Optimal. Leaf size=48 \[ \frac{\text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{6 b}-\frac{\cos ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.0357419, antiderivative size = 48, normalized size of antiderivative = 1., 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, 2641} \[ \frac{F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{6 b}-\frac{\cos ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4295
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx &=-\frac{\cos ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{1}{6} \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{F\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{6 b}-\frac{\cos ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}\\ \end{align*}
Mathematica [A] time = 1.00654, size = 82, normalized size = 1.71 \[ -\frac{\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) \text{EllipticF}\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x)),\frac{1}{2}\right )}{\sqrt{\sin (2 (a+b x))+1}}+\sqrt{\sin (2 (a+b x))} \csc ^2(a+b x)}{12 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 11.904, size = 123, normalized size = 2.6 \begin{align*}{\frac{1}{12\,\cos \left ( 2\,bx+2\,a \right ) b} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \sin \left ( 2\,bx+2\,a \right ) -2\, \left ( \cos \left ( 2\,bx+2\,a \right ) \right ) ^{2}-2\,\cos \left ( 2\,bx+2\,a \right ) \right ) \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cos \left (b x + a\right )^{2}}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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