Optimal. Leaf size=109 \[ \frac{2 \left (3 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e} \]
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Rubi [A] time = 0.128876, 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, 2642, 2641} \[ \frac{2 \left (3 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^2}{\sqrt{e \cos (c+d x)}} \, dx &=-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}+\frac{2}{3} \int \frac{\frac{3 a^2}{2}+b^2+\frac{5}{2} a b \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}+\frac{1}{3} \left (3 a^2+2 b^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}+\frac{\left (\left (3 a^2+2 b^2\right ) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 \sqrt{e \cos (c+d x)}}\\ &=-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}+\frac{2 \left (3 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}\\ \end{align*}
Mathematica [A] time = 0.392046, size = 75, normalized size = 0.69 \[ \frac{2 \left (3 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-2 b \cos (c+d x) (6 a+b \sin (c+d x))}{3 d \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.777, size = 210, normalized size = 1.9 \begin{align*} -{\frac{2}{3\,d} \left ( -4\,{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+3\,\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 EllipticF} \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 EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{2}-12\,ab \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+2\,{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+6\,\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 \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\sqrt{e \cos \left (d x + c\right )}}\,{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 \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 \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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