Optimal. Leaf size=124 \[ \frac {i c^2}{12 a^3 f (c+i c \tan (e+f x))^3}+\frac {i c}{8 a^3 f (c+i c \tan (e+f x))^2}-\frac {i}{16 a^3 f (c-i c \tan (e+f x))}+\frac {3 i}{16 a^3 f (c+i c \tan (e+f x))}+\frac {x}{4 a^3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3522, 3487, 44, 206} \[ \frac {i c^2}{12 a^3 f (c+i c \tan (e+f x))^3}+\frac {i c}{8 a^3 f (c+i c \tan (e+f x))^2}-\frac {i}{16 a^3 f (c-i c \tan (e+f x))}+\frac {3 i}{16 a^3 f (c+i c \tan (e+f x))}+\frac {x}{4 a^3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 206
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx &=\frac {\int \cos ^6(e+f x) (c-i c \tan (e+f x))^2 \, dx}{a^3 c^3}\\ &=\frac {\left (i c^4\right ) \operatorname {Subst}\left (\int \frac {1}{(c-x)^4 (c+x)^2} \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=\frac {\left (i c^4\right ) \operatorname {Subst}\left (\int \left (\frac {1}{4 c^2 (c-x)^4}+\frac {1}{4 c^3 (c-x)^3}+\frac {3}{16 c^4 (c-x)^2}+\frac {1}{16 c^4 (c+x)^2}+\frac {1}{4 c^4 \left (c^2-x^2\right )}\right ) \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=-\frac {i}{16 a^3 f (c-i c \tan (e+f x))}+\frac {i c^2}{12 a^3 f (c+i c \tan (e+f x))^3}+\frac {i c}{8 a^3 f (c+i c \tan (e+f x))^2}+\frac {3 i}{16 a^3 f (c+i c \tan (e+f x))}+\frac {i \operatorname {Subst}\left (\int \frac {1}{c^2-x^2} \, dx,x,-i c \tan (e+f x)\right )}{4 a^3 f}\\ &=\frac {x}{4 a^3 c}-\frac {i}{16 a^3 f (c-i c \tan (e+f x))}+\frac {i c^2}{12 a^3 f (c+i c \tan (e+f x))^3}+\frac {i c}{8 a^3 f (c+i c \tan (e+f x))^2}+\frac {3 i}{16 a^3 f (c+i c \tan (e+f x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.19, size = 101, normalized size = 0.81 \[ -\frac {\sec ^2(e+f x) (12 i f x \sin (2 (e+f x))+3 \sin (2 (e+f x))+2 \sin (4 (e+f x))+3 (4 f x+i) \cos (2 (e+f x))-i \cos (4 (e+f x))+9 i)}{48 a^3 c f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 68, normalized size = 0.55 \[ \frac {{\left (24 \, f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (8 i \, f x + 8 i \, e\right )} + 18 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 6 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, a^{3} c f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.03, size = 123, normalized size = 0.99 \[ -\frac {-\frac {6 i \, \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c} + \frac {6 i \, \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c} + \frac {3 \, {\left (2 i \, \tan \left (f x + e\right ) - 3\right )}}{a^{3} c {\left (\tan \left (f x + e\right ) + i\right )}} + \frac {-11 i \, \tan \left (f x + e\right )^{3} - 42 \, \tan \left (f x + e\right )^{2} + 57 i \, \tan \left (f x + e\right ) + 30}{a^{3} c {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.35, size = 135, normalized size = 1.09 \[ \frac {i \ln \left (\tan \left (f x +e \right )+i\right )}{8 f \,a^{3} c}+\frac {1}{16 f \,a^{3} c \left (\tan \left (f x +e \right )+i\right )}-\frac {i \ln \left (\tan \left (f x +e \right )-i\right )}{8 f \,a^{3} c}-\frac {i}{8 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{12 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {3}{16 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.90, size = 77, normalized size = 0.62 \[ \frac {x}{4\,a^3\,c}-\frac {\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{4}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{2}-\frac {\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{12}+\frac {1}{3}}{a^3\,c\,f\,{\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.44, size = 216, normalized size = 1.74 \[ \begin {cases} \frac {\left (- 24576 i a^{9} c^{3} f^{3} e^{14 i e} e^{2 i f x} + 147456 i a^{9} c^{3} f^{3} e^{10 i e} e^{- 2 i f x} + 49152 i a^{9} c^{3} f^{3} e^{8 i e} e^{- 4 i f x} + 8192 i a^{9} c^{3} f^{3} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{786432 a^{12} c^{4} f^{4}} & \text {for}\: 786432 a^{12} c^{4} f^{4} e^{12 i e} \neq 0 \\x \left (\frac {\left (e^{8 i e} + 4 e^{6 i e} + 6 e^{4 i e} + 4 e^{2 i e} + 1\right ) e^{- 6 i e}}{16 a^{3} c} - \frac {1}{4 a^{3} c}\right ) & \text {otherwise} \end {cases} + \frac {x}{4 a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________