Optimal. Leaf size=137 \[ \frac {256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
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Rubi [A]
time = 0.20, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2753,
2752} \begin {gather*} \frac {256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rule 2815
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=(a c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx\\ &=\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac {1}{3} \left (4 a c^2\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac {1}{21} \left (32 a c^3\right ) \int \cos ^2(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac {1}{105} \left (128 a c^4\right ) \int \frac {\cos ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 104, normalized size = 0.76 \begin {gather*} \frac {a c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c-c \sin (e+f x)} (1606-330 \cos (2 (e+f x))-1389 \sin (e+f x)+35 \sin (3 (e+f x)))}{630 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.00, size = 79, normalized size = 0.58
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{2} a \left (35 \left (\sin ^{3}\left (f x +e \right )\right )-165 \left (\sin ^{2}\left (f x +e \right )\right )+321 \sin \left (f x +e \right )-319\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(79\) |
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 = 192, normalized size = 1.40 \begin {gather*} -\frac {2 \, {\left (35 \, a c^{3} \cos \left (f x + e\right )^{5} - 95 \, a c^{3} \cos \left (f x + e\right )^{4} - 226 \, a c^{3} \cos \left (f x + e\right )^{3} + 32 \, a c^{3} \cos \left (f x + e\right )^{2} - 128 \, a c^{3} \cos \left (f x + e\right ) - 256 \, a c^{3} + {\left (35 \, a c^{3} \cos \left (f x + e\right )^{4} + 130 \, a c^{3} \cos \left (f x + e\right )^{3} - 96 \, a c^{3} \cos \left (f x + e\right )^{2} - 128 \, a c^{3} \cos \left (f x + e\right ) - 256 \, a c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{315 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\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 [A]
time = 0.53, size = 144, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {2} {\left (4410 \, a c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 504 \, a c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 225 \, a c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 35 \, a c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{2520 \, 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 \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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