Optimal. Leaf size=89 \[ \frac{35 \sec ^3(a+b x)}{24 b}+\frac{35 \sec (a+b x)}{8 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b} \]
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Rubi [A] time = 0.0462182, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2622, 288, 302, 207} \[ \frac{35 \sec ^3(a+b x)}{24 b}+\frac{35 \sec (a+b x)}{8 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 288
Rule 302
Rule 207
Rubi steps
\begin{align*} \int \csc ^5(a+b x) \sec ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^8}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac{7 \operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{4 b}\\ &=-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac{35 \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{8 b}\\ &=-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac{35 \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{8 b}\\ &=\frac{35 \sec (a+b x)}{8 b}+\frac{35 \sec ^3(a+b x)}{24 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}+\frac{35 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{8 b}\\ &=-\frac{35 \tanh ^{-1}(\cos (a+b x))}{8 b}+\frac{35 \sec (a+b x)}{8 b}+\frac{35 \sec ^3(a+b x)}{24 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}\\ \end{align*}
Mathematica [B] time = 0.504024, size = 268, normalized size = 3.01 \[ -\frac{\csc ^{10}(a+b x) \left (658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )-105 \cos (5 (a+b x)) \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+105 \cos (7 (a+b x)) \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+3 \cos (a+b x) \left (-105 \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )+105 \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+76\right )+315 \cos (3 (a+b x)) \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )+105 \cos (5 (a+b x)) \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )-105 \cos (7 (a+b x)) \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )-204\right )}{24 b \left (\csc ^2\left (\frac{1}{2} (a+b x)\right )-\sec ^2\left (\frac{1}{2} (a+b x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 99, normalized size = 1.1 \begin{align*} -{\frac{1}{4\,b \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}+{\frac{7}{12\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}-{\frac{35}{24\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}\cos \left ( bx+a \right ) }}+{\frac{35}{8\,b\cos \left ( bx+a \right ) }}+{\frac{35\,\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{8\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973315, size = 123, normalized size = 1.38 \begin{align*} \frac{\frac{2 \,{\left (105 \, \cos \left (b x + a\right )^{6} - 175 \, \cos \left (b x + a\right )^{4} + 56 \, \cos \left (b x + a\right )^{2} + 8\right )}}{\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}} - 105 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 105 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89678, size = 416, normalized size = 4.67 \begin{align*} \frac{210 \, \cos \left (b x + a\right )^{6} - 350 \, \cos \left (b x + a\right )^{4} + 112 \, \cos \left (b x + a\right )^{2} - 105 \,{\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 105 \,{\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 16}{48 \,{\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22033, size = 282, normalized size = 3.17 \begin{align*} \frac{\frac{3 \,{\left (\frac{24 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{210 \,{\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{72 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{3 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{256 \,{\left (\frac{9 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{6 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 5\right )}}{{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 420 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{192 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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