Optimal. Leaf size=181 \[ \frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{65 a^2 d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.183031, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2681, 2683, 2636, 2640, 2639} \[ \frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{65 a^2 d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2681
Rule 2683
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx &=-\frac{2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}+\frac{9 \int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx}{13 a}\\ &=-\frac{2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac{7 \int \frac{1}{(e \cos (c+d x))^{7/2}} \, dx}{13 a^2}\\ &=\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}+\frac{21 \int \frac{1}{(e \cos (c+d x))^{3/2}} \, dx}{65 a^2 e^2}\\ &=\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}-\frac{21 \int \sqrt{e \cos (c+d x)} \, dx}{65 a^2 e^4}\\ &=\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}-\frac{\left (21 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{65 a^2 e^4 \sqrt{\cos (c+d x)}}\\ &=-\frac{42 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{65 a^2 d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac{2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.105818, size = 66, normalized size = 0.36 \[ \frac{(\sin (c+d x)+1)^{5/4} \, _2F_1\left (-\frac{5}{4},\frac{17}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{20 \sqrt [4]{2} a^2 d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.316, size = 670, normalized size = 3.7 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e \cos \left (d x + c\right )}}{a^{2} e^{4} \cos \left (d x + c\right )^{6} - 2 \, a^{2} e^{4} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} e^{4} \cos \left (d x + c\right )^{4}}, 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{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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