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]

-(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)

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Rubi [A]  time = 0.326534, antiderivative size = 284, normalized size of antiderivative = 1., 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 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))^{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*}

Mathematica [B]  time = 15.958, 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]

((-I/8)*(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*Se
c[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*Sin
[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])*S
in[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])*Ta
n[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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Maple [B]  time = 0.1, size = 478, normalized size = 1.7 \begin{align*}{\frac{{a}^{3}{c}^{2}}{240\,f}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) } \left ( 40\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{5}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+48\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{4}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+70\,iB\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \left ( \tan \left ( fx+e \right ) \right ) ^{3}+48\,B \left ( \tan \left ( fx+e \right ) \right ) ^{4}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+96\,iA\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \left ( \tan \left ( fx+e \right ) \right ) ^{2}+60\,A \left ( \tan \left ( fx+e \right ) \right ) ^{3}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}-15\,iB\ln \left ({ \left ( ac\tan \left ( fx+e \right ) +\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}} \right ) ac+15\,iB\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}\tan \left ( fx+e \right ) +96\,B \left ( \tan \left ( fx+e \right ) \right ) ^{2}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+48\,iA\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+90\,A\ln \left ({\frac{ac\tan \left ( fx+e \right ) +\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}}{\sqrt{ac}}} \right ) ac+150\,A\sqrt{ac}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\tan \left ( fx+e \right ) +48\,B\sqrt{ac}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) } \right ){\frac{1}{\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }}}{\frac{1}{\sqrt{ac}}}} \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))^(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*(a*c*(1+tan(f*x+e
)^2))^(1/2)*(a*c)^(1/2)+48*I*A*tan(f*x+e)^4*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+70*I*B*(a*c*(1+tan(f*x+e)
^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^3+48*B*tan(f*x+e)^4*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+96*I*A*(a*c*(1+
tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2+60*A*tan(f*x+e)^3*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)-15*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+15*I*B*(a*c*(1+tan(f*x+e)^2))^
(1/2)*(a*c)^(1/2)*tan(f*x+e)+96*B*tan(f*x+e)^2*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+48*I*A*(a*c*(1+tan(f*x
+e)^2))^(1/2)*(a*c)^(1/2)+90*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+1
50*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)+48*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 = 33.223, size = 2724, normalized size = 9.59 \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))^(5/2),x, algorithm="maxima")

[Out]

-((1382400*A - 230400*I*B)*a^3*c^2*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + (7833600*A - 130560
0*I*B)*a^3*c^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (5345280*A - 20551680*I*B)*a^3*c^2*cos(7
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - (18247680*A - 3041280*I*B)*a^3*c^2*cos(5/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) - (7833600*A - 1305600*I*B)*a^3*c^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) - (1382400*A - 230400*I*B)*a^3*c^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 230400*(-6*I*
A - B)*a^3*c^2*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 1305600*(-6*I*A - B)*a^3*c^2*sin(9/2*ar
ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 92160*(58*I*A + 223*B)*a^3*c^2*sin(7/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) - 3041280*(6*I*A + B)*a^3*c^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 130560
0*(6*I*A + B)*a^3*c^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 230400*(6*I*A + B)*a^3*c^2*sin(1/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + ((691200*A - 115200*I*B)*a^3*c^2*cos(12*f*x + 12*e) + (414720
0*A - 691200*I*B)*a^3*c^2*cos(10*f*x + 10*e) + (10368000*A - 1728000*I*B)*a^3*c^2*cos(8*f*x + 8*e) + (13824000
*A - 2304000*I*B)*a^3*c^2*cos(6*f*x + 6*e) + (10368000*A - 1728000*I*B)*a^3*c^2*cos(4*f*x + 4*e) + (4147200*A
- 691200*I*B)*a^3*c^2*cos(2*f*x + 2*e) - 115200*(-6*I*A - B)*a^3*c^2*sin(12*f*x + 12*e) - 691200*(-6*I*A - B)*
a^3*c^2*sin(10*f*x + 10*e) - 1728000*(-6*I*A - B)*a^3*c^2*sin(8*f*x + 8*e) - 2304000*(-6*I*A - B)*a^3*c^2*sin(
6*f*x + 6*e) - 1728000*(-6*I*A - B)*a^3*c^2*sin(4*f*x + 4*e) - 691200*(-6*I*A - B)*a^3*c^2*sin(2*f*x + 2*e) +
(691200*A - 115200*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) + ((691200*A - 115200*I*B)*a^3*c^2*cos(12*f*x + 12*e) + (4147200*A
- 691200*I*B)*a^3*c^2*cos(10*f*x + 10*e) + (10368000*A - 1728000*I*B)*a^3*c^2*cos(8*f*x + 8*e) + (13824000*A -
2304000*I*B)*a^3*c^2*cos(6*f*x + 6*e) + (10368000*A - 1728000*I*B)*a^3*c^2*cos(4*f*x + 4*e) + (4147200*A - 69
1200*I*B)*a^3*c^2*cos(2*f*x + 2*e) - 115200*(-6*I*A - B)*a^3*c^2*sin(12*f*x + 12*e) - 691200*(-6*I*A - B)*a^3*
c^2*sin(10*f*x + 10*e) - 1728000*(-6*I*A - B)*a^3*c^2*sin(8*f*x + 8*e) - 2304000*(-6*I*A - B)*a^3*c^2*sin(6*f*
x + 6*e) - 1728000*(-6*I*A - B)*a^3*c^2*sin(4*f*x + 4*e) - 691200*(-6*I*A - B)*a^3*c^2*sin(2*f*x + 2*e) + (691
200*A - 115200*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(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - (57600*(-6*I*A - B)*a^3*c^2*cos(12*f*x + 12*e) + 345600*(-6*I*A - B)
*a^3*c^2*cos(10*f*x + 10*e) + 864000*(-6*I*A - B)*a^3*c^2*cos(8*f*x + 8*e) + 1152000*(-6*I*A - B)*a^3*c^2*cos(
6*f*x + 6*e) + 864000*(-6*I*A - B)*a^3*c^2*cos(4*f*x + 4*e) + 345600*(-6*I*A - B)*a^3*c^2*cos(2*f*x + 2*e) + (
345600*A - 57600*I*B)*a^3*c^2*sin(12*f*x + 12*e) + (2073600*A - 345600*I*B)*a^3*c^2*sin(10*f*x + 10*e) + (5184
000*A - 864000*I*B)*a^3*c^2*sin(8*f*x + 8*e) + (6912000*A - 1152000*I*B)*a^3*c^2*sin(6*f*x + 6*e) + (5184000*A
- 864000*I*B)*a^3*c^2*sin(4*f*x + 4*e) + (2073600*A - 345600*I*B)*a^3*c^2*sin(2*f*x + 2*e) + 57600*(-6*I*A -
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) - (57600*(6*I*A + B)*a^3*c^2*c
os(12*f*x + 12*e) + 345600*(6*I*A + B)*a^3*c^2*cos(10*f*x + 10*e) + 864000*(6*I*A + B)*a^3*c^2*cos(8*f*x + 8*e
) + 1152000*(6*I*A + B)*a^3*c^2*cos(6*f*x + 6*e) + 864000*(6*I*A + B)*a^3*c^2*cos(4*f*x + 4*e) + 345600*(6*I*A
+ B)*a^3*c^2*cos(2*f*x + 2*e) - (345600*A - 57600*I*B)*a^3*c^2*sin(12*f*x + 12*e) - (2073600*A - 345600*I*B)*
a^3*c^2*sin(10*f*x + 10*e) - (5184000*A - 864000*I*B)*a^3*c^2*sin(8*f*x + 8*e) - (6912000*A - 1152000*I*B)*a^3
*c^2*sin(6*f*x + 6*e) - (5184000*A - 864000*I*B)*a^3*c^2*sin(4*f*x + 4*e) - (2073600*A - 345600*I*B)*a^3*c^2*s
in(2*f*x + 2*e) + 57600*(6*I*A + 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))*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) - 36864000*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 + 12*e) + 11059200*sin(10*f*x + 10*e) + 27648000*sin(8*f*x + 8*e) + 36864000*sin(6*f*x + 6*e) + 2764800
0*sin(4*f*x + 4*e) + 11059200*sin(2*f*x + 2*e) - 1843200*I))

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Fricas [B]  time = 1.81744, size = 1991, normalized size = 7.01 \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))^(5/2),x, algorithm="fricas")

[Out]

1/480*(4*((-90*I*A - 15*B)*a^3*c^2*e^(10*I*f*x + 10*I*e) + (-510*I*A - 85*B)*a^3*c^2*e^(8*I*f*x + 8*I*e) + (34
8*I*A + 1338*B)*a^3*c^2*e^(6*I*f*x + 6*I*e) + (1188*I*A + 198*B)*a^3*c^2*e^(4*I*f*x + 4*I*e) + (510*I*A + 85*B
)*a^3*c^2*e^(2*I*f*x + 2*I*e) + (90*I*A + 15*B)*a^3*c^2)*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*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^(2*I*f*x + 2*I*e) + (24*I*A + 4*B)*a^3*c^2)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqr
t(c/(e^(2*I*f*x + 2*I*e) + 1))*e^(I*f*x + I*e) + 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^(2*I*f*x + 2*I*e) + (24*I*A + 4*B
)*a^3*c^2)*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((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)))/(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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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))**(5/2),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{5}{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))^(5/2),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)^(5/2), x)