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 = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 207
Rule 288
Rule 302
Rule 2621
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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 0.48, size = 140, normalized size = 1.57 \[ -\frac {210 \, \cos \left (b x + a\right )^{6} - 280 \, \cos \left (b x + a\right )^{4} - 105 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 105 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 42 \, \cos \left (b x + a\right )^{2} + 12}{48 \, {\left (b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 85, normalized size = 0.96 \[ -\frac {\frac {6 \, {\left (11 \, \sin \left (b x + a\right )^{3} - 13 \, \sin \left (b x + a\right )\right )}}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{2}} + \frac {16 \, {\left (9 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 105 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 105 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 97, normalized size = 1.09 \[ \frac {1}{4 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{4}}-\frac {7}{12 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{2}}+\frac {35}{24 b \sin \left (b x +a \right ) \cos \left (b x +a \right )^{2}}-\frac {35}{8 b \sin \left (b x +a \right )}+\frac {35 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 91, normalized size = 1.02 \[ -\frac {\frac {2 \, {\left (105 \, \sin \left (b x + a\right )^{6} - 175 \, \sin \left (b x + a\right )^{4} + 56 \, \sin \left (b x + a\right )^{2} + 8\right )}}{\sin \left (b x + a\right )^{7} - 2 \, \sin \left (b x + a\right )^{5} + \sin \left (b x + a\right )^{3}} - 105 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 105 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 79, normalized size = 0.89 \[ \frac {35\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{8\,b}-\frac {\frac {35\,{\sin \left (a+b\,x\right )}^6}{8}-\frac {175\,{\sin \left (a+b\,x\right )}^4}{24}+\frac {7\,{\sin \left (a+b\,x\right )}^2}{3}+\frac {1}{3}}{b\,\left ({\sin \left (a+b\,x\right )}^7-2\,{\sin \left (a+b\,x\right )}^5+{\sin \left (a+b\,x\right )}^3\right )} \]
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 \[ \int \frac {\sec ^{5}{\left (a + b x \right )}}{\sin ^{4}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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