Optimal. Leaf size=124 \[ -\frac {7 c^4 x}{a^3}-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2759,
2761, 8} \begin {gather*} -\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac {7 c^4 x}{a^3}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a \sin (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2759
Rule 2761
Rule 2815
Rubi steps
\begin {align*} \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx &=\left (a^4 c^4\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^7} \, dx\\ &=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}-\frac {1}{5} \left (7 a^2 c^4\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^5} \, dx\\ &=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}+\frac {1}{3} \left (7 c^4\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}-\frac {\left (7 c^4\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a^2}\\ &=-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}-\frac {\left (7 c^4\right ) \int 1 \, dx}{a^3}\\ &=-\frac {7 c^4 x}{a^3}-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(124)=248\).
time = 0.41, size = 270, normalized size = 2.18 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (96 \sin \left (\frac {1}{2} (e+f x)\right )-48 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-256 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+464 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-105 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-15 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5\right ) (c-c \sin (e+f x))^4}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (a+a \sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 102, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {2 c^{4} \left (-\frac {64}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-7 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,a^{3}}\) | \(102\) |
default | \(\frac {2 c^{4} \left (-\frac {64}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-7 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,a^{3}}\) | \(102\) |
risch | \(-\frac {7 c^{4} x}{a^{3}}-\frac {c^{4} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{3} f}-\frac {c^{4} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{3} f}-\frac {16 \left (120 i c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+45 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}-100 i c^{4} {\mathrm e}^{i \left (f x +e \right )}-170 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+29 c^{4}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(137\) |
norman | \(\frac {-\frac {334 c^{4}}{15 a f}-\frac {7 c^{4} x}{a}-\frac {35 c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}-\frac {35 c^{4} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {7 c^{4} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {66 c^{4} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {16 c^{4} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {286 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a f}-\frac {588 c^{4} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {497 c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {357 c^{4} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {3686 c^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}-\frac {1366 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {860 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {98 c^{4} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {638 c^{4} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {210 c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {1102 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {2404 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {98 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {210 c^{4} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {357 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {497 c^{4} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {588 c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {16036 c^{4} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}-\frac {10814 c^{4} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}-\frac {11524 c^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(574\) |
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. 1192 vs.
\(2 (125) = 250\).
time = 0.58, size = 1192, normalized size = 9.61 \begin {gather*} -\frac {2 \, {\left (3 \, c^{4} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {189 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {160 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {75 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 24}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {11 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {11 \, a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} + \frac {15 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + 4 \, c^{4} {\left (\frac {\frac {95 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {145 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 22}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {15 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac {c^{4} {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {12 \, c^{4} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} - \frac {12 \, c^{4} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, 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. 270 vs.
\(2 (125) = 250\).
time = 0.33, size = 270, normalized size = 2.18 \begin {gather*} -\frac {15 \, c^{4} \cos \left (f x + e\right )^{4} - 420 \, c^{4} f x - 48 \, c^{4} + {\left (105 \, c^{4} f x + 277 \, c^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (315 \, c^{4} f x - 134 \, c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (35 \, c^{4} f x + 74 \, c^{4}\right )} \cos \left (f x + e\right ) + {\left (15 \, c^{4} \cos \left (f x + e\right )^{3} - 420 \, c^{4} f x + 48 \, c^{4} + {\left (105 \, c^{4} f x - 262 \, c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (35 \, c^{4} f x + 66 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} 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. 2314 vs.
\(2 (119) = 238\).
time = 17.26, size = 2314, normalized size = 18.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 135, normalized size = 1.09 \begin {gather*} -\frac {\frac {105 \, {\left (f x + e\right )} c^{4}}{a^{3}} + \frac {30 \, c^{4}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3}} + \frac {16 \, {\left (15 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 60 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 130 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 80 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19 \, c^{4}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.90, size = 290, normalized size = 2.34 \begin {gather*} \frac {7\,c^4\,\left (e+f\,x\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (35\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (525\,e+525\,f\,x+1430\right )}{15}\right )-\frac {c^4\,\left (105\,e+105\,f\,x+334\right )}{15}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (35\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (525\,e+525\,f\,x+240\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (77\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1155\,e+1155\,f\,x+990\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (77\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1155\,e+1155\,f\,x+2684\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (105\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1575\,e+1575\,f\,x+2470\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (105\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1575\,e+1575\,f\,x+2540\right )}{15}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {7\,c^4\,x}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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