Optimal. Leaf size=135 \[ -\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 a d e^2 \sqrt {e \sin (c+d x)}} \]
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Rubi [A]
time = 0.18, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3957, 2918,
2644, 30, 2647, 2716, 2721, 2720} \begin {gather*} \frac {4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 a d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rule 2647
Rule 2716
Rule 2720
Rule 2721
Rule 2918
Rule 3957
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx &=-\int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx\\ &=\frac {e^2 \int \frac {\cos (c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a}-\frac {e^2 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{9/2}} \, dx}{a}\\ &=\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 \int \frac {1}{(e \sin (c+d x))^{5/2}} \, dx}{7 a}+\frac {e \text {Subst}\left (\int \frac {1}{x^{9/2}} \, dx,x,e \sin (c+d x)\right )}{a d}\\ &=-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 a e^2}\\ &=-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {\left (2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac {2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac {4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}+\frac {4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 a d e^2 \sqrt {e \sin (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.85, size = 91, normalized size = 0.67 \begin {gather*} -\frac {2 \left (4+2 \cos (c+d x)+\cos (2 (c+d x))+\csc ^2\left (\frac {1}{2} (c+d x)\right ) F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sin ^{\frac {7}{2}}(c+d x)\right )}{21 a d e (1+\cos (c+d x)) (e \sin (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 136, normalized size = 1.01
method | result | size |
default | \(\frac {-\frac {2 e}{7 a \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {2 \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {9}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{5}\left (d x +c \right )\right )+5 \left (\sin ^{3}\left (d x +c \right )\right )-3 \sin \left (d x +c \right )\right )}{21 e^{2} a \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.79, size = 205, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (\sqrt {-i} {\left (\sqrt {2} \cos \left (d x + c\right )^{3} + \sqrt {2} \cos \left (d x + c\right )^{2} - \sqrt {2} \cos \left (d x + c\right ) - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {i} {\left (\sqrt {2} \cos \left (d x + c\right )^{3} + \sqrt {2} \cos \left (d x + c\right )^{2} - \sqrt {2} \cos \left (d x + c\right ) - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (2 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 3\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{21 \, {\left (a d \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} + a d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} - a d \cos \left (d x + c\right ) e^{\frac {5}{2}} - a d e^{\frac {5}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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