Optimal. Leaf size=109 \[ \frac{2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e} \]
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Rubi [A] time = 0.129601, antiderivative size = 109, 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 = {2692, 2669, 2640, 2639} \[ \frac{2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2 \, dx &=-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}+\frac{2}{5} \int \sqrt{e \cos (c+d x)} \left (\frac{5 a^2}{2}+b^2+\frac{7}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}+\frac{1}{5} \left (5 a^2+2 b^2\right ) \int \sqrt{e \cos (c+d x)} \, dx\\ &=-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}+\frac{\left (\left (5 a^2+2 b^2\right ) \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}+\frac{2 \left (5 a^2+2 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\\ \end{align*}
Mathematica [A] time = 0.312287, size = 80, normalized size = 0.73 \[ \frac{\sqrt{e \cos (c+d x)} \left (6 \left (5 a^2+2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-2 b \cos ^{\frac{3}{2}}(c+d x) (10 a+3 b \sin (c+d x))\right )}{15 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.132, size = 251, normalized size = 2.3 \begin{align*}{\frac{2\,e}{15\,d} \left ( -24\,{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-40\,ab \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+24\,{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+15\,\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}+6\,\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}+40\,ab \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-6\,{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-10\,\sin \left ( 1/2\,dx+c/2 \right ) ab \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \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 \sqrt{e \cos \left (d x + c\right )}{\left (b \sin \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\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 )}, 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 \sqrt{e \cos \left (d x + c\right )}{\left (b \sin \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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