Optimal. Leaf size=110 \[ \frac {1}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {i x}{16 a^4}+\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {1}{12 a d (a+i a \tan (c+d x))^3}-\frac {1}{8 d (a+i a \tan (c+d x))^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3526, 3479, 8} \[ \frac {1}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {i x}{16 a^4}+\frac {1}{12 a d (a+i a \tan (c+d x))^3}-\frac {1}{8 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3479
Rule 3526
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac {1}{8 d (a+i a \tan (c+d x))^4}-\frac {i \int \frac {1}{(a+i a \tan (c+d x))^3} \, dx}{2 a}\\ &=-\frac {1}{8 d (a+i a \tan (c+d x))^4}+\frac {1}{12 a d (a+i a \tan (c+d x))^3}-\frac {i \int \frac {1}{(a+i a \tan (c+d x))^2} \, dx}{4 a^2}\\ &=-\frac {1}{8 d (a+i a \tan (c+d x))^4}+\frac {1}{12 a d (a+i a \tan (c+d x))^3}+\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {i \int \frac {1}{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=-\frac {1}{8 d (a+i a \tan (c+d x))^4}+\frac {1}{12 a d (a+i a \tan (c+d x))^3}+\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {1}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {i \int 1 \, dx}{16 a^4}\\ &=-\frac {i x}{16 a^4}-\frac {1}{8 d (a+i a \tan (c+d x))^4}+\frac {1}{12 a d (a+i a \tan (c+d x))^3}+\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {1}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.37, size = 94, normalized size = 0.85 \[ \frac {\sec ^4(c+d x) (32 i \sin (2 (c+d x))+24 d x \sin (4 (c+d x))+3 i \sin (4 (c+d x))+16 \cos (2 (c+d x))+(-3-24 i d x) \cos (4 (c+d x)))}{384 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 54, normalized size = 0.49 \[ \frac {{\left (-24 i \, d x e^{\left (8 i \, d x + 8 i \, c\right )} + 24 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 8 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.89, size = 88, normalized size = 0.80 \[ \frac {\frac {12 \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} - \frac {12 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} + \frac {25 \, \tan \left (d x + c\right )^{4} - 124 i \, \tan \left (d x + c\right )^{3} - 246 \, \tan \left (d x + c\right )^{2} + 252 i \, \tan \left (d x + c\right ) + 57}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 116, normalized size = 1.05 \[ \frac {\ln \left (\tan \left (d x +c \right )+i\right )}{32 d \,a^{4}}+\frac {i}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {i}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}-\frac {1}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {1}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {\ln \left (\tan \left (d x +c \right )-i\right )}{32 d \,a^{4}} \]
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 = 3.90, size = 60, normalized size = 0.55 \[ -\frac {x\,1{}\mathrm {i}}{16\,a^4}+\frac {-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}}{16}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{4}+\frac {\mathrm {tan}\left (c+d\,x\right )\,19{}\mathrm {i}}{48}+\frac {1}{12}}{a^4\,d\,{\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.77, size = 158, normalized size = 1.44 \[ \begin {cases} \frac {\left (6144 a^{8} d^{2} e^{14 i c} e^{- 2 i d x} - 2048 a^{8} d^{2} e^{10 i c} e^{- 6 i d x} - 768 a^{8} d^{2} e^{8 i c} e^{- 8 i d x}\right ) e^{- 16 i c}}{98304 a^{12} d^{3}} & \text {for}\: 98304 a^{12} d^{3} e^{16 i c} \neq 0 \\x \left (\frac {\left (- i e^{8 i c} - 2 i e^{6 i c} + 2 i e^{2 i c} + i\right ) e^{- 8 i c}}{16 a^{4}} + \frac {i}{16 a^{4}}\right ) & \text {otherwise} \end {cases} - \frac {i x}{16 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________