Optimal. Leaf size=116 \[ \frac{2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0910747, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {16, 2748, 2640, 2639, 2635, 2642, 2641} \[ \frac{2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2748
Rule 2640
Rule 2639
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx}{b^3}\\ &=\frac{A \int \sqrt{b \cos (c+d x)} \, dx}{b^3}+\frac{B \int (b \cos (c+d x))^{3/2} \, dx}{b^4}\\ &=\frac{2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 b^3 d}+\frac{B \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{3 b^2}+\frac{\left (A \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{b^3 \sqrt{\cos (c+d x)}}\\ &=\frac{2 A \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 b^3 d}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{2 A \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{b \cos (c+d x)} \sin (c+d x)}{3 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.0766783, size = 78, normalized size = 0.67 \[ \frac{2 \sqrt{\cos (c+d x)} \left (3 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+B \left (F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)}\right )\right )}{3 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.504, size = 240, normalized size = 2.1 \begin{align*}{\frac{2}{3\,{b}^{2}d}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -4\,B\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+3\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -B\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) +2\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/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 \frac{{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\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{{\left (B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )}}{b^{3}}, 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{{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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