Optimal. Leaf size=178 \[ -\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{10 \left (a^2 \sin (c+d x)+a^2\right )^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac{22 \left (a^4 \sin (c+d x)+a^4\right ) (e \cos (c+d x))^{3/2}}{21 d e}+\frac{22 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e} \]
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Rubi [A] time = 0.200701, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2678, 2669, 2640, 2639} \[ -\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{10 \left (a^2 \sin (c+d x)+a^2\right )^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac{22 \left (a^4 \sin (c+d x)+a^4\right ) (e \cos (c+d x))^{3/2}}{21 d e}+\frac{22 a^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx &=-\frac{2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}+\frac{1}{3} (5 a) \int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac{10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}+\frac{1}{21} \left (55 a^2\right ) \int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac{10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac{22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac{1}{3} \left (11 a^3\right ) \int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x)) \, dx\\ &=-\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac{10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac{22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac{1}{3} \left (11 a^4\right ) \int \sqrt{e \cos (c+d x)} \, dx\\ &=-\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac{10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac{22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac{\left (11 a^4 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{3 \sqrt{\cos (c+d x)}}\\ &=-\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}+\frac{22 a^4 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac{10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac{22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}\\ \end{align*}
Mathematica [C] time = 0.0668708, size = 66, normalized size = 0.37 \[ -\frac{32\ 2^{3/4} a^4 (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac{15}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e (\sin (c+d x)+1)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.519, size = 258, normalized size = 1.5 \begin{align*}{\frac{2\,{a}^{4}e}{63\,d} \left ( 224\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}\cos \left ( 1/2\,dx+c/2 \right ) -448\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+576\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-392\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -1152\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+616\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +192\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+231\,\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 ) -168\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +384\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-132\,\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 \sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{4}\,{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^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\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 \sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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