Optimal. Leaf size=98 \[ \frac{4 d^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{5/2}}{7 b d}+\frac{4 d \sin (a+b x) \sqrt{d \cos (a+b x)}}{21 b} \]
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Rubi [A] time = 0.0788385, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2568, 2635, 2642, 2641} \[ \frac{4 d^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{5/2}}{7 b d}+\frac{4 d \sin (a+b x) \sqrt{d \cos (a+b x)}}{21 b} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{3/2} \sin ^2(a+b x) \, dx &=-\frac{2 (d \cos (a+b x))^{5/2} \sin (a+b x)}{7 b d}+\frac{2}{7} \int (d \cos (a+b x))^{3/2} \, dx\\ &=\frac{4 d \sqrt{d \cos (a+b x)} \sin (a+b x)}{21 b}-\frac{2 (d \cos (a+b x))^{5/2} \sin (a+b x)}{7 b d}+\frac{1}{21} \left (2 d^2\right ) \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx\\ &=\frac{4 d \sqrt{d \cos (a+b x)} \sin (a+b x)}{21 b}-\frac{2 (d \cos (a+b x))^{5/2} \sin (a+b x)}{7 b d}+\frac{\left (2 d^2 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{21 \sqrt{d \cos (a+b x)}}\\ &=\frac{4 d^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{d \cos (a+b x)}}+\frac{4 d \sqrt{d \cos (a+b x)} \sin (a+b x)}{21 b}-\frac{2 (d \cos (a+b x))^{5/2} \sin (a+b x)}{7 b d}\\ \end{align*}
Mathematica [C] time = 0.0707112, size = 57, normalized size = 0.58 \[ \frac{\cos ^2(a+b x)^{3/4} \tan ^3(a+b x) (d \cos (a+b x))^{3/2} \, _2F_1\left (-\frac{1}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 208, normalized size = 2.1 \begin{align*}{\frac{4\,{d}^{2}}{21\,b}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 24\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}-60\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}+50\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-15\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-\sqrt{ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) ,\sqrt{2} \right ) +\cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ){\frac{1}{\sqrt{-d \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \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 \left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} \sin \left (b x + a\right )^{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 (-{\left (d \cos \left (b x + a\right )^{3} - d \cos \left (b x + a\right )\right )} \sqrt{d \cos \left (b x + 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(-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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