Optimal. Leaf size=90 \[ \frac {5 c^3 x}{a^2}+\frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^4}+\frac {5 c^3 x}{a^2}+\frac {10 c^3 \cos ^3(e+f x)}{3 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))^3}{(a+a \sin (e+f x))^2} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^5} \, dx\\ &=-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}-\frac {1}{3} \left (5 a c^3\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2}+\frac {\left (5 c^3\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2}+\frac {\left (5 c^3\right ) \int 1 \, dx}{a^2}\\ &=\frac {5 c^3 x}{a^2}+\frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 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. \(210\) vs. \(2(90)=180\).
time = 0.25, size = 210, normalized size = 2.33 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (16 \sin \left (\frac {1}{2} (e+f x)\right )-8 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-56 \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+15 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+3 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3\right ) (c-c \sin (e+f x))^3}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (a+a \sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 85, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {2 c^{3} \left (\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+5 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(85\) |
default | \(\frac {2 c^{3} \left (\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+5 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(85\) |
risch | \(\frac {5 c^{3} x}{a^{2}}+\frac {c^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{2} f}+\frac {c^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{2} f}+\frac {32 i c^{3} {\mathrm e}^{i \left (f x +e \right )}+24 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {56 c^{3}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(108\) |
norman | \(\frac {\frac {8 c^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {38 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {5 c^{3} x}{a}+\frac {46 c^{3}}{3 a f}+\frac {15 c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {30 c^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {50 c^{3} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {60 c^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {60 c^{3} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {50 c^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {30 c^{3} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {15 c^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {5 c^{3} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {110 c^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {78 c^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {130 c^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {34 c^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {58 c^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {106 c^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(406\) |
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. 640 vs.
\(2 (91) = 182\).
time = 0.55, size = 640, normalized size = 7.11 \begin {gather*} \frac {2 \, {\left (2 \, c^{3} {\left (\frac {\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {11 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 5}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + 3 \, c^{3} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} - \frac {c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, 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. 195 vs.
\(2 (91) = 182\).
time = 0.32, size = 195, normalized size = 2.17 \begin {gather*} \frac {3 \, c^{3} \cos \left (f x + e\right )^{3} - 30 \, c^{3} f x + 8 \, c^{3} + {\left (15 \, c^{3} f x - 31 \, c^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (15 \, c^{3} f x + 26 \, c^{3}\right )} \cos \left (f x + e\right ) - {\left (30 \, c^{3} f x + 3 \, c^{3} \cos \left (f x + e\right )^{2} + 8 \, c^{3} + {\left (15 \, c^{3} f x + 34 \, c^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} 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. 1282 vs.
\(2 (87) = 174\).
time = 4.94, size = 1282, normalized size = 14.24 \begin {gather*} \begin {cases} \frac {15 c^{3} f x \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {45 c^{3} f x \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {60 c^{3} f x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {60 c^{3} f x \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {45 c^{3} f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {15 c^{3} f x}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {24 c^{3} \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {102 c^{3} \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {82 c^{3} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {114 c^{3} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} + \frac {46 c^{3}}{3 a^{2} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 12 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (- c \sin {\left (e \right )} + c\right )^{3}}{\left (a \sin {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 101, normalized size = 1.12 \begin {gather*} \frac {\frac {15 \, {\left (f x + e\right )} c^{3}}{a^{2}} + \frac {6 \, c^{3}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{2}} + \frac {8 \, {\left (3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, c^{3}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.62, size = 217, normalized size = 2.41 \begin {gather*} \frac {5\,c^3\,x}{a^2}-\frac {5\,c^3\,\left (e+f\,x\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (15\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (45\,e+45\,f\,x+114\right )}{3}\right )-\frac {c^3\,\left (15\,e+15\,f\,x+46\right )}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (15\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (45\,e+45\,f\,x+24\right )}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (20\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (60\,e+60\,f\,x+82\right )}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (20\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (60\,e+60\,f\,x+102\right )}{3}\right )}{a^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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