Optimal. Leaf size=73 \[ \frac {8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \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))^3 (c-c \sin (e+f x))^{3/2} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}+\frac {1}{9} \left (4 a^3 c^4\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac {8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.87, size = 84, normalized size = 1.15 \begin {gather*} -\frac {2 a^3 c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (-11+7 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{63 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 = 2.54, size = 61, normalized size = 0.84
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (7 \sin \left (f x +e \right )-11\right )}{63 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(61\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs.
\(2 (69) = 138\).
time = 0.36, size = 192, normalized size = 2.63 \begin {gather*} -\frac {2 \, {\left (7 \, a^{3} c \cos \left (f x + e\right )^{5} + 17 \, a^{3} c \cos \left (f x + e\right )^{4} - 2 \, a^{3} c \cos \left (f x + e\right )^{3} + 4 \, a^{3} c \cos \left (f x + e\right )^{2} - 16 \, a^{3} c \cos \left (f x + e\right ) - 32 \, a^{3} c + {\left (7 \, a^{3} c \cos \left (f x + e\right )^{4} - 10 \, a^{3} c \cos \left (f x + e\right )^{3} - 12 \, a^{3} c \cos \left (f x + e\right )^{2} - 16 \, a^{3} c \cos \left (f x + e\right ) - 32 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{63 \, {\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int c \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int 2 c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\right )\, dx\right ) \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. 144 vs.
\(2 (69) = 138\).
time = 0.58, size = 144, normalized size = 1.97 \begin {gather*} -\frac {\sqrt {2} {\left (378 \, a^{3} c \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 ) + 168 \, a^{3} c \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 27 \, a^{3} c \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 ) - 7 \, a^{3} c \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}}{504 \, 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 )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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