Optimal. Leaf size=34 \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc (a+b x)}{16 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.040758, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4287, 3768, 3770} \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc (a+b x)}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4287
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \cos ^3(a+b x) \csc ^3(2 a+2 b x) \, dx &=\frac{1}{8} \int \csc ^3(a+b x) \, dx\\ &=-\frac{\cot (a+b x) \csc (a+b x)}{16 b}+\frac{1}{16} \int \csc (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc (a+b x)}{16 b}\\ \end{align*}
Mathematica [B] time = 0.0151075, size = 79, normalized size = 2.32 \[ \frac{1}{8} \left (-\frac{\csc ^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{\sec ^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{\log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{2 b}-\frac{\log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{2 b}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 40, normalized size = 1.2 \begin{align*} -{\frac{\cot \left ( bx+a \right ) \csc \left ( bx+a \right ) }{16\,b}}+{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{16\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.20577, size = 753, normalized size = 22.15 \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 = 0.493301, size = 203, normalized size = 5.97 \begin{align*} -\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) -{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (b x + a\right )}{32 \,{\left (b \cos \left (b x + a\right )^{2} - b\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 [B] time = 1.31529, size = 124, normalized size = 3.65 \begin{align*} -\frac{\frac{{\left (\frac{2 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]