Optimal. Leaf size=131 \[ \frac {i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}+\frac {x}{8 a^3}+\frac {i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac {i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {3082, 8} \[ \frac {i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}+\frac {x}{8 a^3}+\frac {i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac {i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3082
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=\frac {i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx}{2 a}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac {i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}+\frac {\int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx}{4 a^2}\\ &=\frac {i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac {i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}+\frac {i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}+\frac {\int 1 \, dx}{8 a^3}\\ &=\frac {x}{8 a^3}+\frac {i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac {i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}+\frac {i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 84, normalized size = 0.64 \[ \frac {18 \sin (2 (c+d x))+9 \sin (4 (c+d x))+2 \sin (6 (c+d x))+18 i \cos (2 (c+d x))+9 i \cos (4 (c+d x))+2 i \cos (6 (c+d x))+12 c+12 d x}{96 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 54, normalized size = 0.41 \[ \frac {{\left (12 \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 18 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 80, normalized size = 0.61 \[ -\frac {\frac {6 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac {6 i \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac {-11 i \, \tan \left (d x + c\right )^{3} - 45 \, \tan \left (d x + c\right )^{2} + 69 i \, \tan \left (d x + c\right ) + 51}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 98, normalized size = 0.75 \[ \frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{16 a^{3} d}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{16 a^{3} d}-\frac {i}{8 a^{3} d \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{6 a^{3} d \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{8 a^{3} d \left (\tan \left (d x +c \right )-i\right )} \]
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 = 3.49, size = 96, normalized size = 0.73 \[ \frac {x}{8\,a^3}+\frac {\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,9{}\mathrm {i}}{2}-\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,9{}\mathrm {i}}{2}+\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{a^3\,d\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.31, size = 160, normalized size = 1.22 \[ \begin {cases} - \frac {\left (- 4608 i a^{6} d^{2} e^{10 i c} e^{- 2 i d x} - 2304 i a^{6} d^{2} e^{8 i c} e^{- 4 i d x} - 512 i a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (\frac {\left (e^{6 i c} + 3 e^{4 i c} + 3 e^{2 i c} + 1\right ) e^{- 6 i c}}{8 a^{3}} - \frac {1}{8 a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {x}{8 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________