3.818 \(\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=284 \[ -\frac {a^{7/2} c^{5/2} (B+6 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^3 c^2 (6 A-i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 c (6 A-i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f} \]

[Out]

-1/8*a^(7/2)*(6*I*A+B)*c^(5/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c*tan(f*x+e))^(1/2))/f+1/1
6*a^3*(6*A-I*B)*c^2*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)*tan(f*x+e)/f+1/24*a^2*(6*A-I*B)*c*tan(f*
x+e)*(a+I*a*tan(f*x+e))^(3/2)*(c-I*c*tan(f*x+e))^(3/2)/f+1/30*a*(6*I*A+B)*(a+I*a*tan(f*x+e))^(5/2)*(c-I*c*tan(
f*x+e))^(5/2)/f+1/6*B*(a+I*a*tan(f*x+e))^(7/2)*(c-I*c*tan(f*x+e))^(5/2)/f

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Rubi [A]  time = 0.33, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3588, 80, 49, 38, 63, 217, 203} \[ -\frac {a^{7/2} c^{5/2} (B+6 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^3 c^2 (6 A-i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 c (6 A-i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

-(a^(7/2)*((6*I)*A + B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]
])])/(8*f) + (a^3*(6*A - I*B)*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(16*f) +
(a^2*(6*A - I*B)*c*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) + (a*((6*I)
*A + B)*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(30*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c
- I*c*Tan[e + f*x])^(5/2))/(6*f)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 49

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[(2*c*n)/(m + n + 1), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]

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 80

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

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 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]

Rubi steps

\begin {align*} \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^{5/2} (A+B x) (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {(a (6 A-i B) c) \operatorname {Subst}\left (\int (a+i a x)^{5/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {\left (a^2 (6 A-i B) c\right ) \operatorname {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {\left (a^3 (6 A-i B) c^2\right ) \operatorname {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {\left (a^4 (6 A-i B) c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}-\frac {\left (a^3 (6 i A+B) c^3\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 )}{8 f}\\ &=\frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}-\frac {\left (a^3 (6 i A+B) c^3\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 )}{8 f}\\ &=-\frac {a^{7/2} (6 i A+B) c^{5/2} \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 )}{8 f}+\frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}\\ \end {align*}

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Mathematica [B]  time = 17.00, size = 572, normalized size = 2.01 \[ \frac {\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left (\sec (e) \left (\frac {1}{30} c^2 \cos (3 e)-\frac {1}{30} i c^2 \sin (3 e)\right ) \sec ^4(e+f x) (6 i A \cos (e)+5 i B \sin (e)+6 B \cos (e))+\sec (e) \left (\frac {1}{24} \cos (3 e)-\frac {1}{24} i \sin (3 e)\right ) \sec ^3(e+f x) \left (6 A c^2 \sin (f x)-i B c^2 \sin (f x)\right )+\sec (e) \left (\frac {1}{16} \cos (3 e)-\frac {1}{16} i \sin (3 e)\right ) \sec (e+f x) \left (6 A c^2 \sin (f x)-i B c^2 \sin (f x)\right )+(6 A-i B) \tan (e) \left (\frac {1}{24} c^2 \cos (3 e)-\frac {1}{24} i c^2 \sin (3 e)\right ) \sec ^2(e+f x)+(6 A-i B) \tan (e) \left (\frac {1}{16} c^2 \cos (3 e)-\frac {1}{16} i c^2 \sin (3 e)\right )+i B c^2 \sec (e) \left (\frac {1}{6} \cos (3 e)-\frac {1}{6} i \sin (3 e)\right ) \sin (f x) \sec ^5(e+f x)\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac {i c^3 (6 A-i B) \sqrt {e^{i f x}} e^{-i (4 e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{8 f \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \sec ^{\frac {9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

((-1/8*I)*(6*A - I*B)*c^3*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x
))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*
Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*Sqrt[Sec
[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^4*((6*I)*A*Cos[e] + 6*B*Cos[e] + (5*I)*B*S
in[e])*((c^2*Cos[3*e])/30 - (I/30)*c^2*Sin[3*e]) + I*B*c^2*Sec[e]*Sec[e + f*x]^5*(Cos[3*e]/6 - (I/6)*Sin[3*e])
*Sin[f*x] + Sec[e]*Sec[e + f*x]^3*(Cos[3*e]/24 - (I/24)*Sin[3*e])*(6*A*c^2*Sin[f*x] - I*B*c^2*Sin[f*x]) + Sec[
e]*Sec[e + f*x]*(Cos[3*e]/16 - (I/16)*Sin[3*e])*(6*A*c^2*Sin[f*x] - I*B*c^2*Sin[f*x]) + (6*A - I*B)*Sec[e + f*
x]^2*((c^2*Cos[3*e])/24 - (I/24)*c^2*Sin[3*e])*Tan[e] + (6*A - I*B)*((c^2*Cos[3*e])/16 - (I/16)*c^2*Sin[3*e])*
Tan[e])*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*Si
n[e + f*x]))

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fricas [B]  time = 0.70, size = 753, normalized size = 2.65 \[ \frac {15 \, \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {2 \, {\left ({\left ({\left (24 i \, A + 4 \, B\right )} a^{3} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (24 i \, A + 4 \, B\right )} a^{3} c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + 2 \, \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (6 i \, A + B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 i \, A + B\right )} a^{3} c^{2}}\right ) - 15 \, \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {2 \, {\left ({\left ({\left (24 i \, A + 4 \, B\right )} a^{3} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (24 i \, A + 4 \, B\right )} a^{3} c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - 2 \, \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (6 i \, A + B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 i \, A + B\right )} a^{3} c^{2}}\right ) + 4 \, {\left ({\left (-90 i \, A - 15 \, B\right )} a^{3} c^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + {\left (-510 i \, A - 85 \, B\right )} a^{3} c^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (348 i \, A + 1338 \, B\right )} a^{3} c^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (1188 i \, A + 198 \, B\right )} a^{3} c^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (510 i \, A + 85 \, B\right )} a^{3} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (90 i \, A + 15 \, B\right )} a^{3} c^{2} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{480 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/480*(15*sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2)*(f*e^(10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*
f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)*log(2*(((24*I*A + 4*B)*a^3*c^2
*e^(3*I*f*x + 3*I*e) + (24*I*A + 4*B)*a^3*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*
I*f*x + 2*I*e) + 1)) + 2*sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((6*I*A + B)
*a^3*c^2*e^(2*I*f*x + 2*I*e) + (6*I*A + B)*a^3*c^2)) - 15*sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2)*(f*e^(10
*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*
f*x + 2*I*e) + f)*log(2*(((24*I*A + 4*B)*a^3*c^2*e^(3*I*f*x + 3*I*e) + (24*I*A + 4*B)*a^3*c^2*e^(I*f*x + I*e))
*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - 2*sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^
5/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((6*I*A + B)*a^3*c^2*e^(2*I*f*x + 2*I*e) + (6*I*A + B)*a^3*c^2)) + 4*((-90
*I*A - 15*B)*a^3*c^2*e^(11*I*f*x + 11*I*e) + (-510*I*A - 85*B)*a^3*c^2*e^(9*I*f*x + 9*I*e) + (348*I*A + 1338*B
)*a^3*c^2*e^(7*I*f*x + 7*I*e) + (1188*I*A + 198*B)*a^3*c^2*e^(5*I*f*x + 5*I*e) + (510*I*A + 85*B)*a^3*c^2*e^(3
*I*f*x + 3*I*e) + (90*I*A + 15*B)*a^3*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*
x + 2*I*e) + 1)))/(f*e^(10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*
f*x + 4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 0.63, size = 478, normalized size = 1.68 \[ \frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{3} c^{2} \left (40 i B \left (\tan ^{5}\left (f x +e \right )\right ) \sqrt {c a}\, \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}+48 i A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (\tan ^{4}\left (f x +e \right )\right )+70 i B \left (\tan ^{3}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+48 B \left (\tan ^{4}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+96 i A \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+60 A \left (\tan ^{3}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-15 i B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) a c +15 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )+96 B \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+48 i A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+90 A \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) a c +150 A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )+48 B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\right )}{240 f \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x)

[Out]

1/240/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(-1+I*tan(f*x+e)))^(1/2)*a^3*c^2*(40*I*B*tan(f*x+e)^5*(c*a)^(1/2)*(c*a*
(1+tan(f*x+e)^2))^(1/2)+48*I*A*tan(f*x+e)^4*(c*a)^(1/2)*(c*a*(1+tan(f*x+e)^2))^(1/2)+70*I*B*(c*a)^(1/2)*(c*a*(
1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^3+48*B*tan(f*x+e)^4*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)+96*I*A*(c*a)^(1
/2)*(c*a*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+60*A*tan(f*x+e)^3*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)-15*I*
B*ln((c*a*tan(f*x+e)+(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2))/(c*a)^(1/2))*a*c+15*I*B*(c*a)^(1/2)*(c*a*(1+tan
(f*x+e)^2))^(1/2)*tan(f*x+e)+96*B*tan(f*x+e)^2*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)+48*I*A*(c*a)^(1/2)*(c*
a*(1+tan(f*x+e)^2))^(1/2)+90*A*ln((c*a*tan(f*x+e)+(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2))/(c*a)^(1/2))*a*c+1
50*A*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)*tan(f*x+e)+48*B*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2))/(c*a*(
1+tan(f*x+e)^2))^(1/2)/(c*a)^(1/2)

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maxima [B]  time = 9.02, size = 2026, normalized size = 7.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

-(230400*(6*A - I*B)*a^3*c^2*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1305600*(6*A - I*B)*a^3*c
^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 92160*(58*A - 223*I*B)*a^3*c^2*cos(7/2*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) - 3041280*(6*A - I*B)*a^3*c^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) - 1305600*(6*A - I*B)*a^3*c^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 230400*(6*A - I*B)*a
^3*c^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (1382400*I*A + 230400*B)*a^3*c^2*sin(11/2*arctan
2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (7833600*I*A + 1305600*B)*a^3*c^2*sin(9/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e))) + (-5345280*I*A - 20551680*B)*a^3*c^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) +
(-18247680*I*A - 3041280*B)*a^3*c^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (-7833600*I*A - 13
05600*B)*a^3*c^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (-1382400*I*A - 230400*B)*a^3*c^2*sin(
1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (115200*(6*A - I*B)*a^3*c^2*cos(12*f*x + 12*e) + 691200*(6*
A - I*B)*a^3*c^2*cos(10*f*x + 10*e) + 1728000*(6*A - I*B)*a^3*c^2*cos(8*f*x + 8*e) + 2304000*(6*A - I*B)*a^3*c
^2*cos(6*f*x + 6*e) + 1728000*(6*A - I*B)*a^3*c^2*cos(4*f*x + 4*e) + 691200*(6*A - I*B)*a^3*c^2*cos(2*f*x + 2*
e) + (691200*I*A + 115200*B)*a^3*c^2*sin(12*f*x + 12*e) + (4147200*I*A + 691200*B)*a^3*c^2*sin(10*f*x + 10*e)
+ (10368000*I*A + 1728000*B)*a^3*c^2*sin(8*f*x + 8*e) + (13824000*I*A + 2304000*B)*a^3*c^2*sin(6*f*x + 6*e) +
(10368000*I*A + 1728000*B)*a^3*c^2*sin(4*f*x + 4*e) + (4147200*I*A + 691200*B)*a^3*c^2*sin(2*f*x + 2*e) + 1152
00*(6*A - I*B)*a^3*c^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), sin(1/2*arctan2(sin(2*f*
x + 2*e), cos(2*f*x + 2*e))) + 1) + (115200*(6*A - I*B)*a^3*c^2*cos(12*f*x + 12*e) + 691200*(6*A - I*B)*a^3*c^
2*cos(10*f*x + 10*e) + 1728000*(6*A - I*B)*a^3*c^2*cos(8*f*x + 8*e) + 2304000*(6*A - I*B)*a^3*c^2*cos(6*f*x +
6*e) + 1728000*(6*A - I*B)*a^3*c^2*cos(4*f*x + 4*e) + 691200*(6*A - I*B)*a^3*c^2*cos(2*f*x + 2*e) + (691200*I*
A + 115200*B)*a^3*c^2*sin(12*f*x + 12*e) + (4147200*I*A + 691200*B)*a^3*c^2*sin(10*f*x + 10*e) + (10368000*I*A
+ 1728000*B)*a^3*c^2*sin(8*f*x + 8*e) + (13824000*I*A + 2304000*B)*a^3*c^2*sin(6*f*x + 6*e) + (10368000*I*A +
1728000*B)*a^3*c^2*sin(4*f*x + 4*e) + (4147200*I*A + 691200*B)*a^3*c^2*sin(2*f*x + 2*e) + 115200*(6*A - I*B)*
a^3*c^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) + 1) + ((345600*I*A + 57600*B)*a^3*c^2*cos(12*f*x + 12*e) + (2073600*I*A + 345600*B)*a^3*c^2*co
s(10*f*x + 10*e) + (5184000*I*A + 864000*B)*a^3*c^2*cos(8*f*x + 8*e) + (6912000*I*A + 1152000*B)*a^3*c^2*cos(6
*f*x + 6*e) + (5184000*I*A + 864000*B)*a^3*c^2*cos(4*f*x + 4*e) + (2073600*I*A + 345600*B)*a^3*c^2*cos(2*f*x +
2*e) - 57600*(6*A - I*B)*a^3*c^2*sin(12*f*x + 12*e) - 345600*(6*A - I*B)*a^3*c^2*sin(10*f*x + 10*e) - 864000*
(6*A - I*B)*a^3*c^2*sin(8*f*x + 8*e) - 1152000*(6*A - I*B)*a^3*c^2*sin(6*f*x + 6*e) - 864000*(6*A - I*B)*a^3*c
^2*sin(4*f*x + 4*e) - 345600*(6*A - I*B)*a^3*c^2*sin(2*f*x + 2*e) + (345600*I*A + 57600*B)*a^3*c^2)*log(cos(1/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*
sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + ((-345600*I*A - 57600*B)*a^3*c^2*cos(12*f*x + 12*e
) + (-2073600*I*A - 345600*B)*a^3*c^2*cos(10*f*x + 10*e) + (-5184000*I*A - 864000*B)*a^3*c^2*cos(8*f*x + 8*e)
+ (-6912000*I*A - 1152000*B)*a^3*c^2*cos(6*f*x + 6*e) + (-5184000*I*A - 864000*B)*a^3*c^2*cos(4*f*x + 4*e) + (
-2073600*I*A - 345600*B)*a^3*c^2*cos(2*f*x + 2*e) + 57600*(6*A - I*B)*a^3*c^2*sin(12*f*x + 12*e) + 345600*(6*A
- I*B)*a^3*c^2*sin(10*f*x + 10*e) + 864000*(6*A - I*B)*a^3*c^2*sin(8*f*x + 8*e) + 1152000*(6*A - I*B)*a^3*c^2
*sin(6*f*x + 6*e) + 864000*(6*A - I*B)*a^3*c^2*sin(4*f*x + 4*e) + 345600*(6*A - I*B)*a^3*c^2*sin(2*f*x + 2*e)
+ (-345600*I*A - 57600*B)*a^3*c^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan
2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1))*sqrt(a
)*sqrt(c)/(f*(-1843200*I*cos(12*f*x + 12*e) - 11059200*I*cos(10*f*x + 10*e) - 27648000*I*cos(8*f*x + 8*e) - 36
864000*I*cos(6*f*x + 6*e) - 27648000*I*cos(4*f*x + 4*e) - 11059200*I*cos(2*f*x + 2*e) + 1843200*sin(12*f*x + 1
2*e) + 11059200*sin(10*f*x + 10*e) + 27648000*sin(8*f*x + 8*e) + 36864000*sin(6*f*x + 6*e) + 27648000*sin(4*f*
x + 4*e) + 11059200*sin(2*f*x + 2*e) - 1843200*I))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(7/2)*(c - c*tan(e + f*x)*1i)^(5/2),x)

[Out]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(7/2)*(c - c*tan(e + f*x)*1i)^(5/2), x)

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sympy [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((a+I*a*tan(f*x+e))**(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(5/2),x)

[Out]

Timed out

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