Optimal. Leaf size=134 \[ -\frac {a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f} \]
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Rubi [A]
time = 0.18, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817}
\begin {gather*} -\frac {a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 2819
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx &=-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}+\frac {1}{3} (2 a) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}+\frac {1}{15} \left (4 a^2\right ) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx\\ &=-\frac {a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 156, normalized size = 1.16 \begin {gather*} -\frac {c^3 (-1+\sin (e+f x))^3 (a (1+\sin (e+f x)))^{5/2} \sqrt {c-c \sin (e+f x)} (75 \cos (2 (e+f x))+30 \cos (4 (e+f x))+5 \cos (6 (e+f x))+600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x)))}{960 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 16.49, size = 116, normalized size = 0.87
method | result | size |
default | \(\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \left (5 \left (\cos ^{6}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+6 \left (\cos ^{4}\left (f x +e \right )\right )+3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+8 \left (\cos ^{2}\left (f x +e \right )\right )+11 \sin \left (f x +e \right )+11\right )}{30 f \cos \left (f x +e \right )^{7}}\) | \(116\) |
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 [A]
time = 0.36, size = 119, normalized size = 0.89 \begin {gather*} \frac {{\left (5 \, a^{2} c^{3} \cos \left (f x + e\right )^{6} - 5 \, a^{2} c^{3} + 2 \, {\left (3 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} + 4 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} + 8 \, a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, 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(-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 [A]
time = 0.51, size = 216, normalized size = 1.61 \begin {gather*} \frac {16 \, {\left (10 \, a^{2} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 36 \, a^{2} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 45 \, a^{2} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 20 \, a^{2} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.94, size = 124, normalized size = 0.93 \begin {gather*} \frac {a^2\,c^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (75\,\cos \left (e+f\,x\right )+105\,\cos \left (3\,e+3\,f\,x\right )+35\,\cos \left (5\,e+5\,f\,x\right )+5\,\cos \left (7\,e+7\,f\,x\right )+700\,\sin \left (2\,e+2\,f\,x\right )+112\,\sin \left (4\,e+4\,f\,x\right )+12\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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