Optimal. Leaf size=113 \[ -\frac{2 \left (a^2+2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 0.135237, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2691, 2669, 2640, 2639} \[ -\frac{2 \left (a^2+2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}-\frac{2 \int \sqrt{e \cos (c+d x)} \left (\frac{a^2}{2}+b^2+\frac{3}{2} a b \sin (c+d x)\right ) \, dx}{e^2}\\ &=\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}-\frac{\left (a^2+2 b^2\right ) \int \sqrt{e \cos (c+d x)} \, dx}{e^2}\\ &=\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}-\frac{\left (\left (a^2+2 b^2\right ) \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{e^2 \sqrt{\cos (c+d x)}}\\ &=\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}-\frac{2 \left (a^2+2 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.233122, size = 71, normalized size = 0.63 \[ \frac{2 \left (a^2+b^2\right ) \sin (c+d x)-2 \left (a^2+2 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+4 a b}{d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.283, size = 197, normalized size = 1.7 \begin{align*} -2\,{\frac{\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 ){a}^{2}+2\,\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}^{2}-2\,{a}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-2\,{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-2\,\sin \left ( 1/2\,dx+c/2 \right ) ab}{e\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}\sin \left ( 1/2\,dx+c/2 \right ) d}} \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 \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{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^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt{e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, 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 \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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