Optimal. Leaf size=145 \[ -\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^4 d}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.152142, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2680, 2635, 2642, 2641} \[ -\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^4 d}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2680
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a+a \sin (c+d x))^3}-\frac{\left (3 e^2\right ) \int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^2} \, dx}{a^2}\\ &=-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a+a \sin (c+d x))^3}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (15 e^4\right ) \int (e \cos (c+d x))^{3/2} \, dx}{a^4}\\ &=-\frac{10 e^5 \sqrt{e \cos (c+d x)} \sin (c+d x)}{a^4 d}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a+a \sin (c+d x))^3}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (5 e^6\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{a^4}\\ &=-\frac{10 e^5 \sqrt{e \cos (c+d x)} \sin (c+d x)}{a^4 d}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a+a \sin (c+d x))^3}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (5 e^6 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{a^4 \sqrt{e \cos (c+d x)}}\\ &=-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d \sqrt{e \cos (c+d x)}}-\frac{10 e^5 \sqrt{e \cos (c+d x)} \sin (c+d x)}{a^4 d}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a+a \sin (c+d x))^3}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.171778, size = 66, normalized size = 0.46 \[ -\frac{\sqrt [4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left (\frac{7}{4},\frac{13}{4};\frac{17}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{13 a^4 d e (\sin (c+d x)+1)^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.204, size = 263, normalized size = 1.8 \begin{align*}{\frac{2\,{e}^{6}}{3\,{a}^{4}d} \left ( -8\, \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}+8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +48\, \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 ) -18\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -48\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+20\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( 2\, \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{\left (e \cos \left (d x + c\right )\right )^{\frac{11}{2}}}{{\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 (\frac{\sqrt{e \cos \left (d x + c\right )} e^{5} \cos \left (d x + c\right )^{5}}{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 )}, 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(-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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