Optimal. Leaf size=34 \[ \frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2815, 2750}
\begin {gather*} \frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2815
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).
time = 0.52, size = 93, normalized size = 2.74 \begin {gather*} \frac {a^3 \left (35 \cos \left (\frac {1}{2} (e+f x)\right )-21 \cos \left (\frac {3}{2} (e+f x)\right )-7 \cos \left (\frac {5}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{28 c^4 f (-1+\sin (e+f x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs.
\(2(32)=64\).
time = 0.42, size = 118, normalized size = 3.47
method | result | size |
risch | \(-\frac {2 \left (7 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-35 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+21 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-a^{3}\right )}{7 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\) | \(71\) |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\right )}{f \,c^{4}}\) | \(118\) |
default | \(\frac {2 a^{3} \left (-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\right )}{f \,c^{4}}\) | \(118\) |
norman | \(\frac {-\frac {2 a^{3}}{7 c f}-\frac {48 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {202 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {352 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {42 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {16 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(177\) |
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. 1137 vs.
\(2 (35) = 70\).
time = 0.35, size = 1137, normalized size = 33.44 \begin {gather*} \frac {2 \, {\left (\frac {a^{3} {\left (\frac {91 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {168 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {175 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - 13\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} - \frac {a^{3} {\left (\frac {49 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {147 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {210 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 12\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} - \frac {4 \, a^{3} {\left (\frac {14 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {42 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 2\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} + \frac {2 \, a^{3} {\left (\frac {7 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - 1\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}}\right )}}{35 \, f} \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. 238 vs.
\(2 (35) = 70\).
time = 0.32, size = 238, normalized size = 7.00 \begin {gather*} \frac {a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} - {\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )}{7 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \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. 619 vs.
\(2 (29) = 58\).
time = 19.96, size = 619, normalized size = 18.21 \begin {gather*} \begin {cases} - \frac {14 a^{3} \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {70 a^{3} \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {42 a^{3} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {2 a^{3}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )^{3}}{\left (- c \sin {\left (e \right )} + c\right )^{4}} & \text {otherwise} \end {cases} \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. 77 vs.
\(2 (35) = 70\).
time = 0.48, size = 77, normalized size = 2.26 \begin {gather*} -\frac {2 \, {\left (7 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 21 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}\right )}}{7 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.01, size = 116, normalized size = 3.41 \begin {gather*} \frac {2\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+21\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{7\,c^4\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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