Optimal. Leaf size=181 \[ -\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{65 a^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {2}{13 d e \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2636
Rule 2639
Rule 2640
Rule 2681
Rule 2683
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*}
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Mathematica [C] time = 0.11, 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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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )}}{a^{2} e^{4} \cos \left (d x + c\right )^{6} - 2 \, a^{2} e^{4} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} e^{4} \cos \left (d x + c\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.59, size = 670, normalized size = 3.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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