Optimal. Leaf size=34 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3770} \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \, dx &=-\frac {\cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int \csc (a+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 75, normalized size = 2.21 \[ -\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 72, normalized size = 2.12 \[ -\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (b x + a\right )}{4 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 92, normalized size = 2.71 \[ -\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.85, size = 40, normalized size = 1.18 \[ -\frac {\cot \left (b x +a \right ) \csc \left (b x +a \right )}{2 b}+\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 46, normalized size = 1.35 \[ \frac {\frac {2 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 36, normalized size = 1.06 \[ \frac {\cos \left (a+b\,x\right )}{2\,b\,\left ({\cos \left (a+b\,x\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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