Optimal. Leaf size=85 \[ \frac{6 \sin ^5(c+d x)}{35 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin (c+d x)}{7 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0553903, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3502, 2633} \[ \frac{6 \sin ^5(c+d x)}{35 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin (c+d x)}{7 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3502
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{a+i a \tan (c+d x)} \, dx &=\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}+\frac{6 \int \cos ^5(c+d x) \, dx}{7 a}\\ &=\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}-\frac{6 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{7 a d}\\ &=\frac{6 \sin (c+d x)}{7 a d}-\frac{4 \sin ^3(c+d x)}{7 a d}+\frac{6 \sin ^5(c+d x)}{35 a d}+\frac{i \cos ^5(c+d x)}{7 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.172666, size = 94, normalized size = 1.11 \[ -\frac{\sec (c+d x) (350 i \sin (2 (c+d x))+56 i \sin (4 (c+d x))+6 i \sin (6 (c+d x))+175 \cos (2 (c+d x))+14 \cos (4 (c+d x))+\cos (6 (c+d x))-350)}{1120 a d (\tan (c+d x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.085, size = 207, normalized size = 2.4 \begin{align*} 2\,{\frac{1}{ad} \left ( -1/7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-7}+{\frac{i/2}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{{\frac{15\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{{\frac{11\,i}{8}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{21}{20\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{11}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{21}{32\,\tan \left ( 1/2\,dx+c/2 \right ) -32\,i}}+{\frac{i/8}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{4}}}-{\frac{i/4}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}+1/20\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-5}-1/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-3}+{\frac{11}{32\,\tan \left ( 1/2\,dx+c/2 \right ) +32\,i}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.03141, size = 284, normalized size = 3.34 \begin{align*} \frac{{\left (-7 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 70 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 525 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 700 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 175 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 42 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{2240 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.32842, size = 265, normalized size = 3.12 \begin{align*} \begin{cases} \frac{\left (- 150323855360 i a^{6} d^{6} e^{21 i c} e^{5 i d x} - 1503238553600 i a^{6} d^{6} e^{19 i c} e^{3 i d x} - 11274289152000 i a^{6} d^{6} e^{17 i c} e^{i d x} + 15032385536000 i a^{6} d^{6} e^{15 i c} e^{- i d x} + 3758096384000 i a^{6} d^{6} e^{13 i c} e^{- 3 i d x} + 901943132160 i a^{6} d^{6} e^{11 i c} e^{- 5 i d x} + 107374182400 i a^{6} d^{6} e^{9 i c} e^{- 7 i d x}\right ) e^{- 16 i c}}{48103633715200 a^{7} d^{7}} & \text{for}\: 48103633715200 a^{7} d^{7} e^{16 i c} \neq 0 \\\frac{x \left (e^{12 i c} + 6 e^{10 i c} + 15 e^{8 i c} + 20 e^{6 i c} + 15 e^{4 i c} + 6 e^{2 i c} + 1\right ) e^{- 7 i c}}{64 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15593, size = 231, normalized size = 2.72 \begin{align*} \frac{\frac{7 \,{\left (55 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 180 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 250 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 160 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 43\right )}}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{5}} + \frac{735 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3360 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 7315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 8820 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6321 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2492 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 461}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{7}}}{560 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]