Optimal. Leaf size=35 \[ -\frac {a^3 B c^3 \cos ^7(e+f x) (a-a \sin (e+f x))^{-3+m}}{f} \]
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Rubi [A]
time = 0.15, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3046, 2933}
\begin {gather*} -\frac {a^3 B c^3 \cos ^7(e+f x) (a-a \sin (e+f x))^{m-3}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2933
Rule 3046
Rubi steps
\begin {align*} \int (a-a \sin (e+f x))^m (c+c \sin (e+f x))^3 (B (-3+m)+B (4+m) \sin (e+f x)) \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (a-a \sin (e+f x))^{-3+m} (B (-3+m)+B (4+m) \sin (e+f x)) \, dx\\ &=-\frac {a^3 B c^3 \cos ^7(e+f x) (a-a \sin (e+f x))^{-3+m}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 61, normalized size = 1.74 \begin {gather*} \frac {B c^3 (a-a \sin (e+f x))^m (-14 \cos (e+f x)+6 \cos (3 (e+f x))-14 \sin (2 (e+f x))+\sin (4 (e+f x)))}{8 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.09, size = 0, normalized size = 0.00 \[\int \left (a -a \sin \left (f x +e \right )\right )^{m} \left (c +c \sin \left (f x +e \right )\right )^{3} \left (B \left (-3+m \right )+B \left (4+m \right ) \sin \left (f x +e \right )\right )\, dx\]
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. 83 vs.
\(2 (37) = 74\).
time = 0.37, size = 83, normalized size = 2.37 \begin {gather*} \frac {{\left (3 \, B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right ) + {\left (B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} {\left (-a \sin \left (f x + e\right ) + a\right )}^{m}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 898 vs.
\(2 (32) = 64\).
time = 81.10, size = 898, normalized size = 25.66 \begin {gather*} \begin {cases} \frac {B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} + \frac {6 B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} + \frac {14 B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} + \frac {14 B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {14 B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {14 B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {6 B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} - \frac {B c^{3} \left (a - \frac {2 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1}\right )^{m}}{f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 4 f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + f} & \text {for}\: f \neq 0 \\x \left (B \left (m - 3\right ) + B \left (m + 4\right ) \sin {\left (e \right )}\right ) \left (- a \sin {\left (e \right )} + a\right )^{m} \left (c \sin {\left (e \right )} + c\right )^{3} & \text {otherwise} \end {cases} \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 [B]
time = 14.36, size = 64, normalized size = 1.83 \begin {gather*} -\frac {B\,c^3\,{\left (-a\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,\left (14\,\cos \left (e+f\,x\right )-6\,\cos \left (3\,e+3\,f\,x\right )+14\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )}{8\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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