Optimal. Leaf size=119 \[ \frac{2 \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.139395, antiderivative size = 119, 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, 2642, 2641} \[ \frac{2 \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}-\frac{2 \int \frac{-\frac{a^2}{2}+b^2+\frac{1}{2} a b \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}+\frac{\left (a^2-2 b^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}+\frac{\left (\left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 e^2 \sqrt{e \cos (c+d x)}}\\ &=\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 \left (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 e^2 \sqrt{e \cos (c+d x)}}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.255964, size = 72, normalized size = 0.61 \[ \frac{2 \left (\left (a^2+b^2\right ) \sin (c+d x)+\left (a^2-2 b^2\right ) \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 a b\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.688, size = 333, normalized size = 2.8 \begin{align*} -{\frac{2}{3\,d{e}^{2}} \left ( 2\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){a}^{2} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-4\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){b}^{2} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-\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 ){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}+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 \right ) \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \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}}{\left (e \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^{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^{3} \cos \left (d x + c\right )^{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 \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \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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