Optimal. Leaf size=77 \[ -\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{10 b}+\frac {\sin ^2(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {3 \cos (2 a+2 b x)}{10 b \sqrt {\sin (2 a+2 b x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 77, 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, 2716,
2719} \begin {gather*} \frac {\sin ^2(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{10 b}-\frac {3 \cos (2 a+2 b x)}{10 b \sqrt {\sin (2 a+2 b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 4381
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^2(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {3}{10} \int \frac {1}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\\ &=\frac {\sin ^2(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {3 \cos (2 a+2 b x)}{10 b \sqrt {\sin (2 a+2 b x)}}-\frac {3}{10} \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{10 b}+\frac {\sin ^2(a+b x)}{5 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {3 \cos (2 a+2 b x)}{10 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 66, normalized size = 0.86 \begin {gather*} -\frac {12 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\frac {4 (1+6 \cos (2 (a+b x))+3 \cos (4 (a+b x))) \sin ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 (a+b x))}}{40 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs.
\(2(92)=184\).
time = 247.57, size = 227, normalized size = 2.95
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\frac {8 \sqrt {2}}{5 \sin \left (2 x b +2 a \right )^{\frac {5}{2}}}+\frac {4 \sqrt {2}\, \left (6 \sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \left (\sin ^{2}\left (2 x b +2 a \right )\right ) \EllipticE \left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \left (\sin ^{2}\left (2 x b +2 a \right )\right ) \EllipticF \left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )+6 \left (\sin ^{4}\left (2 x b +2 a \right )\right )-4 \left (\sin ^{2}\left (2 x b +2 a \right )\right )-2\right )}{5 \sin \left (2 x b +2 a \right )^{\frac {5}{2}} \cos \left (2 x b +2 a \right )}\right )}{32 b}\) | \(227\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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