Optimal. Leaf size=191 \[ -\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4 \sin (c+d x)+a^4\right )}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4} \]
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Rubi [A] time = 0.23, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2681, 2683, 2640, 2639} \[ -\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4 \sin (c+d x)+a^4\right )}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2681
Rule 2683
Rubi steps
\begin {align*} \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}+\frac {5 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx}{13 a}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}+\frac {5 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx}{39 a^2}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {\int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{39 a^3}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\int \sqrt {e \cos (c+d x)} \, dx}{39 a^4}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{39 a^4 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 66, normalized size = 0.35 \[ -\frac {(e \cos (c+d x))^{3/2} \, _2F_1\left (\frac {3}{4},\frac {17}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{12 \sqrt [4]{2} a^4 d e (\sin (c+d x)+1)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.34, size = 694, normalized size = 3.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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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