Optimal. Leaf size=45 \[ -\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{2 b}+\frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, 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 = {4381, 2719}
\begin {gather*} \frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 4381
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx &=\frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {1}{2} \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{2 b}+\frac {\sin ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 41, normalized size = 0.91 \begin {gather*} \frac {-E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\sqrt {\sin (2 (a+b x))} \tan (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 12.41, size = 96383272, normalized size = 2141850.49
method | result | size |
default | \(\text {Expression too large to display}\) | \(96383272\) |
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 = -0.02 \begin {gather*} \int \frac {{\sin \left (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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