Optimal. Leaf size=124 \[ \frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d} \]
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Rubi [A] time = 0.0842286, antiderivative size = 124, 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, 2635, 2642, 2641} \[ \frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx &=-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+a \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} \left (5 a e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{10 a e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 a e^4\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{10 a e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{\left (5 a e^4 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 \sqrt{e \cos (c+d x)}}\\ &=-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d \sqrt{e \cos (c+d x)}}+\frac{10 a e^3 \sqrt{e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.583878, size = 98, normalized size = 0.79 \[ -\frac{a e^3 \sqrt{e \cos (c+d x)} \left (\sqrt{\cos (c+d x)} (-138 \sin (c+d x)-18 \sin (3 (c+d x))+28 \cos (2 (c+d x))+7 \cos (4 (c+d x))+21)-120 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{252 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.454, size = 249, normalized size = 2. \begin{align*} -{\frac{2\,a{e}^{4}}{63\,d} \left ( -224\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}+144\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+560\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-216\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -560\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+168\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +280\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+15\,\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 ) -48\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -70\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+7\,\sin \left ( 1/2\,dx+c/2 \right ) \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 \left (e \cos \left (d x + c\right )\right )^{\frac{7}{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 ({\left (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}\right )} \sqrt{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 \left (e \cos \left (d x + c\right )\right )^{\frac{7}{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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