Optimal. Leaf size=23 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\csc (a+b x)}{b} \]
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Rubi [A] time = 0.0217056, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2621, 321, 207} \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sec (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac{\csc (a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\csc (a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 0.0146672, size = 27, normalized size = 1.17 \[ -\frac{\csc (a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\sin ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 33, normalized size = 1.4 \begin{align*} -{\frac{1}{b\sin \left ( bx+a \right ) }}+{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989706, size = 49, normalized size = 2.13 \begin{align*} -\frac{\frac{2}{\sin \left (b x + a\right )} - \log \left (\sin \left (b x + a\right ) + 1\right ) + \log \left (\sin \left (b x + a\right ) - 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9893, size = 136, normalized size = 5.91 \begin{align*} \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 2}{2 \, b \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{\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 = 1.19789, size = 51, normalized size = 2.22 \begin{align*} -\frac{\frac{2}{\sin \left (b x + a\right )} - \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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