Optimal. Leaf size=143 \[ -\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2683, 2636, 2640, 2639} \[ \frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rule 2683
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx &=-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx}{9 a}\\ &=\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{15 a e^2}\\ &=\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}-\frac {7 \int \sqrt {e \cos (c+d x)} \, dx}{15 a e^4}\\ &=\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}-\frac {\left (7 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 a e^4 \sqrt {\cos (c+d x)}}\\ &=-\frac {14 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 66, normalized size = 0.46 \[ \frac {(\sin (c+d x)+1)^{5/4} \, _2F_1\left (-\frac {5}{4},\frac {13}{4};-\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{10 \sqrt [4]{2} a d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a e^{4} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + a e^{4} \cos \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.72, size = 488, normalized size = 3.41 \[ -\frac {2 \left (336 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-672 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-672 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1344 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1064 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-168 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+392 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+21 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-66 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d} \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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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