Optimal. Leaf size=164 \[ -\frac {5 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{4 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b} \]
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Rubi [A]
time = 0.12, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4385, 4393,
4386, 4387, 4391} \begin {gather*} -\frac {5 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))}{4 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (a+b x)}{3 b}+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}-\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 4385
Rule 4386
Rule 4387
Rule 4391
Rule 4393
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx &=\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+4 \int \csc (a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx\\ &=\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+8 \int \cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx\\ &=\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+\frac {20}{3} \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx\\ &=-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+5 \int \cos (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\\ &=\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+\frac {5}{2} \int \frac {\sin (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{4 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 84, normalized size = 0.51 \begin {gather*} \frac {-5 \left (\text {ArcSin}(\cos (a+b x)-\sin (a+b x))+\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )\right )+2 \sqrt {\sin (2 (a+b x))} (6 \sin (a+b x)+\sin (3 (a+b x)))}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 129.38, size = 973, normalized size = 5.93
method | result | size |
default | \(\text {Expression too large to display}\) | \(973\) |
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 = 2.19, size = 280, normalized size = 1.71 \begin {gather*} \frac {8 \, \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 )} \sin \left (b x + a\right ) + 10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{16 \, b} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{7/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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