Optimal. Leaf size=81 \[ \frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{21 b \sqrt {\sin (2 a+2 b x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4381, 4389,
4376} \begin {gather*} \frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{21 b \sqrt {\sin (2 a+2 b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4376
Rule 4381
Rule 4389
Rubi steps
\begin {align*} \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2}{7} \int \frac {\sin (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx\\ &=\frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {4}{21} \int \frac {\cos (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\\ &=\frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{21 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 55, normalized size = 0.68 \begin {gather*} -\frac {(5+12 \cos (2 (a+b x))+4 \cos (4 (a+b x))) \csc (a+b x) \sec ^4(a+b x) \sqrt {\sin (2 (a+b x))}}{336 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sin ^{3}\left (x b +a \right )}{\sin \left (2 x b +2 a \right )^{\frac {9}{2}}}\, dx\]
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 = 3.57, size = 79, normalized size = 0.98 \begin {gather*} -\frac {32 \, \cos \left (b x + a\right )^{4} \sin \left (b x + a\right ) + \sqrt {2} {\left (32 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{336 \, b \cos \left (b x + a\right )^{4} \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(-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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Mupad [B]
time = 3.71, size = 300, normalized size = 3.70 \begin {gather*} -\frac {{\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}}\,5{}\mathrm {i}}{84\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2}+\frac {3\,{\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}}}{14\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^3}-\frac {{\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}}\,1{}\mathrm {i}}{7\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^4}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {5}{84\,b}+\frac {4\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}}{21\,b}\right )\,\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}}-1\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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