Optimal. Leaf size=178 \[ -\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{26 \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt{e \cos (c+d x)}}{35 d e}-\frac{78 \left (a^4 \sin (c+d x)+a^4\right ) \sqrt{e \cos (c+d x)}}{35 d e}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}{7 d e} \]
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Rubi [A] time = 0.21279, 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, 2642, 2641} \[ -\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{26 \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt{e \cos (c+d x)}}{35 d e}-\frac{78 \left (a^4 \sin (c+d x)+a^4\right ) \sqrt{e \cos (c+d x)}}{35 d e}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}{7 d e} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^4}{\sqrt{e \cos (c+d x)}} \, dx &=-\frac{2 a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}+\frac{1}{7} (13 a) \int \frac{(a+a \sin (c+d x))^3}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac{26 \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}+\frac{1}{35} \left (117 a^2\right ) \int \frac{(a+a \sin (c+d x))^2}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac{26 \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac{78 \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac{1}{7} \left (39 a^3\right ) \int \frac{a+a \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{2 a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac{26 \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac{78 \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac{1}{7} \left (39 a^4\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{2 a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac{26 \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac{78 \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac{\left (39 a^4 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{7 \sqrt{e \cos (c+d x)}}\\ &=-\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 a \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac{26 \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac{78 \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}\\ \end{align*}
Mathematica [C] time = 0.0598027, size = 64, normalized size = 0.36 \[ -\frac{32 \sqrt [4]{2} a^4 \sqrt{e \cos (c+d x)} \, _2F_1\left (-\frac{13}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{d e \sqrt [4]{\sin (c+d x)+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.615, size = 222, normalized size = 1.3 \begin{align*} -{\frac{2\,{a}^{4}}{35\,d} \left ( 80\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-120\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +224\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}-280\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -336\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+195\,\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 ) +160\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -392\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+252\,\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 \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\sqrt{e \cos \left (d x + c\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{{\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 )}}{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 \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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