Optimal. Leaf size=55 \[ -\frac {3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3770} \[ -\frac {3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^5(a+b x) \, dx &=-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {3}{4} \int \csc ^3(a+b x) \, dx\\ &=-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {3}{8} \int \csc (a+b x) \, dx\\ &=-\frac {3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 113, normalized size = 2.05 \[ -\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {3 \csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {3 \sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {3 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {3 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 112, normalized size = 2.04 \[ \frac {6 \, \cos \left (b x + a\right )^{3} - 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 10 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 138, normalized size = 2.51 \[ \frac {\frac {{\left (\frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 12 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.83, size = 59, normalized size = 1.07 \[ -\frac {\cot \left (b x +a \right ) \left (\csc ^{3}\left (b x +a \right )\right )}{4 b}-\frac {3 \cot \left (b x +a \right ) \csc \left (b x +a \right )}{8 b}+\frac {3 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 71, normalized size = 1.29 \[ \frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} - 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 59, normalized size = 1.07 \[ -\frac {3\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{8\,b}-\frac {\frac {5\,\cos \left (a+b\,x\right )}{8}-\frac {3\,{\cos \left (a+b\,x\right )}^3}{8}}{b\,\left ({\cos \left (a+b\,x\right )}^4-2\,{\cos \left (a+b\,x\right )}^2+1\right )} \]
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 \[ \int \csc ^{5}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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