Optimal. Leaf size=132 \[ -\frac {256 c^3 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{5 a f}+\frac {64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}+\frac {8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f} \]
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Rubi [A]
time = 0.25, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753,
2752} \begin {gather*} -\frac {256 c^3 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{5 a f}+\frac {64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac {8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rule 2815
Rubi steps
\begin {align*} \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx &=\frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{a c}\\ &=\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac {12 \int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{5 a}\\ &=\frac {8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac {(32 c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{5 a}\\ &=\frac {64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}+\frac {8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac {\left (128 c^2\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{5 a}\\ &=-\frac {256 c^3 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{5 a f}+\frac {64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}+\frac {8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 112, normalized size = 0.85 \begin {gather*} \frac {c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (-350-14 \cos (2 (e+f x))-175 \sin (e+f x)+\sin (3 (e+f x)))}{10 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.62, size = 69, normalized size = 0.52
method | result | size |
default | \(\frac {2 c^{4} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{3}\left (f x +e \right )-7 \left (\sin ^{2}\left (f x +e \right )\right )+43 \sin \left (f x +e \right )+91\right )}{5 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 258 vs.
\(2 (124) = 248\).
time = 0.60, size = 258, normalized size = 1.95 \begin {gather*} \frac {2 \, {\left (91 \, c^{\frac {7}{2}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {490 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {91 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{5 \, {\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 79, normalized size = 0.60 \begin {gather*} -\frac {2 \, {\left (7 \, c^{3} \cos \left (f x + e\right )^{2} + 84 \, c^{3} - {\left (c^{3} \cos \left (f x + e\right )^{2} - 44 \, c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{5 \, a f \cos \left (f x + e\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs.
\(2 (124) = 248\).
time = 0.57, size = 343, normalized size = 2.60 \begin {gather*} \frac {16 \, \sqrt {2} {\left (\frac {5 \, c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}} - \frac {11 \, c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {50 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {80 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {30 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {5 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}}{a {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{5}}\right )} \sqrt {c}}{5 \, f} \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 {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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