Optimal. Leaf size=225 \[ -\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}-\frac {18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.30, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}-\frac {18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}} \]
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))^{3/2} (a+a \sin (c+d x))^4} \, dx &=-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}+\frac {9 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx}{17 a}\\ &=-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}+\frac {63 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx}{221 a^2}\\ &=-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {35 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx}{221 a^3}\\ &=-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}+\frac {21 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{221 a^4}\\ &=\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac {21 \int \sqrt {e \cos (c+d x)} \, dx}{221 a^4 e^2}\\ &=\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (21 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{221 a^4 e^2 \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 )}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 66, normalized size = 0.29 \[ \frac {\sqrt [4]{\sin (c+d x)+1} \, _2F_1\left (-\frac {1}{4},\frac {21}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{8 \sqrt [4]{2} a^4 d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a^{4} e^{2} \cos \left (d x + c\right )^{6} - 8 \, a^{4} e^{2} \cos \left (d x + c\right )^{4} + 8 \, a^{4} e^{2} \cos \left (d x + c\right )^{2} - 4 \, {\left (a^{4} e^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{4} e^{2} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.76, size = 878, normalized size = 3.90 \[ \text {result too large to display} \]
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 = -0.00 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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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