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

Optimal. Leaf size=267 \[ -\frac{5 i a^{7/2} A c^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{8 f}+\frac{5 a^3 A c^3 \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{16 f}+\frac{5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac{a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac{B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f} \]

[Out]

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

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Rubi [A]  time = 0.297291, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3588, 80, 38, 63, 217, 203} \[ -\frac{5 i a^{7/2} A c^{7/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right )}{8 f}+\frac{5 a^3 A c^3 \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{16 f}+\frac{5 a^2 A c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac{a A c \tan (e+f x) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac{B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{7/2}}{7 f} \]

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])^(7/2),x]

[Out]

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

Mathematica [B]  time = 17.073, size = 535, normalized size = 2. \[ \frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left (A c^3 \sec (e) \left (\frac{1}{6} \cos (3 e)-\frac{1}{6} i \sin (3 e)\right ) \sin (f x) \sec ^5(e+f x)+A c^3 \sec (e) \left (\frac{5}{24} \cos (3 e)-\frac{5}{24} i \sin (3 e)\right ) \sin (f x) \sec ^3(e+f x)+A c^3 \sec (e) \left (\frac{5}{16} \cos (3 e)-\frac{5}{16} i \sin (3 e)\right ) \sin (f x) \sec (e+f x)+\tan (e) \sec ^4(e+f x) \left (\frac{1}{6} A c^3 \cos (3 e)-\frac{1}{6} i A c^3 \sin (3 e)\right )+\tan (e) \sec ^2(e+f x) \left (\frac{5}{24} A c^3 \cos (3 e)-\frac{5}{24} i A c^3 \sin (3 e)\right )+\tan (e) \left (\frac{5}{16} A c^3 \cos (3 e)-\frac{5}{16} i A c^3 \sin (3 e)\right )+\sec ^6(e+f x) \left (\frac{1}{7} B c^3 \cos (3 e)-\frac{1}{7} i B c^3 \sin (3 e)\right )\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac{5 i A c^4 \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{8 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))} \]

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])^(7/2),x]

[Out]

(((-5*I)/8)*A*c^4*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 + f*x]^6*((B*c^3*Cos[3*e])/7 - (I/7)*B*c^3*Sin[3*e]) + A*c^3*Sec
[e]*Sec[e + f*x]^5*(Cos[3*e]/6 - (I/6)*Sin[3*e])*Sin[f*x] + A*c^3*Sec[e]*Sec[e + f*x]^3*((5*Cos[3*e])/24 - ((5
*I)/24)*Sin[3*e])*Sin[f*x] + A*c^3*Sec[e]*Sec[e + f*x]*((5*Cos[3*e])/16 - ((5*I)/16)*Sin[3*e])*Sin[f*x] + Sec[
e + f*x]^4*((A*c^3*Cos[3*e])/6 - (I/6)*A*c^3*Sin[3*e])*Tan[e] + Sec[e + f*x]^2*((5*A*c^3*Cos[3*e])/24 - ((5*I)
/24)*A*c^3*Sin[3*e])*Tan[e] + ((5*A*c^3*Cos[3*e])/16 - ((5*I)/16)*A*c^3*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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Maple [A]  time = 0.105, size = 314, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}{c}^{3}}{336\,f}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) } \left ( 48\,B \left ( \tan \left ( fx+e \right ) \right ) ^{6}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+56\,A \left ( \tan \left ( fx+e \right ) \right ) ^{5}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+144\,B \left ( \tan \left ( fx+e \right ) \right ) ^{4}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+182\,A \left ( \tan \left ( fx+e \right ) \right ) ^{3}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+144\,B \left ( \tan \left ( fx+e \right ) \right ) ^{2}\sqrt{ac \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{ac}+105\,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+231\,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))^(7/2),x)

[Out]

1/336/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(-1+I*tan(f*x+e)))^(1/2)*a^3*c^3*(48*B*tan(f*x+e)^6*(a*c*(1+tan(f*x+e)^
2))^(1/2)*(a*c)^(1/2)+56*A*tan(f*x+e)^5*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+144*B*tan(f*x+e)^4*(a*c*(1+ta
n(f*x+e)^2))^(1/2)*(a*c)^(1/2)+182*A*tan(f*x+e)^3*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+144*B*tan(f*x+e)^2*
(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+105*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+231*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 = 19.374, size = 2570, normalized size = 9.63 \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))^(7/2),x, algorithm="maxima")

[Out]

-(420*A*a^3*c^3*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2800*A*a^3*c^3*cos(11/2*arctan2(sin(2*
f*x + 2*e), cos(2*f*x + 2*e))) + 7924*A*a^3*c^3*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12288*I
*B*a^3*c^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 7924*A*a^3*c^3*cos(5/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e))) - 2800*A*a^3*c^3*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 420*A*a^3*c^3*
cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 420*I*A*a^3*c^3*sin(13/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) + 2800*I*A*a^3*c^3*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 7924*I*A*a^3*c^3*sin
(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12288*B*a^3*c^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) - 7924*I*A*a^3*c^3*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2800*I*A*a^3*c^3*sin(3/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 420*I*A*a^3*c^3*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) + (210*A*a^3*c^3*cos(14*f*x + 14*e) + 1470*A*a^3*c^3*cos(12*f*x + 12*e) + 4410*A*a^3*c^3*cos(10*f*x + 10
*e) + 7350*A*a^3*c^3*cos(8*f*x + 8*e) + 7350*A*a^3*c^3*cos(6*f*x + 6*e) + 4410*A*a^3*c^3*cos(4*f*x + 4*e) + 14
70*A*a^3*c^3*cos(2*f*x + 2*e) + 210*I*A*a^3*c^3*sin(14*f*x + 14*e) + 1470*I*A*a^3*c^3*sin(12*f*x + 12*e) + 441
0*I*A*a^3*c^3*sin(10*f*x + 10*e) + 7350*I*A*a^3*c^3*sin(8*f*x + 8*e) + 7350*I*A*a^3*c^3*sin(6*f*x + 6*e) + 441
0*I*A*a^3*c^3*sin(4*f*x + 4*e) + 1470*I*A*a^3*c^3*sin(2*f*x + 2*e) + 210*A*a^3*c^3)*arctan2(cos(1/2*arctan2(si
n(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) + (210*A*a^3*c^3
*cos(14*f*x + 14*e) + 1470*A*a^3*c^3*cos(12*f*x + 12*e) + 4410*A*a^3*c^3*cos(10*f*x + 10*e) + 7350*A*a^3*c^3*c
os(8*f*x + 8*e) + 7350*A*a^3*c^3*cos(6*f*x + 6*e) + 4410*A*a^3*c^3*cos(4*f*x + 4*e) + 1470*A*a^3*c^3*cos(2*f*x
+ 2*e) + 210*I*A*a^3*c^3*sin(14*f*x + 14*e) + 1470*I*A*a^3*c^3*sin(12*f*x + 12*e) + 4410*I*A*a^3*c^3*sin(10*f
*x + 10*e) + 7350*I*A*a^3*c^3*sin(8*f*x + 8*e) + 7350*I*A*a^3*c^3*sin(6*f*x + 6*e) + 4410*I*A*a^3*c^3*sin(4*f*
x + 4*e) + 1470*I*A*a^3*c^3*sin(2*f*x + 2*e) + 210*A*a^3*c^3)*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) - (-105*I*A*a^3*c^3*cos(14*f*x + 14*e
) - 735*I*A*a^3*c^3*cos(12*f*x + 12*e) - 2205*I*A*a^3*c^3*cos(10*f*x + 10*e) - 3675*I*A*a^3*c^3*cos(8*f*x + 8*
e) - 3675*I*A*a^3*c^3*cos(6*f*x + 6*e) - 2205*I*A*a^3*c^3*cos(4*f*x + 4*e) - 735*I*A*a^3*c^3*cos(2*f*x + 2*e)
+ 105*A*a^3*c^3*sin(14*f*x + 14*e) + 735*A*a^3*c^3*sin(12*f*x + 12*e) + 2205*A*a^3*c^3*sin(10*f*x + 10*e) + 36
75*A*a^3*c^3*sin(8*f*x + 8*e) + 3675*A*a^3*c^3*sin(6*f*x + 6*e) + 2205*A*a^3*c^3*sin(4*f*x + 4*e) + 735*A*a^3*
c^3*sin(2*f*x + 2*e) - 105*I*A*a^3*c^3)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*a
rctan2(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) -
(105*I*A*a^3*c^3*cos(14*f*x + 14*e) + 735*I*A*a^3*c^3*cos(12*f*x + 12*e) + 2205*I*A*a^3*c^3*cos(10*f*x + 10*e)
+ 3675*I*A*a^3*c^3*cos(8*f*x + 8*e) + 3675*I*A*a^3*c^3*cos(6*f*x + 6*e) + 2205*I*A*a^3*c^3*cos(4*f*x + 4*e) +
735*I*A*a^3*c^3*cos(2*f*x + 2*e) - 105*A*a^3*c^3*sin(14*f*x + 14*e) - 735*A*a^3*c^3*sin(12*f*x + 12*e) - 2205
*A*a^3*c^3*sin(10*f*x + 10*e) - 3675*A*a^3*c^3*sin(8*f*x + 8*e) - 3675*A*a^3*c^3*sin(6*f*x + 6*e) - 2205*A*a^3
*c^3*sin(4*f*x + 4*e) - 735*A*a^3*c^3*sin(2*f*x + 2*e) + 105*I*A*a^3*c^3)*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*(-672*I*cos(14*f*x + 14*e) - 4704*I*cos(12*f*x + 12*e) - 1
4112*I*cos(10*f*x + 10*e) - 23520*I*cos(8*f*x + 8*e) - 23520*I*cos(6*f*x + 6*e) - 14112*I*cos(4*f*x + 4*e) - 4
704*I*cos(2*f*x + 2*e) + 672*sin(14*f*x + 14*e) + 4704*sin(12*f*x + 12*e) + 14112*sin(10*f*x + 10*e) + 23520*s
in(8*f*x + 8*e) + 23520*sin(6*f*x + 6*e) + 14112*sin(4*f*x + 4*e) + 4704*sin(2*f*x + 2*e) - 672*I))

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Fricas [B]  time = 1.56795, size = 1883, normalized size = 7.05 \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))^(7/2),x, algorithm="fricas")

[Out]

1/672*(4*(-105*I*A*a^3*c^3*e^(12*I*f*x + 12*I*e) - 700*I*A*a^3*c^3*e^(10*I*f*x + 10*I*e) - 1981*I*A*a^3*c^3*e^
(8*I*f*x + 8*I*e) + 3072*B*a^3*c^3*e^(6*I*f*x + 6*I*e) + 1981*I*A*a^3*c^3*e^(4*I*f*x + 4*I*e) + 700*I*A*a^3*c^
3*e^(2*I*f*x + 2*I*e) + 105*I*A*a^3*c^3)*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) - 105*sqrt(A^2*a^7*c^7/f^2)*(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*
f*x + 8*I*e) + 20*f*e^(6*I*f*x + 6*I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x + 2*I*e) + f)*log(1/8*(64*
(A*a^3*c^3*e^(2*I*f*x + 2*I*e) + A*a^3*c^3)*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) + sqrt(A^2*a^7*c^7/f^2)*(32*I*f*e^(2*I*f*x + 2*I*e) - 32*I*f))/(A*a^3*c^3*e^(2*I*f*x + 2*I*e
) + A*a^3*c^3)) + 105*sqrt(A^2*a^7*c^7/f^2)*(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I
*f*x + 8*I*e) + 20*f*e^(6*I*f*x + 6*I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x + 2*I*e) + f)*log(1/8*(64
*(A*a^3*c^3*e^(2*I*f*x + 2*I*e) + A*a^3*c^3)*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) + sqrt(A^2*a^7*c^7/f^2)*(-32*I*f*e^(2*I*f*x + 2*I*e) + 32*I*f))/(A*a^3*c^3*e^(2*I*f*x + 2*I
*e) + A*a^3*c^3)))/(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*f*x + 8*I*e) + 20*f*e^(6
*I*f*x + 6*I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*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))**(7/2),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{7}{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))^(7/2),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)^(7/2), x)