Optimal. Leaf size=139 \[ -\frac {4 e^4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 a d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d} \]
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Rubi [A]
time = 0.19, antiderivative size = 139, 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, 2648, 2649, 2721, 2720} \begin {gather*} -\frac {4 e^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 \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rule 2648
Rule 2649
Rule 2720
Rule 2721
Rule 2918
Rule 3957
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{-a-a \cos (c+d x)} \, dx\\ &=\frac {e^2 \int \cos (c+d x) (e \sin (c+d x))^{3/2} \, dx}{a}-\frac {e^2 \int \cos ^2(c+d x) (e \sin (c+d x))^{3/2} \, dx}{a}\\ &=\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {e \text {Subst}\left (\int x^{3/2} \, dx,x,e \sin (c+d x)\right )}{a d}-\frac {e^4 \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{7 a}\\ &=-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (2 e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 a}\\ &=-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (2 e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a \sqrt {e \sin (c+d x)}}\\ &=-\frac {4 e^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 \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a d}+\frac {2 e (e \sin (c+d x))^{5/2}}{5 a d}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 122, normalized size = 0.88 \begin {gather*} \frac {e^3 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (40 F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+(42+25 \cos (c+d x)-42 \cos (2 (c+d x))+15 \cos (3 (c+d x))) \sqrt {\sin (c+d x)}\right ) \sqrt {e \sin (c+d x)}}{105 a d (1+\sec (c+d x)) \sqrt {\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 128, normalized size = 0.92
method | result | size |
default | \(\frac {\frac {2 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 a}+\frac {2 e^{4} \left (3 \left (\sin ^{5}\left (d x +c \right )\right )+\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-5 \left (\sin ^{3}\left (d x +c \right )\right )+2 \sin \left (d x +c \right )\right )}{21 a \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(128\) |
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 \begin {gather*} \text {Failed to integrate} \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 = 0.95, size = 114, normalized size = 0.82 \begin {gather*} -\frac {2 \, {\left (5 \, \sqrt {2} \sqrt {-i} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} \sqrt {i} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (15 \, \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} - 21 \, \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 5 \, \cos \left (d x + c\right ) e^{\frac {7}{2}} + 21 \, e^{\frac {7}{2}}\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{105 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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 )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{a\,\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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