Optimal. Leaf size=169 \[ -\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left (a^4-a^4 \sin (c+d x)\right )}-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2}+\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{77 d e^6 \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 0.259435, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2670, 2680, 2681, 2683, 2642, 2641} \[ -\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left (a^4-a^4 \sin (c+d x)\right )}-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2}+\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{77 d e^6 \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2670
Rule 2680
Rule 2681
Rule 2683
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx &=\frac{a^8 \int \frac{(e \cos (c+d x))^{3/2}}{(a-a \sin (c+d x))^4} \, dx}{e^8}\\ &=\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{a^6 \int \frac{1}{\sqrt{e \cos (c+d x)} (a-a \sin (c+d x))^2} \, dx}{11 e^6}\\ &=\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac{\left (3 a^5\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} (a-a \sin (c+d x))} \, dx}{77 e^6}\\ &=\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac{2 a^5 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}-\frac{a^4 \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{77 e^6}\\ &=\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac{2 a^5 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}-\frac{\left (a^4 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{77 e^6 \sqrt{e \cos (c+d x)}}\\ &=-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{77 d e^6 \sqrt{e \cos (c+d x)}}+\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac{2 a^5 \sqrt{e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.211174, size = 66, normalized size = 0.39 \[ \frac{4 \sqrt [4]{2} a^4 (\sin (c+d x)+1)^{11/4} \, _2F_1\left (-\frac{11}{4},-\frac{1}{4};-\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{11 d e (e \cos (c+d x))^{11/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.272, size = 583, normalized size = 3.5 \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 )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac{13}{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^{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^{7} \cos \left (d x + c\right )^{7}}, 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}}{\left (e \cos \left (d x + c\right )\right )^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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