Optimal. Leaf size=43 \[ -\frac{\csc ^3(a+b x)}{48 b}-\frac{\csc (a+b x)}{16 b}+\frac{\tanh ^{-1}(\sin (a+b x))}{16 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0513725, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4287, 2621, 302, 207} \[ -\frac{\csc ^3(a+b x)}{48 b}-\frac{\csc (a+b x)}{16 b}+\frac{\tanh ^{-1}(\sin (a+b x))}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4287
Rule 2621
Rule 302
Rule 207
Rubi steps
\begin{align*} \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac{1}{16} \int \csc ^4(a+b x) \sec (a+b x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac{\csc (a+b x)}{16 b}-\frac{\csc ^3(a+b x)}{48 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac{\csc (a+b x)}{16 b}-\frac{\csc ^3(a+b x)}{48 b}\\ \end{align*}
Mathematica [C] time = 0.0181735, size = 31, normalized size = 0.72 \[ -\frac{\csc ^3(a+b x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\sin ^2(a+b x)\right )}{48 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 47, normalized size = 1.1 \begin{align*} -{\frac{1}{48\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}}}-{\frac{1}{16\,b\sin \left ( bx+a \right ) }}+{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{16\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.41644, size = 1126, normalized size = 26.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.501965, size = 254, normalized size = 5.91 \begin{align*} \frac{3 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 8}{96 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23286, size = 70, normalized size = 1.63 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]