Optimal. Leaf size=191 \[ -\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4 \sin (c+d x)+a^4\right )}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{39 a^4 d \sqrt{\cos (c+d x)}}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4} \]
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Rubi [A] time = 0.228498, antiderivative size = 191, 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 = {2681, 2683, 2640, 2639} \[ -\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4 \sin (c+d x)+a^4\right )}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{39 a^4 d \sqrt{\cos (c+d x)}}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2681
Rule 2683
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx &=-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}+\frac{5 \int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx}{13 a}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}+\frac{5 \int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx}{39 a^2}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{\int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{39 a^3}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\int \sqrt{e \cos (c+d x)} \, dx}{39 a^4}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\sqrt{e \cos (c+d x)} \int \sqrt{\cos (c+d x)} \, dx}{39 a^4 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.0461394, size = 66, normalized size = 0.35 \[ -\frac{(e \cos (c+d x))^{3/2} \, _2F_1\left (\frac{3}{4},\frac{17}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{12 \sqrt [4]{2} a^4 d e (\sin (c+d x)+1)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.932, size = 694, normalized size = 3.6 \begin{align*} \text{result too large to display} \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{\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 (\frac{\sqrt{e \cos \left (d x + c\right )}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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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