Optimal. Leaf size=78 \[ \frac{2 a^2 B \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}-\frac{2 a B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d} \]
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Rubi [A] time = 0.166554, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {21, 2804, 2639, 2803, 2641, 2805} \[ \frac{2 a^2 B \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}-\frac{2 a B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2804
Rule 2639
Rule 2803
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx &=B \int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx\\ &=\frac{B \int \sqrt{\cos (c+d x)} \, dx}{b}-\frac{(a B) \int \frac{\sqrt{\cos (c+d x)}}{a+b \cos (c+d x)} \, dx}{b}\\ &=\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d}-\frac{(a B) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{b^2}+\frac{\left (a^2 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^2}\\ &=\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d}-\frac{2 a B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 a^2 B \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}\\ \end{align*}
Mathematica [A] time = 0.102399, size = 85, normalized size = 1.09 \[ -\frac{2 B \sin (c+d x) \left (-(a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-a \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{b^2 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.178, size = 228, normalized size = 2.9 \begin{align*} 2\,{\frac{\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}}{ \left ( a-b \right ){b}^{2}\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d} \left ({\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){a}^{2}-{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) ab+{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) ab-{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{2}-{a}^{2}{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,-2\,{\frac{b}{a-b}},\sqrt{2} \right ) \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 \frac{{\left (B b \cos \left (d x + c\right ) + B a\right )} \cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{{\left (B b \cos \left (d x + c\right ) + B a\right )} \cos \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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