\(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx\) [1088]

Optimal result

Integrand size = 28, antiderivative size = 234 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )} \]

[Out]

1/8*(c^4+4*I*c^3*d-6*c^2*d^2-4*I*c*d^3-7*d^4)*x/a^3/(c-I*d)/(c+I*d)^4+d^4*ln(c*cos(f*x+e)+d*sin(f*x+e))/a^3/(c
+I*d)^4/(I*c+d)/f-1/6/(I*c-d)/f/(a+I*a*tan(f*x+e))^3+1/8*(I*c-3*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2+1/8*(c^2
+4*I*c*d-7*d^2)/(I*c-d)^3/f/(a^3+I*a^3*tan(f*x+e))

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3640, 3677, 3612, 3611} \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {c^2+4 i c d-7 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {x \left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right )}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^4 (d+i c)}+\frac {-3 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]

[In]

Int[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])),x]

[Out]

((c^4 + (4*I)*c^3*d - 6*c^2*d^2 - (4*I)*c*d^3 - 7*d^4)*x)/(8*a^3*(c - I*d)*(c + I*d)^4) + (d^4*Log[c*Cos[e + f
*x] + d*Sin[e + f*x]])/(a^3*(c + I*d)^4*(I*c + d)*f) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (I*c - 3*d
)/(8*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2) + (c^2 + (4*I)*c*d - 7*d^2)/(8*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e
+ f*x]))

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3640

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3677

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-3 a (i c-2 d)-3 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx}{6 a^2 (i c-d)} \\ & = -\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-6 a^2 \left (c^2+3 i c d-4 d^2\right )-6 a^2 (c+3 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{24 a^4 (c+i d)^2} \\ & = -\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {\int \frac {6 a^3 \left (i c^3-4 c^2 d-7 i c d^2+8 d^3\right )+6 a^3 d \left (i c^2-4 c d-7 i d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{48 a^6 (i c-d)^3} \\ & = \frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {d^4 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^3 (c+i d)^4 (i c+d)} \\ & = \frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=-\frac {\frac {3 i \left (\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-15 d^4\right ) \log (i-\tan (e+f x))-(c+i d)^4 \log (i+\tan (e+f x))+16 d^4 \log (c+d \tan (e+f x))\right )}{(c-i d) (c+i d)^2}+\frac {8 (c+i d)}{(-i+\tan (e+f x))^3}+\frac {6 i (c+3 i d)}{(-i+\tan (e+f x))^2}-\frac {6 \left (c^2+4 i c d-7 d^2\right )}{(c+i d) (-i+\tan (e+f x))}}{48 a^3 (c+i d)^2 f} \]

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])),x]

[Out]

-1/48*(((3*I)*((c^4 + (4*I)*c^3*d - 6*c^2*d^2 - (4*I)*c*d^3 - 15*d^4)*Log[I - Tan[e + f*x]] - (c + I*d)^4*Log[
I + Tan[e + f*x]] + 16*d^4*Log[c + d*Tan[e + f*x]]))/((c - I*d)*(c + I*d)^2) + (8*(c + I*d))/(-I + Tan[e + f*x
])^3 + ((6*I)*(c + (3*I)*d))/(-I + Tan[e + f*x])^2 - (6*(c^2 + (4*I)*c*d - 7*d^2))/((c + I*d)*(-I + Tan[e + f*
x])))/(a^3*(c + I*d)^2*f)

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.58

method	result	size
risch	\(-\frac {x}{8 a^{3} \left (i d -c \right )}-\frac {5 \,{\mathrm e}^{-2 i \left (f x +e \right )} c d}{8 a^{3} \left (i d +c \right )^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {11 i {\mathrm e}^{-2 i \left (f x +e \right )} d^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {5 \,{\mathrm e}^{-4 i \left (f x +e \right )} d}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{48 a^{3} \left (i d +c \right ) f}-\frac {2 d^{4} x}{a^{3} \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {2 d^{4} e}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}\)	\(370\)
derivativedivides	\(\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right ) \left (i d +c \right )^{4}}+\frac {11 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {5 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {15 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {11 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {7 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {5 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {15 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {11 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {5 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3} \left (16 i d -16 c \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3} \left (16 i d -16 c \right )}\)	\(702\)
default	\(\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right ) \left (i d +c \right )^{4}}+\frac {11 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {5 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {15 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {11 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {7 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {5 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {15 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {11 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {5 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3} \left (16 i d -16 c \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3} \left (16 i d -16 c \right )}\)	\(702\)
norman	\(\frac {\frac {-16 i c d -5 c^{2}+17 d^{2}}{12 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {\left (20 i c d +7 c^{2}-17 d^{2}\right ) \tan \left (f x +e \right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (5 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {i d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{2 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (c^{2}+5 d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{4 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{6}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}-\frac {d^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}\)	\(763\)

[In]

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

-1/8*x/a^3/(I*d-c)-5/8/a^3/(c+I*d)^3/f*exp(-2*I*(f*x+e))*c*d+3/16*I/a^3/(c+I*d)^3/f*exp(-2*I*(f*x+e))*c^2-11/1
6*I/a^3/(c+I*d)^3/f*exp(-2*I*(f*x+e))*d^2-5/32/a^3/(c+I*d)^2/f*exp(-4*I*(f*x+e))*d+3/32*I/a^3/(c+I*d)^2/f*exp(
-4*I*(f*x+e))*c+1/48*I/a^3/(c+I*d)/f*exp(-6*I*(f*x+e))-2*d^4/a^3/(3*I*c^4*d+2*I*c^2*d^3-I*d^5+c^5-2*c^3*d^2-3*
c*d^4)*x-2*d^4/a^3/f/(3*I*c^4*d+2*I*c^2*d^3-I*d^5+c^5-2*c^3*d^2-3*c*d^4)*e-I*d^4/a^3/f/(3*I*c^4*d+2*I*c^2*d^3-
I*d^5+c^5-2*c^3*d^2-3*c*d^4)*ln(exp(2*I*(f*x+e))-(c+I*d)/(I*d-c))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=-\frac {{\left (96 \, d^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) - 2 \, c^{4} - 4 i \, c^{3} d - 4 i \, c d^{3} + 2 \, d^{4} - 12 \, {\left (-i \, c^{4} + 4 \, c^{3} d + 6 i \, c^{2} d^{2} - 4 \, c d^{3} + 15 i \, d^{4}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (3 \, c^{4} + 10 i \, c^{3} d - 8 \, c^{2} d^{2} + 10 i \, c d^{3} - 11 \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, {\left (3 \, c^{4} + 8 i \, c^{3} d - 2 \, c^{2} d^{2} + 8 i \, c d^{3} - 5 \, d^{4}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, {\left (-i \, a^{3} c^{5} + 3 \, a^{3} c^{4} d + 2 i \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} + 3 i \, a^{3} c d^{4} - a^{3} d^{5}\right )} f} \]

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

-1/96*(96*d^4*e^(6*I*f*x + 6*I*e)*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)) - 2*c^4 - 4*I*c^3*d
- 4*I*c*d^3 + 2*d^4 - 12*(-I*c^4 + 4*c^3*d + 6*I*c^2*d^2 - 4*c*d^3 + 15*I*d^4)*f*x*e^(6*I*f*x + 6*I*e) - 6*(3
*c^4 + 10*I*c^3*d - 8*c^2*d^2 + 10*I*c*d^3 - 11*d^4)*e^(4*I*f*x + 4*I*e) - 3*(3*c^4 + 8*I*c^3*d - 2*c^2*d^2 +
8*I*c*d^3 - 5*d^4)*e^(2*I*f*x + 2*I*e))*e^(-6*I*f*x - 6*I*e)/((-I*a^3*c^5 + 3*a^3*c^4*d + 2*I*a^3*c^3*d^2 + 2*
a^3*c^2*d^3 + 3*I*a^3*c*d^4 - a^3*d^5)*f)

Sympy [A] (verification not implemented)

Time = 11.75 (sec) , antiderivative size = 1192, normalized size of antiderivative = 5.09 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {x \left (c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}\right )}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \begin {cases} \frac {\left (512 i a^{6} c^{5} f^{2} e^{6 i e} - 2560 a^{6} c^{4} d f^{2} e^{6 i e} - 5120 i a^{6} c^{3} d^{2} f^{2} e^{6 i e} + 5120 a^{6} c^{2} d^{3} f^{2} e^{6 i e} + 2560 i a^{6} c d^{4} f^{2} e^{6 i e} - 512 a^{6} d^{5} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{5} f^{2} e^{8 i e} - 13056 a^{6} c^{4} d f^{2} e^{8 i e} - 29184 i a^{6} c^{3} d^{2} f^{2} e^{8 i e} + 32256 a^{6} c^{2} d^{3} f^{2} e^{8 i e} + 17664 i a^{6} c d^{4} f^{2} e^{8 i e} - 3840 a^{6} d^{5} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{5} f^{2} e^{10 i e} - 29184 a^{6} c^{4} d f^{2} e^{10 i e} - 76800 i a^{6} c^{3} d^{2} f^{2} e^{10 i e} + 101376 a^{6} c^{2} d^{3} f^{2} e^{10 i e} + 66048 i a^{6} c d^{4} f^{2} e^{10 i e} - 16896 a^{6} d^{5} f^{2} e^{10 i e}\right ) e^{- 2 i f x}}{24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e}} & \text {for}\: 24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \frac {c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3} + 5 i c^{2} d e^{6 i e} + 13 i c^{2} d e^{4 i e} + 11 i c^{2} d e^{2 i e} + 3 i c^{2} d - 11 c d^{2} e^{6 i e} - 21 c d^{2} e^{4 i e} - 13 c d^{2} e^{2 i e} - 3 c d^{2} - 15 i d^{3} e^{6 i e} - 11 i d^{3} e^{4 i e} - 5 i d^{3} e^{2 i e} - i d^{3}}{8 a^{3} c^{4} e^{6 i e} + 32 i a^{3} c^{3} d e^{6 i e} - 48 a^{3} c^{2} d^{2} e^{6 i e} - 32 i a^{3} c d^{3} e^{6 i e} + 8 a^{3} d^{4} e^{6 i e}}\right ) & \text {otherwise} \end {cases} - \frac {i d^{4} \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{a^{3} f \left (c - i d\right ) \left (c + i d\right )^{4}} \]

[In]

integrate(1/(a+I*a*tan(f*x+e))**3/(c+d*tan(f*x+e)),x)

[Out]

x*(c**3 + 5*I*c**2*d - 11*c*d**2 - 15*I*d**3)/(8*a**3*c**4 + 32*I*a**3*c**3*d - 48*a**3*c**2*d**2 - 32*I*a**3*
c*d**3 + 8*a**3*d**4) + Piecewise((((512*I*a**6*c**5*f**2*exp(6*I*e) - 2560*a**6*c**4*d*f**2*exp(6*I*e) - 5120
*I*a**6*c**3*d**2*f**2*exp(6*I*e) + 5120*a**6*c**2*d**3*f**2*exp(6*I*e) + 2560*I*a**6*c*d**4*f**2*exp(6*I*e) -
512*a**6*d**5*f**2*exp(6*I*e))*exp(-6*I*f*x) + (2304*I*a**6*c**5*f**2*exp(8*I*e) - 13056*a**6*c**4*d*f**2*exp
(8*I*e) - 29184*I*a**6*c**3*d**2*f**2*exp(8*I*e) + 32256*a**6*c**2*d**3*f**2*exp(8*I*e) + 17664*I*a**6*c*d**4*
f**2*exp(8*I*e) - 3840*a**6*d**5*f**2*exp(8*I*e))*exp(-4*I*f*x) + (4608*I*a**6*c**5*f**2*exp(10*I*e) - 29184*a
**6*c**4*d*f**2*exp(10*I*e) - 76800*I*a**6*c**3*d**2*f**2*exp(10*I*e) + 101376*a**6*c**2*d**3*f**2*exp(10*I*e)
+ 66048*I*a**6*c*d**4*f**2*exp(10*I*e) - 16896*a**6*d**5*f**2*exp(10*I*e))*exp(-2*I*f*x))/(24576*a**9*c**6*f*
*3*exp(12*I*e) + 147456*I*a**9*c**5*d*f**3*exp(12*I*e) - 368640*a**9*c**4*d**2*f**3*exp(12*I*e) - 491520*I*a**
9*c**3*d**3*f**3*exp(12*I*e) + 368640*a**9*c**2*d**4*f**3*exp(12*I*e) + 147456*I*a**9*c*d**5*f**3*exp(12*I*e)
- 24576*a**9*d**6*f**3*exp(12*I*e)), Ne(24576*a**9*c**6*f**3*exp(12*I*e) + 147456*I*a**9*c**5*d*f**3*exp(12*I*
e) - 368640*a**9*c**4*d**2*f**3*exp(12*I*e) - 491520*I*a**9*c**3*d**3*f**3*exp(12*I*e) + 368640*a**9*c**2*d**4
*f**3*exp(12*I*e) + 147456*I*a**9*c*d**5*f**3*exp(12*I*e) - 24576*a**9*d**6*f**3*exp(12*I*e), 0)), (x*(-(c**3
+ 5*I*c**2*d - 11*c*d**2 - 15*I*d**3)/(8*a**3*c**4 + 32*I*a**3*c**3*d - 48*a**3*c**2*d**2 - 32*I*a**3*c*d**3 +
8*a**3*d**4) + (c**3*exp(6*I*e) + 3*c**3*exp(4*I*e) + 3*c**3*exp(2*I*e) + c**3 + 5*I*c**2*d*exp(6*I*e) + 13*I
*c**2*d*exp(4*I*e) + 11*I*c**2*d*exp(2*I*e) + 3*I*c**2*d - 11*c*d**2*exp(6*I*e) - 21*c*d**2*exp(4*I*e) - 13*c*
d**2*exp(2*I*e) - 3*c*d**2 - 15*I*d**3*exp(6*I*e) - 11*I*d**3*exp(4*I*e) - 5*I*d**3*exp(2*I*e) - I*d**3)/(8*a*
*3*c**4*exp(6*I*e) + 32*I*a**3*c**3*d*exp(6*I*e) - 48*a**3*c**2*d**2*exp(6*I*e) - 32*I*a**3*c*d**3*exp(6*I*e)
+ 8*a**3*d**4*exp(6*I*e))), True)) - I*d**4*log((c + I*d)/(c*exp(2*I*e) - I*d*exp(2*I*e)) + exp(2*I*f*x))/(a**
3*f*(c - I*d)*(c + I*d)**4)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (198) = 396\).

Time = 0.60 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {\frac {192 \, d^{5} \log \left (d \tan \left (f x + e\right ) + c\right )}{2 i \, a^{3} c^{5} d - 6 \, a^{3} c^{4} d^{2} - 4 i \, a^{3} c^{3} d^{3} - 4 \, a^{3} c^{2} d^{4} - 6 i \, a^{3} c d^{5} + 2 \, a^{3} d^{6}} + \frac {6 \, {\left (-i \, c^{3} + 5 \, c^{2} d + 11 i \, c d^{2} - 15 \, d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}} + \frac {192 \, \log \left (\tan \left (f x + e\right ) + i\right )}{-32 i \, a^{3} c - 32 \, a^{3} d} + \frac {11 i \, c^{3} \tan \left (f x + e\right )^{3} - 55 \, c^{2} d \tan \left (f x + e\right )^{3} - 121 i \, c d^{2} \tan \left (f x + e\right )^{3} + 165 \, d^{3} \tan \left (f x + e\right )^{3} + 45 \, c^{3} \tan \left (f x + e\right )^{2} + 225 i \, c^{2} d \tan \left (f x + e\right )^{2} - 495 \, c d^{2} \tan \left (f x + e\right )^{2} - 579 i \, d^{3} \tan \left (f x + e\right )^{2} - 69 i \, c^{3} \tan \left (f x + e\right ) + 345 \, c^{2} d \tan \left (f x + e\right ) + 711 i \, c d^{2} \tan \left (f x + e\right ) - 699 \, d^{3} \tan \left (f x + e\right ) - 51 \, c^{3} - 223 i \, c^{2} d + 385 \, c d^{2} + 301 i \, d^{3}}{{\left (a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \]

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e)),x, algorithm="giac")

[Out]

1/96*(192*d^5*log(d*tan(f*x + e) + c)/(2*I*a^3*c^5*d - 6*a^3*c^4*d^2 - 4*I*a^3*c^3*d^3 - 4*a^3*c^2*d^4 - 6*I*a
^3*c*d^5 + 2*a^3*d^6) + 6*(-I*c^3 + 5*c^2*d + 11*I*c*d^2 - 15*d^3)*log(tan(f*x + e) - I)/(a^3*c^4 + 4*I*a^3*c^
3*d - 6*a^3*c^2*d^2 - 4*I*a^3*c*d^3 + a^3*d^4) + 192*log(tan(f*x + e) + I)/(-32*I*a^3*c - 32*a^3*d) + (11*I*c^
3*tan(f*x + e)^3 - 55*c^2*d*tan(f*x + e)^3 - 121*I*c*d^2*tan(f*x + e)^3 + 165*d^3*tan(f*x + e)^3 + 45*c^3*tan(
f*x + e)^2 + 225*I*c^2*d*tan(f*x + e)^2 - 495*c*d^2*tan(f*x + e)^2 - 579*I*d^3*tan(f*x + e)^2 - 69*I*c^3*tan(f
*x + e) + 345*c^2*d*tan(f*x + e) + 711*I*c*d^2*tan(f*x + e) - 699*d^3*tan(f*x + e) - 51*c^3 - 223*I*c^2*d + 38
5*c*d^2 + 301*I*d^3)/((a^3*c^4 + 4*I*a^3*c^3*d - 6*a^3*c^2*d^2 - 4*I*a^3*c*d^3 + a^3*d^4)*(tan(f*x + e) - I)^3
))/f

Mupad [B] (verification not implemented)

Time = 10.89 (sec) , antiderivative size = 1952, normalized size of antiderivative = 8.34 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\text {Too large to display} \]

[In]

int(1/((a + a*tan(e + f*x)*1i)^3*(c + d*tan(e + f*x))),x)

[Out]

symsum(log(- (81*c*d^5 + c^5*d + d^6*56i - c^2*d^4*64i - 30*c^3*d^3 + c^4*d^2*8i)*(a^3*d^8 + a^3*c*d^7*6i - 15
*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2) - root(a^9*c^5*d^5*e^3*7168i +
3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^3 - 3328*a^9*c^2*d^8*e^3 + a^9*c^7*d^3*e^3*2
048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3*1536i + 256*a^9*d^10*e^3 - 256*a^9*c^10*e^3
- a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d^3*e*56i - 54*a^3*c^4*d^4*e + 28*a^3*c^6*d^2
*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5*c^2*d^5 + c*d^6*11i - 15*d^7, e, k)*((a^3*d
^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)*(8*a^3*c
^7 - a^3*d^7*56i - 264*a^3*c*d^6 + a^3*c^6*d*56i + a^3*c^2*d^5*520i + 568*a^3*c^3*d^4 - a^3*c^4*d^3*392i - 184
*a^3*c^5*d^2) + root(a^9*c^5*d^5*e^3*7168i + 3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^
3 - 3328*a^9*c^2*d^8*e^3 + a^9*c^7*d^3*e^3*2048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3
*1536i + 256*a^9*d^10*e^3 - 256*a^9*c^10*e^3 - a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d
^3*e*56i - 54*a^3*c^4*d^4*e + 28*a^3*c^6*d^2*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5
*c^2*d^5 + c*d^6*11i - 15*d^7, e, k)*((512*a^6*c^7*d - 512*a^6*c*d^7 + a^6*c^2*d^6*3072i + 7680*a^6*c^3*d^5 -
a^6*c^4*d^4*10240i - 7680*a^6*c^5*d^3 + a^6*c^6*d^2*3072i)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*
d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2) - tan(e + f*x)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d
^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)*(128*a^6*c^8 + 384*a^6*d^8 - a^6*c*d^7*2
304i + a^6*c^7*d*768i - 5888*a^6*c^2*d^6 + a^6*c^3*d^5*8448i + 7680*a^6*c^4*d^4 - a^6*c^5*d^3*4864i - 2304*a^6
*c^6*d^2)) + tan(e + f*x)*(a^3*c*d^6*688i - 192*a^3*d^7 + 16*a^3*c^6*d + 976*a^3*c^2*d^5 - a^3*c^3*d^4*736i -
352*a^3*c^4*d^3 + a^3*c^5*d^2*112i)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^
4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)) - tan(e + f*x)*(49*d^6 - c*d^5*56i - 30*c^2*d^4 + c^3*d^3*8i + c^4*d^2)*(a^
3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2))*root
(a^9*c^5*d^5*e^3*7168i + 3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^3 - 3328*a^9*c^2*d^8
*e^3 + a^9*c^7*d^3*e^3*2048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3*1536i + 256*a^9*d^1
0*e^3 - 256*a^9*c^10*e^3 - a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d^3*e*56i - 54*a^3*c^
4*d^4*e + 28*a^3*c^6*d^2*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5*c^2*d^5 + c*d^6*11i
- 15*d^7, e, k), k, 1, 3)/f - ((c*d*32i + 10*c^2 - 34*d^2)/(24*a^3*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)) + (ta
n(e + f*x)*(c*d*12i + 3*c^2 - 17*d^2)*1i)/(8*a^3*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)) - (tan(e + f*x)^2*(c*d*4
i + c^2 - 7*d^2))/(8*a^3*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)))/(f*(3*tan(e + f*x) + tan(e + f*x)^2*3i - tan(e
+ f*x)^3 - 1i))