ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(
a*c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)^4*a*c+15*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^5-2940*B*
ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)^3*a*c-150*B*tan(f*x+e)^4*
(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+30*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^4-840*I*A*
ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)*a*c+1260*A*ln((a*c*tan(f*
x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)^2*a*c+584*A*tan(f*x+e)^3*(a*c*(1+tan(f*
x+e)^2))^(1/2)*(a*c)^(1/2)-735*I*B*ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a
*c+334*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+2940*B*ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*
c)^(1/2))/(a*c)^(1/2))*tan(f*x+e)*a*c+4576*B*tan(f*x+e)^2*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)-1316*I*A*(a
*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2-210*A*ln((a*c*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c
)^(1/2))/(a*c)^(1/2))*a*c-1096*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-1154*B*(a*c)^(1/2)*(a*c*(
1+tan(f*x+e)^2))^(1/2))/(a*c*(1+tan(f*x+e)^2))^(1/2)/(a*c)^(1/2)/(-tan(f*x+e)+I)^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(9/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.91753, size = 1820, normalized size = 5.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(9/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/60*(2*((-334*I*A + 1154*B)*c^4*e^(9*I*f*x + 9*I*e) + (420*I*A - 1470*B)*c^4*e^(8*I*f*x + 8*I*e) + (-668*I*A
+ 2308*B)*c^4*e^(7*I*f*x + 7*I*e) + (700*I*A - 2450*B)*c^4*e^(6*I*f*x + 6*I*e) + (-334*I*A + 1154*B)*c^4*e^(5*
I*f*x + 5*I*e) + (224*I*A - 784*B)*c^4*e^(4*I*f*x + 4*I*e) + (-32*I*A + 112*B)*c^4*e^(2*I*f*x + 2*I*e) + (24*I
*A - 24*B)*c^4)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*e^(I*f*x + I*e) - 15*(a^3*
f*e^(8*I*f*x + 8*I*e) + a^3*f*e^(6*I*f*x + 6*I*e))*sqrt((196*A^2 + 1372*I*A*B - 2401*B^2)*c^9/(a^5*f^2))*log((
2*((14*I*A - 49*B)*c^4*e^(2*I*f*x + 2*I*e) + (14*I*A - 49*B)*c^4)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^
(2*I*f*x + 2*I*e) + 1))*e^(I*f*x + I*e) + (a^3*f*e^(2*I*f*x + 2*I*e) - a^3*f)*sqrt((196*A^2 + 1372*I*A*B - 240
1*B^2)*c^9/(a^5*f^2)))/((-28*I*A + 98*B)*c^4*e^(2*I*f*x + 2*I*e) + (-28*I*A + 98*B)*c^4)) + 15*(a^3*f*e^(8*I*f
*x + 8*I*e) + a^3*f*e^(6*I*f*x + 6*I*e))*sqrt((196*A^2 + 1372*I*A*B - 2401*B^2)*c^9/(a^5*f^2))*log((2*((14*I*A
- 49*B)*c^4*e^(2*I*f*x + 2*I*e) + (14*I*A - 49*B)*c^4)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x +
2*I*e) + 1))*e^(I*f*x + I*e) - (a^3*f*e^(2*I*f*x + 2*I*e) - a^3*f)*sqrt((196*A^2 + 1372*I*A*B - 2401*B^2)*c^9
/(a^5*f^2)))/((-28*I*A + 98*B)*c^4*e^(2*I*f*x + 2*I*e) + (-28*I*A + 98*B)*c^4)))/(a^3*f*e^(8*I*f*x + 8*I*e) +
a^3*f*e^(6*I*f*x + 6*I*e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(9/2)/(a+I*a*tan(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (f x + e\right ) + A\right )}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(9/2)/(a+I*a*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((B*tan(f*x + e) + A)*(-I*c*tan(f*x + e) + c)^(9/2)/(I*a*tan(f*x + e) + a)^(5/2), x)