Optimal. Leaf size=76 \[ -\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b} \]
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Rubi [A] time = 0.0363369, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3770} \[ -\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \csc ^7(a+b x) \, dx &=-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac{5}{6} \int \csc ^5(a+b x) \, dx\\ &=-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac{5}{8} \int \csc ^3(a+b x) \, dx\\ &=-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac{5}{16} \int \csc (a+b x) \, dx\\ &=-\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.013899, size = 151, normalized size = 1.99 \[ -\frac{\csc ^6\left (\frac{1}{2} (a+b x)\right )}{384 b}-\frac{\csc ^4\left (\frac{1}{2} (a+b x)\right )}{64 b}-\frac{5 \csc ^2\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{\sec ^6\left (\frac{1}{2} (a+b x)\right )}{384 b}+\frac{\sec ^4\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{5 \sec ^2\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{5 \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{16 b}-\frac{5 \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{16 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 78, normalized size = 1. \begin{align*} -{\frac{\cot \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{5}}{6\,b}}-{\frac{5\,\cot \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{3}}{24\,b}}-{\frac{5\,\csc \left ( bx+a \right ) \cot \left ( bx+a \right ) }{16\,b}}+{\frac{5\,\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.
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Maxima [A] time = 1.04107, size = 123, normalized size = 1.62 \begin{align*} \frac{\frac{2 \,{\left (15 \, \cos \left (b x + a\right )^{5} - 40 \, \cos \left (b x + a\right )^{3} + 33 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{96 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.497951, size = 424, normalized size = 5.58 \begin{align*} \frac{30 \, \cos \left (b x + a\right )^{5} - 80 \, \cos \left (b x + a\right )^{3} - 15 \,{\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 15 \,{\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 66 \, \cos \left (b x + a\right )}{96 \,{\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, b \cos \left (b x + a\right )^{2} - b\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 \csc ^{7}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29354, size = 246, normalized size = 3.24 \begin{align*} -\frac{\frac{{\left (\frac{9 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{45 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{110 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}} + \frac{45 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{9 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 60 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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