Optimal. Leaf size=126 \[ \frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.0863084, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2636, 2640, 2639} \[ \frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx &=\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+a \int \frac{1}{(e \cos (c+d x))^{7/2}} \, dx\\ &=\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{(3 a) \int \frac{1}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{(3 a) \int \sqrt{e \cos (c+d x)} \, dx}{5 e^4}\\ &=\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{\left (3 a \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 e^4 \sqrt{\cos (c+d x)}}\\ &=\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}-\frac{6 a \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.22272, size = 144, normalized size = 1.14 \[ \frac{2 a e^{i (c+d x)} \left (i \sqrt{1+e^{2 i (c+d x)}} \left (e^{i (c+d x)}-i\right )^2 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-6 e^{i (c+d x)}-3 i e^{2 i (c+d x)}+i\right )}{5 d e^3 \left (e^{i (c+d x)}-i\right )^2 \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.962, size = 304, normalized size = 2.4 \begin{align*} -{\frac{2\,a}{5\,d{e}^{3}} \left ( 12\,{\it EllipticE} \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}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -12\,{\it EllipticE} \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}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +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 EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \left ( 4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\, \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{a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{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 (a \sin \left (d x + c\right ) + a\right )}}{e^{4} \cos \left (d x + c\right )^{4}}, 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{a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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