Optimal. Leaf size=49 \[ -\frac{3 \csc (a+b x)}{2 b}+\frac{3 \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{\csc (a+b x) \sec ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0431818, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2621, 288, 321, 207} \[ -\frac{3 \csc (a+b x)}{2 b}+\frac{3 \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{\csc (a+b x) \sec ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 288
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sec ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{\csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=-\frac{3 \csc (a+b x)}{2 b}+\frac{\csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=\frac{3 \tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{3 \csc (a+b x)}{2 b}+\frac{\csc (a+b x) \sec ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [C] time = 0.0131511, size = 27, normalized size = 0.55 \[ -\frac{\csc (a+b x) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\sin ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 55, normalized size = 1.1 \begin{align*}{\frac{1}{2\,b\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}-{\frac{3}{2\,b\sin \left ( bx+a \right ) }}+{\frac{3\,\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01712, size = 82, normalized size = 1.67 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \sin \left (b x + a\right )^{2} - 2\right )}}{\sin \left (b x + a\right )^{3} - \sin \left (b x + a\right )} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98169, size = 228, normalized size = 4.65 \begin{align*} \frac{3 \, \cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 2}{4 \, b \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{3}{\left (a + b x \right )}}{\sin ^{2}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21955, size = 85, normalized size = 1.73 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \sin \left (b x + a\right )^{2} - 2\right )}}{\sin \left (b x + a\right )^{3} - \sin \left (b x + a\right )} - 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 )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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