Optimal. Leaf size=126 \[ -\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2669, 2636, 2640, 2639} \[ \frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rule 2669
Rubi steps
\begin {align*} \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}+a \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx\\ &=\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {(3 a) \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {(3 a) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^4}\\ &=\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (3 a \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^4 \sqrt {\cos (c+d x)}}\\ &=\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 1.27, size = 144, normalized size = 1.14 \[ \frac {2 a e^{i (c+d x)} \left (i \sqrt {1+e^{2 i (c+d x)}} \left (e^{i (c+d x)}-i\right )^2 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-6 e^{i (c+d x)}-3 i e^{2 i (c+d x)}+i\right )}{5 d e^3 \left (e^{i (c+d x)}-i\right )^2 \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}{e^{4} \cos \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.40, size = 304, normalized size = 2.41 \[ -\frac {2 \left (12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d} \]
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 \[ \int \frac {a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {a+a\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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