3.816 \(\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2} \, dx\)

Optimal. Leaf size=350 \[ -\frac {5 a^{7/2} c^{9/2} (-B+8 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{64 f}+\frac {5 a^3 c^4 (8 A+i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{128 f}+\frac {5 a^2 c^3 (8 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a c^2 (8 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {c (-B+8 i A) (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f} \]

[Out]

-5/64*a^(7/2)*(8*I*A-B)*c^(9/2)*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+5/
128*a^3*(8*A+I*B)*c^4*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)*tan(f*x+e)/f+5/192*a^2*(8*A+I*B)*c^3*t
an(f*x+e)*(a+I*a*tan(f*x+e))^(3/2)*(c-I*c*tan(f*x+e))^(3/2)/f+1/48*a*(8*A+I*B)*c^2*tan(f*x+e)*(a+I*a*tan(f*x+e
))^(5/2)*(c-I*c*tan(f*x+e))^(5/2)/f-1/56*(8*I*A-B)*c*(a+I*a*tan(f*x+e))^(7/2)*(c-I*c*tan(f*x+e))^(7/2)/f+1/8*B
*(a+I*a*tan(f*x+e))^(7/2)*(c-I*c*tan(f*x+e))^(9/2)/f

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Rubi [A]  time = 0.37, antiderivative size = 350, normalized size of antiderivative = 1.00, 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, 80, 49, 38, 63, 217, 203} \[ -\frac {5 a^{7/2} c^{9/2} (-B+8 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{64 f}+\frac {5 a^3 c^4 (8 A+i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{128 f}+\frac {5 a^2 c^3 (8 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a c^2 (8 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {c (-B+8 i A) (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f} \]

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])^(9/2),x]

[Out]

(-5*a^(7/2)*((8*I)*A - 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]])])/(64*f) + (5*a^3*(8*A + I*B)*c^4*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(12
8*f) + (5*a^2*(8*A + I*B)*c^3*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(192*f)
+ (a*(8*A + I*B)*c^2*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(48*f) - (((8*I)*
A - B)*c*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(7/2))/(56*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(
c - I*c*Tan[e + f*x])^(9/2))/(8*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))^{9/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^{5/2} (A+B x) (c-i c x)^{7/2} \, 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))^{9/2}}{8 f}+\frac {(a (8 A+i B) c) \operatorname {Subst}\left (\int (a+i a x)^{5/2} (c-i c x)^{7/2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}+\frac {\left (a (8 A+i B) c^2\right ) \operatorname {Subst}\left (\int (a+i a x)^{5/2} (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {a (8 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}+\frac {\left (5 a^2 (8 A+i B) c^3\right ) \operatorname {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{48 f}\\ &=\frac {5 a^2 (8 A+i B) c^3 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a (8 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}+\frac {\left (5 a^3 (8 A+i B) c^4\right ) \operatorname {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{64 f}\\ &=\frac {5 a^3 (8 A+i B) c^4 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{128 f}+\frac {5 a^2 (8 A+i B) c^3 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a (8 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}+\frac {\left (5 a^4 (8 A+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 )}{128 f}\\ &=\frac {5 a^3 (8 A+i B) c^4 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{128 f}+\frac {5 a^2 (8 A+i B) c^3 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a (8 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}-\frac {\left (5 a^3 (8 i A-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 )}{64 f}\\ &=\frac {5 a^3 (8 A+i B) c^4 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{128 f}+\frac {5 a^2 (8 A+i B) c^3 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a (8 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}-\frac {\left (5 a^3 (8 i A-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 )}{64 f}\\ &=-\frac {5 a^{7/2} (8 i A-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 )}{64 f}+\frac {5 a^3 (8 A+i B) c^4 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{128 f}+\frac {5 a^2 (8 A+i B) c^3 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{192 f}+\frac {a (8 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{48 f}-\frac {(8 i A-B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{56 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{9/2}}{8 f}\\ \end {align*}

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Mathematica [A]  time = 17.55, size = 666, normalized size = 1.90 \[ \frac {5 c^5 (B-8 i A) \sqrt {e^{i f x}} e^{-i (4 e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{64 f \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \sec ^{\frac {9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}+\frac {\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left (\sec (e) \left (\frac {1}{56} c^4 \cos (3 e)-\frac {1}{56} i c^4 \sin (3 e)\right ) \sec ^6(e+f x) (-8 i A \cos (e)-7 i B \sin (e)+8 B \cos (e))+\sec (e) \left (\frac {1}{48} \cos (3 e)-\frac {1}{48} i \sin (3 e)\right ) \sec ^5(e+f x) \left (8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right )+\sec (e) \left (\frac {5}{192} \cos (3 e)-\frac {5}{192} i \sin (3 e)\right ) \sec ^3(e+f x) \left (8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right )+\sec (e) \left (\frac {5}{128} \cos (3 e)-\frac {5}{128} i \sin (3 e)\right ) \sec (e+f x) \left (8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right )+(8 A+i B) \tan (e) \left (\frac {1}{48} c^4 \cos (3 e)-\frac {1}{48} i c^4 \sin (3 e)\right ) \sec ^4(e+f x)+(8 A+i B) \tan (e) \left (\frac {5}{192} c^4 \cos (3 e)-\frac {5}{192} i c^4 \sin (3 e)\right ) \sec ^2(e+f x)+(8 A+i B) \tan (e) \left (\frac {5}{128} c^4 \cos (3 e)-\frac {5}{128} i c^4 \sin (3 e)\right )-i B c^4 \sec (e) \left (\frac {1}{8} \cos (3 e)-\frac {1}{8} i \sin (3 e)\right ) \sin (f x) \sec ^7(e+f x)\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2),x]

[Out]

(5*((-8*I)*A + B)*c^5*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]))/(64*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*S
ec[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]^6*((-8*I)*A*Cos[e] + 8*B*Cos[e] - (7*I)*B*S
in[e])*((c^4*Cos[3*e])/56 - (I/56)*c^4*Sin[3*e]) - I*B*c^4*Sec[e]*Sec[e + f*x]^7*(Cos[3*e]/8 - (I/8)*Sin[3*e])
*Sin[f*x] + Sec[e]*Sec[e + f*x]^5*(Cos[3*e]/48 - (I/48)*Sin[3*e])*(8*A*c^4*Sin[f*x] + I*B*c^4*Sin[f*x]) + Sec[
e]*Sec[e + f*x]^3*((5*Cos[3*e])/192 - ((5*I)/192)*Sin[3*e])*(8*A*c^4*Sin[f*x] + I*B*c^4*Sin[f*x]) + Sec[e]*Sec
[e + f*x]*((5*Cos[3*e])/128 - ((5*I)/128)*Sin[3*e])*(8*A*c^4*Sin[f*x] + I*B*c^4*Sin[f*x]) + (8*A + I*B)*Sec[e
+ f*x]^4*((c^4*Cos[3*e])/48 - (I/48)*c^4*Sin[3*e])*Tan[e] + (8*A + I*B)*Sec[e + f*x]^2*((5*c^4*Cos[3*e])/192 -
((5*I)/192)*c^4*Sin[3*e])*Tan[e] + (8*A + I*B)*((5*c^4*Cos[3*e])/128 - ((5*I)/128)*c^4*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*Sin[e + f*x]))

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fricas [B]  time = 2.10, size = 879, normalized size = 2.51 \[ \text {result too large to display} \]

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))^(9/2),x, algorithm="fricas")

[Out]

-1/5376*(21*sqrt((1600*A^2 + 400*I*A*B - 25*B^2)*a^7*c^9/f^2)*(f*e^(14*I*f*x + 14*I*e) + 7*f*e^(12*I*f*x + 12*
I*e) + 21*f*e^(10*I*f*x + 10*I*e) + 35*f*e^(8*I*f*x + 8*I*e) + 35*f*e^(6*I*f*x + 6*I*e) + 21*f*e^(4*I*f*x + 4*
I*e) + 7*f*e^(2*I*f*x + 2*I*e) + f)*log(2*(((-160*I*A + 20*B)*a^3*c^4*e^(3*I*f*x + 3*I*e) + (-160*I*A + 20*B)*
a^3*c^4*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((1600*A^
2 + 400*I*A*B - 25*B^2)*a^7*c^9/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((-40*I*A + 5*B)*a^3*c^4*e^(2*I*f*x + 2*I*e)
+ (-40*I*A + 5*B)*a^3*c^4)) - 21*sqrt((1600*A^2 + 400*I*A*B - 25*B^2)*a^7*c^9/f^2)*(f*e^(14*I*f*x + 14*I*e) +
7*f*e^(12*I*f*x + 12*I*e) + 21*f*e^(10*I*f*x + 10*I*e) + 35*f*e^(8*I*f*x + 8*I*e) + 35*f*e^(6*I*f*x + 6*I*e)
+ 21*f*e^(4*I*f*x + 4*I*e) + 7*f*e^(2*I*f*x + 2*I*e) + f)*log(2*(((-160*I*A + 20*B)*a^3*c^4*e^(3*I*f*x + 3*I*e
) + (-160*I*A + 20*B)*a^3*c^4*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((1600*A^2 + 400*I*A*B - 25*B^2)*a^7*c^9/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((-40*I*A + 5*B)*a^3*c
^4*e^(2*I*f*x + 2*I*e) + (-40*I*A + 5*B)*a^3*c^4)) - 4*((-840*I*A + 105*B)*a^3*c^4*e^(15*I*f*x + 15*I*e) + (-6
440*I*A + 805*B)*a^3*c^4*e^(13*I*f*x + 13*I*e) + (-21448*I*A + 2681*B)*a^3*c^4*e^(11*I*f*x + 11*I*e) + (-40424
*I*A + 5053*B)*a^3*c^4*e^(9*I*f*x + 9*I*e) + (-8728*I*A + 44099*B)*a^3*c^4*e^(7*I*f*x + 7*I*e) + (21448*I*A -
2681*B)*a^3*c^4*e^(5*I*f*x + 5*I*e) + (6440*I*A - 805*B)*a^3*c^4*e^(3*I*f*x + 3*I*e) + (840*I*A - 105*B)*a^3*c
^4*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^(14*I*f*x + 14*I
*e) + 7*f*e^(12*I*f*x + 12*I*e) + 21*f*e^(10*I*f*x + 10*I*e) + 35*f*e^(8*I*f*x + 8*I*e) + 35*f*e^(6*I*f*x + 6*
I*e) + 21*f*e^(4*I*f*x + 4*I*e) + 7*f*e^(2*I*f*x + 2*I*e) + f)

________________________________________________________________________________________

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))^(9/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [B]  time = 0.61, size = 604, normalized size = 1.73 \[ -\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^{4} \left (1152 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 )+952 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 )}+384 i A \left (\tan ^{6}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-384 B \left (\tan ^{6}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+826 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}-448 A \left (\tan ^{5}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-105 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 -1152 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}+1152 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}-1456 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}+105 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )+336 i B \left (\tan ^{7}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-1152 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}+384 i A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-840 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 -1848 A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )-384 B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\right )}{2688 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))^(9/2),x)

[Out]

-1/2688/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(-1+I*tan(f*x+e)))^(1/2)*a^3*c^4*(1152*I*A*(c*a*(1+tan(f*x+e)^2))^(1/
2)*(c*a)^(1/2)*tan(f*x+e)^4+952*I*B*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)*tan(f*x+e)^5+384*I*A*tan(f*x+e)^6
*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)-384*B*tan(f*x+e)^6*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)+826*I*B*
(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)*tan(f*x+e)^3-448*A*tan(f*x+e)^5*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1
/2)-105*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-1152*B*tan(f*x+e)^4*
(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)+1152*I*A*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)*tan(f*x+e)^2-1456*A
*tan(f*x+e)^3*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)+105*I*B*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)*tan(f*
x+e)+336*I*B*tan(f*x+e)^7*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)-1152*B*tan(f*x+e)^2*(c*a*(1+tan(f*x+e)^2))^
(1/2)*(c*a)^(1/2)+384*I*A*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)-840*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-1848*A*(c*a*(1+tan(f*x+e)^2))^(1/2)*(c*a)^(1/2)*tan(f*x+e)-384*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)

________________________________________________________________________________________

maxima [B]  time = 38.45, size = 2618, normalized size = 7.48 \[ \text {result too large to display} \]

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))^(9/2),x, algorithm="maxima")

[Out]

-(36126720*(8*A + I*B)*a^3*c^4*cos(15/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 276971520*(8*A + I*B)*a
^3*c^4*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 922435584*(8*A + I*B)*a^3*c^4*cos(11/2*arctan2(
sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1738555392*(8*A + I*B)*a^3*c^4*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*
f*x + 2*e))) + 344064*(8728*A + 44099*I*B)*a^3*c^4*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 9224
35584*(8*A + I*B)*a^3*c^4*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 276971520*(8*A + I*B)*a^3*c^4
*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 36126720*(8*A + I*B)*a^3*c^4*cos(1/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) - (-289013760*I*A + 36126720*B)*a^3*c^4*sin(15/2*arctan2(sin(2*f*x + 2*e), cos(2*f
*x + 2*e))) - (-2215772160*I*A + 276971520*B)*a^3*c^4*sin(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) -
(-7379484672*I*A + 922435584*B)*a^3*c^4*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (-13908443136*
I*A + 1738555392*B)*a^3*c^4*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (-3002990592*I*A + 15172878
336*B)*a^3*c^4*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (7379484672*I*A - 922435584*B)*a^3*c^4*s
in(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (2215772160*I*A - 276971520*B)*a^3*c^4*sin(3/2*arctan2(s
in(2*f*x + 2*e), cos(2*f*x + 2*e))) - (289013760*I*A - 36126720*B)*a^3*c^4*sin(1/2*arctan2(sin(2*f*x + 2*e), c
os(2*f*x + 2*e))) + (18063360*(8*A + I*B)*a^3*c^4*cos(16*f*x + 16*e) + 144506880*(8*A + I*B)*a^3*c^4*cos(14*f*
x + 14*e) + 505774080*(8*A + I*B)*a^3*c^4*cos(12*f*x + 12*e) + 1011548160*(8*A + I*B)*a^3*c^4*cos(10*f*x + 10*
e) + 1264435200*(8*A + I*B)*a^3*c^4*cos(8*f*x + 8*e) + 1011548160*(8*A + I*B)*a^3*c^4*cos(6*f*x + 6*e) + 50577
4080*(8*A + I*B)*a^3*c^4*cos(4*f*x + 4*e) + 144506880*(8*A + I*B)*a^3*c^4*cos(2*f*x + 2*e) - (-144506880*I*A +
18063360*B)*a^3*c^4*sin(16*f*x + 16*e) - (-1156055040*I*A + 144506880*B)*a^3*c^4*sin(14*f*x + 14*e) - (-40461
92640*I*A + 505774080*B)*a^3*c^4*sin(12*f*x + 12*e) - (-8092385280*I*A + 1011548160*B)*a^3*c^4*sin(10*f*x + 10
*e) - (-10115481600*I*A + 1264435200*B)*a^3*c^4*sin(8*f*x + 8*e) - (-8092385280*I*A + 1011548160*B)*a^3*c^4*si
n(6*f*x + 6*e) - (-4046192640*I*A + 505774080*B)*a^3*c^4*sin(4*f*x + 4*e) - (-1156055040*I*A + 144506880*B)*a^
3*c^4*sin(2*f*x + 2*e) + 18063360*(8*A + I*B)*a^3*c^4)*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) + (18063360*(8*A + I*B)*a^3*c^4*cos(16*f*x +
16*e) + 144506880*(8*A + I*B)*a^3*c^4*cos(14*f*x + 14*e) + 505774080*(8*A + I*B)*a^3*c^4*cos(12*f*x + 12*e) +
1011548160*(8*A + I*B)*a^3*c^4*cos(10*f*x + 10*e) + 1264435200*(8*A + I*B)*a^3*c^4*cos(8*f*x + 8*e) + 10115481
60*(8*A + I*B)*a^3*c^4*cos(6*f*x + 6*e) + 505774080*(8*A + I*B)*a^3*c^4*cos(4*f*x + 4*e) + 144506880*(8*A + I*
B)*a^3*c^4*cos(2*f*x + 2*e) - (-144506880*I*A + 18063360*B)*a^3*c^4*sin(16*f*x + 16*e) - (-1156055040*I*A + 14
4506880*B)*a^3*c^4*sin(14*f*x + 14*e) - (-4046192640*I*A + 505774080*B)*a^3*c^4*sin(12*f*x + 12*e) - (-8092385
280*I*A + 1011548160*B)*a^3*c^4*sin(10*f*x + 10*e) - (-10115481600*I*A + 1264435200*B)*a^3*c^4*sin(8*f*x + 8*e
) - (-8092385280*I*A + 1011548160*B)*a^3*c^4*sin(6*f*x + 6*e) - (-4046192640*I*A + 505774080*B)*a^3*c^4*sin(4*
f*x + 4*e) - (-1156055040*I*A + 144506880*B)*a^3*c^4*sin(2*f*x + 2*e) + 18063360*(8*A + I*B)*a^3*c^4)*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) - ((-72253440*I*A + 9031680*B)*a^3*c^4*cos(16*f*x + 16*e) + (-578027520*I*A + 72253440*B)*a^3*c^4*cos(14*f*
x + 14*e) + (-2023096320*I*A + 252887040*B)*a^3*c^4*cos(12*f*x + 12*e) + (-4046192640*I*A + 505774080*B)*a^3*c
^4*cos(10*f*x + 10*e) + (-5057740800*I*A + 632217600*B)*a^3*c^4*cos(8*f*x + 8*e) + (-4046192640*I*A + 50577408
0*B)*a^3*c^4*cos(6*f*x + 6*e) + (-2023096320*I*A + 252887040*B)*a^3*c^4*cos(4*f*x + 4*e) + (-578027520*I*A + 7
2253440*B)*a^3*c^4*cos(2*f*x + 2*e) + 9031680*(8*A + I*B)*a^3*c^4*sin(16*f*x + 16*e) + 72253440*(8*A + I*B)*a^
3*c^4*sin(14*f*x + 14*e) + 252887040*(8*A + I*B)*a^3*c^4*sin(12*f*x + 12*e) + 505774080*(8*A + I*B)*a^3*c^4*si
n(10*f*x + 10*e) + 632217600*(8*A + I*B)*a^3*c^4*sin(8*f*x + 8*e) + 505774080*(8*A + I*B)*a^3*c^4*sin(6*f*x +
6*e) + 252887040*(8*A + I*B)*a^3*c^4*sin(4*f*x + 4*e) + 72253440*(8*A + I*B)*a^3*c^4*sin(2*f*x + 2*e) + (-7225
3440*I*A + 9031680*B)*a^3*c^4)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(si
n(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) - ((7225344
0*I*A - 9031680*B)*a^3*c^4*cos(16*f*x + 16*e) + (578027520*I*A - 72253440*B)*a^3*c^4*cos(14*f*x + 14*e) + (202
3096320*I*A - 252887040*B)*a^3*c^4*cos(12*f*x + 12*e) + (4046192640*I*A - 505774080*B)*a^3*c^4*cos(10*f*x + 10
*e) + (5057740800*I*A - 632217600*B)*a^3*c^4*cos(8*f*x + 8*e) + (4046192640*I*A - 505774080*B)*a^3*c^4*cos(6*f
*x + 6*e) + (2023096320*I*A - 252887040*B)*a^3*c^4*cos(4*f*x + 4*e) + (578027520*I*A - 72253440*B)*a^3*c^4*cos
(2*f*x + 2*e) - 9031680*(8*A + I*B)*a^3*c^4*sin(16*f*x + 16*e) - 72253440*(8*A + I*B)*a^3*c^4*sin(14*f*x + 14*
e) - 252887040*(8*A + I*B)*a^3*c^4*sin(12*f*x + 12*e) - 505774080*(8*A + I*B)*a^3*c^4*sin(10*f*x + 10*e) - 632
217600*(8*A + I*B)*a^3*c^4*sin(8*f*x + 8*e) - 505774080*(8*A + I*B)*a^3*c^4*sin(6*f*x + 6*e) - 252887040*(8*A
+ I*B)*a^3*c^4*sin(4*f*x + 4*e) - 72253440*(8*A + I*B)*a^3*c^4*sin(2*f*x + 2*e) + (72253440*I*A - 9031680*B)*a
^3*c^4)*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))*sqrt(a)*sqrt(c)/(f*(-462422016*I*
cos(16*f*x + 16*e) - 3699376128*I*cos(14*f*x + 14*e) - 12947816448*I*cos(12*f*x + 12*e) - 25895632896*I*cos(10
*f*x + 10*e) - 32369541120*I*cos(8*f*x + 8*e) - 25895632896*I*cos(6*f*x + 6*e) - 12947816448*I*cos(4*f*x + 4*e
) - 3699376128*I*cos(2*f*x + 2*e) + 462422016*sin(16*f*x + 16*e) + 3699376128*sin(14*f*x + 14*e) + 12947816448
*sin(12*f*x + 12*e) + 25895632896*sin(10*f*x + 10*e) + 32369541120*sin(8*f*x + 8*e) + 25895632896*sin(6*f*x +
6*e) + 12947816448*sin(4*f*x + 4*e) + 3699376128*sin(2*f*x + 2*e) - 462422016*I))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{9/2} \,d x \]

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)^(9/2),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)^(9/2), 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))**(9/2),x)

[Out]

Timed out

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