Optimal. Leaf size=97 \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.0716748, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2636, 2642, 2641} \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+a \int \frac{1}{(e \cos (c+d x))^{5/2}} \, dx\\ &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{a \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{\left (a \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 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \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 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.145656, size = 55, normalized size = 0.57 \[ \frac{2 \left (a \sin (c+d x)+a \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+b\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.668, size = 193, normalized size = 2. \begin{align*} -{\frac{2}{3\,d{e}^{2}} \left ( 2\,\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 ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}a \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\,a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +b\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \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{b \sin \left (d x + c\right ) + a}{\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{\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\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{b \sin \left (d x + c\right ) + a}{\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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