Optimal. Leaf size=55 \[ -\frac {4 \csc (a+b x) \sqrt {\sin (2 a+2 b x)}}{5 b}-\frac {\csc ^3(a+b x) \sqrt {\sin (2 a+2 b x)}}{5 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4385, 4377}
\begin {gather*} -\frac {\sqrt {\sin (2 a+2 b x)} \csc ^3(a+b x)}{5 b}-\frac {4 \sqrt {\sin (2 a+2 b x)} \csc (a+b x)}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 4377
Rule 4385
Rubi steps
\begin {align*} \int \frac {\csc ^3(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx &=-\frac {\csc ^3(a+b x) \sqrt {\sin (2 a+2 b x)}}{5 b}+\frac {4}{5} \int \frac {\csc (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {4 \csc (a+b x) \sqrt {\sin (2 a+2 b x)}}{5 b}-\frac {\csc ^3(a+b x) \sqrt {\sin (2 a+2 b x)}}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 35, normalized size = 0.64 \begin {gather*} -\frac {\csc (a+b x) \left (4+\csc ^2(a+b x)\right ) \sqrt {\sin (2 (a+b x))}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\csc ^{3}\left (x b +a \right )}{\sqrt {\sin \left (2 x b +2 a \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 76, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} - 5\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 4 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.16, size = 93, normalized size = 1.69 \begin {gather*} -\frac {8\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,3{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{5\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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