Optimal. Leaf size=46 \[ -\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b}-\frac {\cos ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 46, 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 = {4295, 2639} \[ -\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b}-\frac {\cos ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 4295
Rubi steps
\begin {align*} \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx &=-\frac {\cos ^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 {\cos ^2(a+b x)}{b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 39, normalized size = 0.85 \[ -\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )+\sqrt {\sin (2 (a+b x))} \cot (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\cos \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}{\cos \left (2 \, b x + 2 \, a\right )^{2} - 1}, x\right ) \]
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 \[ \int \frac {\cos \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 17.33, size = 94273592, normalized size = 2049425.91 \[ \text {output too large to display} \]
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 \[ \int \frac {\cos \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (a+b\,x\right )}^2}{{\sin \left (2\,a+2\,b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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