Optimal. Leaf size=89 \[ -\frac{35 \csc ^3(a+b x)}{24 b}-\frac{35 \csc (a+b x)}{8 b}+\frac{35 \tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}+\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b} \]
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Rubi [A] time = 0.0493638, 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 = {2621, 288, 302, 207} \[ -\frac{35 \csc ^3(a+b x)}{24 b}-\frac{35 \csc (a+b x)}{8 b}+\frac{35 \tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}+\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 288
Rule 302
Rule 207
Rubi steps
\begin{align*} \int \csc ^4(a+b x) \sec ^5(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^8}{\left (-1+x^2\right )^3} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}-\frac{7 \operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{4 b}\\ &=\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}-\frac{35 \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}-\frac{35 \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=-\frac{35 \csc (a+b x)}{8 b}-\frac{35 \csc ^3(a+b x)}{24 b}+\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}-\frac{35 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=\frac{35 \tanh ^{-1}(\sin (a+b x))}{8 b}-\frac{35 \csc (a+b x)}{8 b}-\frac{35 \csc ^3(a+b x)}{24 b}+\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}\\ \end{align*}
Mathematica [C] time = 0.0139726, size = 31, normalized size = 0.35 \[ -\frac{\csc ^3(a+b x) \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\sin ^2(a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 97, normalized size = 1.1 \begin{align*}{\frac{1}{4\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{4}}}-{\frac{7}{12\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{35}{24\,b\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}-{\frac{35}{8\,b\sin \left ( bx+a \right ) }}+{\frac{35\,\ln \left ( \sec \left ( bx+a \right ) +\tan \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 = 0.996, size = 123, normalized size = 1.38 \begin{align*} -\frac{\frac{2 \,{\left (105 \, \sin \left (b x + a\right )^{6} - 175 \, \sin \left (b x + a\right )^{4} + 56 \, \sin \left (b x + a\right )^{2} + 8\right )}}{\sin \left (b x + a\right )^{7} - 2 \, \sin \left (b x + a\right )^{5} + \sin \left (b x + a\right )^{3}} - 105 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 105 \, \log \left (\sin \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 = 2.33682, size = 375, normalized size = 4.21 \begin{align*} -\frac{210 \, \cos \left (b x + a\right )^{6} - 280 \, \cos \left (b x + a\right )^{4} - 105 \,{\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 105 \,{\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 42 \, \cos \left (b x + a\right )^{2} + 12}{48 \,{\left (b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )} \sin \left (b x + a\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 [A] time = 1.21251, size = 115, normalized size = 1.29 \begin{align*} -\frac{\frac{6 \,{\left (11 \, \sin \left (b x + a\right )^{3} - 13 \, \sin \left (b x + a\right )\right )}}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{2}} + \frac{16 \,{\left (9 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 105 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 105 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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