Optimal. Leaf size=145 \[ \frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{3 d e^6 \sqrt{\cos (c+d x)}}+\frac{4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}} \]
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Rubi [A] time = 0.112072, 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 = {2676, 2636, 2640, 2639} \[ \frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{3 d e^6 \sqrt{\cos (c+d x)}}+\frac{4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2676
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx &=\frac{4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}+\frac{\left (5 a^2\right ) \int \frac{1}{(e \cos (c+d x))^{7/2}} \, dx}{9 e^2}\\ &=\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac{4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}+\frac{a^2 \int \frac{1}{(e \cos (c+d x))^{3/2}} \, dx}{3 e^4}\\ &=\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}-\frac{a^2 \int \sqrt{e \cos (c+d x)} \, dx}{3 e^6}\\ &=\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}-\frac{\left (a^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{3 e^6 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 a^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^6 \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.160718, size = 66, normalized size = 0.46 \[ \frac{2^{3/4} a^2 (\sin (c+d x)+1)^{9/4} \, _2F_1\left (-\frac{9}{4},\frac{5}{4};-\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{9 d e (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.622, size = 488, normalized size = 3.4 \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{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac{11}{2}}}\,{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^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \sqrt{e \cos \left (d x + c\right )}}{e^{6} \cos \left (d x + c\right )^{6}}, 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 )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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