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*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)

________________________________________________________________________________________

Rubi [A]  time = 0.369121, antiderivative size = 350, 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, 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 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 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 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 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 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 (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*}

Mathematica [A]  time = 17.5257, size = 666, normalized size = 1.9 \[ \frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left (\sec (e) \left (\frac{1}{56} c^4 \cos (3 e)-\frac{1}{56} i c^4 \sin (3 e)\right ) \sec ^6(e+f x) (-8 i A \cos (e)-7 i B \sin (e)+8 B \cos (e))+\sec (e) \left (\frac{1}{48} \cos (3 e)-\frac{1}{48} i \sin (3 e)\right ) \sec ^5(e+f x) \left (8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right )+\sec (e) \left (\frac{5}{192} \cos (3 e)-\frac{5}{192} i \sin (3 e)\right ) \sec ^3(e+f x) \left (8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right )+\sec (e) \left (\frac{5}{128} \cos (3 e)-\frac{5}{128} i \sin (3 e)\right ) \sec (e+f x) \left (8 A c^4 \sin (f x)+i B c^4 \sin (f x)\right )+(8 A+i B) \tan (e) \left (\frac{1}{48} c^4 \cos (3 e)-\frac{1}{48} i c^4 \sin (3 e)\right ) \sec ^4(e+f x)+(8 A+i B) \tan (e) \left (\frac{5}{192} c^4 \cos (3 e)-\frac{5}{192} i c^4 \sin (3 e)\right ) \sec ^2(e+f x)+(8 A+i B) \tan (e) \left (\frac{5}{128} c^4 \cos (3 e)-\frac{5}{128} i c^4 \sin (3 e)\right )-i B c^4 \sec (e) \left (\frac{1}{8} \cos (3 e)-\frac{1}{8} i \sin (3 e)\right ) \sin (f x) \sec ^7(e+f x)\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}+\frac{5 c^5 (B-8 i A) \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{64 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))} \]

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]))

________________________________________________________________________________________

Maple [B]  time = 0.111, size = 604, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [B]  time = 124.471, size = 3482, normalized size = 9.95 \begin{align*} \text{result too large to display} \end{align*}

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]

-((289013760*A + 36126720*I*B)*a^3*c^4*cos(15/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (2215772160*A +
276971520*I*B)*a^3*c^4*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (7379484672*A + 922435584*I*B)
*a^3*c^4*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (13908443136*A + 1738555392*I*B)*a^3*c^4*cos(
9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (3002990592*A + 15172878336*I*B)*a^3*c^4*cos(7/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) - (7379484672*A + 922435584*I*B)*a^3*c^4*cos(5/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) - (2215772160*A + 276971520*I*B)*a^3*c^4*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)
)) - (289013760*A + 36126720*I*B)*a^3*c^4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 36126720*(8*I
*A - B)*a^3*c^4*sin(15/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 276971520*(8*I*A - B)*a^3*c^4*sin(13/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 922435584*(8*I*A - B)*a^3*c^4*sin(11/2*arctan2(sin(2*f*x + 2*e
), cos(2*f*x + 2*e))) + 1738555392*(8*I*A - B)*a^3*c^4*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) +
344064*(8728*I*A - 44099*B)*a^3*c^4*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 922435584*(-8*I*A +
B)*a^3*c^4*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 276971520*(-8*I*A + B)*a^3*c^4*sin(3/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 36126720*(-8*I*A + B)*a^3*c^4*sin(1/2*arctan2(sin(2*f*x + 2*e), cos
(2*f*x + 2*e))) + ((144506880*A + 18063360*I*B)*a^3*c^4*cos(16*f*x + 16*e) + (1156055040*A + 144506880*I*B)*a^
3*c^4*cos(14*f*x + 14*e) + (4046192640*A + 505774080*I*B)*a^3*c^4*cos(12*f*x + 12*e) + (8092385280*A + 1011548
160*I*B)*a^3*c^4*cos(10*f*x + 10*e) + (10115481600*A + 1264435200*I*B)*a^3*c^4*cos(8*f*x + 8*e) + (8092385280*
A + 1011548160*I*B)*a^3*c^4*cos(6*f*x + 6*e) + (4046192640*A + 505774080*I*B)*a^3*c^4*cos(4*f*x + 4*e) + (1156
055040*A + 144506880*I*B)*a^3*c^4*cos(2*f*x + 2*e) + 18063360*(8*I*A - B)*a^3*c^4*sin(16*f*x + 16*e) + 1445068
80*(8*I*A - B)*a^3*c^4*sin(14*f*x + 14*e) + 505774080*(8*I*A - B)*a^3*c^4*sin(12*f*x + 12*e) + 1011548160*(8*I
*A - B)*a^3*c^4*sin(10*f*x + 10*e) + 1264435200*(8*I*A - B)*a^3*c^4*sin(8*f*x + 8*e) + 1011548160*(8*I*A - B)*
a^3*c^4*sin(6*f*x + 6*e) + 505774080*(8*I*A - B)*a^3*c^4*sin(4*f*x + 4*e) + 144506880*(8*I*A - B)*a^3*c^4*sin(
2*f*x + 2*e) + (144506880*A + 18063360*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) + ((144506880*A + 18063360*I*B)*a^3*c^4*cos(16*
f*x + 16*e) + (1156055040*A + 144506880*I*B)*a^3*c^4*cos(14*f*x + 14*e) + (4046192640*A + 505774080*I*B)*a^3*c
^4*cos(12*f*x + 12*e) + (8092385280*A + 1011548160*I*B)*a^3*c^4*cos(10*f*x + 10*e) + (10115481600*A + 12644352
00*I*B)*a^3*c^4*cos(8*f*x + 8*e) + (8092385280*A + 1011548160*I*B)*a^3*c^4*cos(6*f*x + 6*e) + (4046192640*A +
505774080*I*B)*a^3*c^4*cos(4*f*x + 4*e) + (1156055040*A + 144506880*I*B)*a^3*c^4*cos(2*f*x + 2*e) + 18063360*(
8*I*A - B)*a^3*c^4*sin(16*f*x + 16*e) + 144506880*(8*I*A - B)*a^3*c^4*sin(14*f*x + 14*e) + 505774080*(8*I*A -
B)*a^3*c^4*sin(12*f*x + 12*e) + 1011548160*(8*I*A - B)*a^3*c^4*sin(10*f*x + 10*e) + 1264435200*(8*I*A - B)*a^3
*c^4*sin(8*f*x + 8*e) + 1011548160*(8*I*A - B)*a^3*c^4*sin(6*f*x + 6*e) + 505774080*(8*I*A - B)*a^3*c^4*sin(4*
f*x + 4*e) + 144506880*(8*I*A - B)*a^3*c^4*sin(2*f*x + 2*e) + (144506880*A + 18063360*I*B)*a^3*c^4)*arctan2(co
s(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)
+ (9031680*(8*I*A - B)*a^3*c^4*cos(16*f*x + 16*e) + 72253440*(8*I*A - B)*a^3*c^4*cos(14*f*x + 14*e) + 2528870
40*(8*I*A - B)*a^3*c^4*cos(12*f*x + 12*e) + 505774080*(8*I*A - B)*a^3*c^4*cos(10*f*x + 10*e) + 632217600*(8*I*
A - B)*a^3*c^4*cos(8*f*x + 8*e) + 505774080*(8*I*A - B)*a^3*c^4*cos(6*f*x + 6*e) + 252887040*(8*I*A - B)*a^3*c
^4*cos(4*f*x + 4*e) + 72253440*(8*I*A - B)*a^3*c^4*cos(2*f*x + 2*e) - (72253440*A + 9031680*I*B)*a^3*c^4*sin(1
6*f*x + 16*e) - (578027520*A + 72253440*I*B)*a^3*c^4*sin(14*f*x + 14*e) - (2023096320*A + 252887040*I*B)*a^3*c
^4*sin(12*f*x + 12*e) - (4046192640*A + 505774080*I*B)*a^3*c^4*sin(10*f*x + 10*e) - (5057740800*A + 632217600*
I*B)*a^3*c^4*sin(8*f*x + 8*e) - (4046192640*A + 505774080*I*B)*a^3*c^4*sin(6*f*x + 6*e) - (2023096320*A + 2528
87040*I*B)*a^3*c^4*sin(4*f*x + 4*e) - (578027520*A + 72253440*I*B)*a^3*c^4*sin(2*f*x + 2*e) + 9031680*(8*I*A -
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), co
s(2*f*x + 2*e)))^2 + 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + (9031680*(-8*I*A + B)*a^3*c
^4*cos(16*f*x + 16*e) + 72253440*(-8*I*A + B)*a^3*c^4*cos(14*f*x + 14*e) + 252887040*(-8*I*A + B)*a^3*c^4*cos(
12*f*x + 12*e) + 505774080*(-8*I*A + B)*a^3*c^4*cos(10*f*x + 10*e) + 632217600*(-8*I*A + B)*a^3*c^4*cos(8*f*x
+ 8*e) + 505774080*(-8*I*A + B)*a^3*c^4*cos(6*f*x + 6*e) + 252887040*(-8*I*A + B)*a^3*c^4*cos(4*f*x + 4*e) + 7
2253440*(-8*I*A + B)*a^3*c^4*cos(2*f*x + 2*e) + (72253440*A + 9031680*I*B)*a^3*c^4*sin(16*f*x + 16*e) + (57802
7520*A + 72253440*I*B)*a^3*c^4*sin(14*f*x + 14*e) + (2023096320*A + 252887040*I*B)*a^3*c^4*sin(12*f*x + 12*e)
+ (4046192640*A + 505774080*I*B)*a^3*c^4*sin(10*f*x + 10*e) + (5057740800*A + 632217600*I*B)*a^3*c^4*sin(8*f*x
+ 8*e) + (4046192640*A + 505774080*I*B)*a^3*c^4*sin(6*f*x + 6*e) + (2023096320*A + 252887040*I*B)*a^3*c^4*sin
(4*f*x + 4*e) + (578027520*A + 72253440*I*B)*a^3*c^4*sin(2*f*x + 2*e) + 9031680*(-8*I*A + 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) - 3
2369541120*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) + 129478164
48*sin(4*f*x + 4*e) + 3699376128*sin(2*f*x + 2*e) - 462422016*I))

________________________________________________________________________________________

Fricas [B]  time = 1.8015, size = 2461, normalized size = 7.03 \begin{align*} \text{result too large to display} \end{align*}

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*(4*((-840*I*A + 105*B)*a^3*c^4*e^(14*I*f*x + 14*I*e) + (-6440*I*A + 805*B)*a^3*c^4*e^(12*I*f*x + 12*I*e
) + (-21448*I*A + 2681*B)*a^3*c^4*e^(10*I*f*x + 10*I*e) + (-40424*I*A + 5053*B)*a^3*c^4*e^(8*I*f*x + 8*I*e) +
(-8728*I*A + 44099*B)*a^3*c^4*e^(6*I*f*x + 6*I*e) + (21448*I*A - 2681*B)*a^3*c^4*e^(4*I*f*x + 4*I*e) + (6440*I
*A - 805*B)*a^3*c^4*e^(2*I*f*x + 2*I*e) + (840*I*A - 105*B)*a^3*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) - 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^(2
*I*f*x + 2*I*e) + (-160*I*A + 20*B)*a^3*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) + 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^(2*I*f*x + 2*I*e) + (-160*I*A + 20*B)*a^3*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) - 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)))/(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)

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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+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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (f x + e\right ) + A\right )}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{7}{2}}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{9}{2}}\,{d x} \end{align*}

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]

integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(7/2)*(-I*c*tan(f*x + e) + c)^(9/2), x)