Optimal. Leaf size=112 \[ \frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.0943606, antiderivative size = 112, 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 = {2683, 2636, 2642, 2641} \[ \frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2683
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \, dx &=-\frac{2}{7 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))}+\frac{5 \int \frac{1}{(e \cos (c+d x))^{5/2}} \, dx}{7 a}\\ &=\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))}+\frac{5 \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{21 a e^2}\\ &=\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))}+\frac{\left (5 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a e^2 \sqrt{e \cos (c+d x)}}\\ &=\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 a d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.0609495, size = 66, normalized size = 0.59 \[ \frac{(\sin (c+d x)+1)^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{11}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3\ 2^{3/4} a d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.511, size = 375, normalized size = 3.4 \begin{align*} -{\frac{2}{21\,{e}^{2}ad} \left ( 40\,\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} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-60\,\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} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+40\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +30\,\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 ) \sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-40\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -5\,\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 ) +16\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -3\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( 8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-12\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+6\, \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{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}{\left (a \sin \left (d x + c\right ) + a\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{\sqrt{e \cos \left (d x + c\right )}}{a e^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a 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{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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