Optimal. Leaf size=112 \[ \frac {5}{16} a^3 c^4 x+\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f} \]
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Rubi [A]
time = 0.08, antiderivative size = 112, 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, 2748,
2715, 8} \begin {gather*} \frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {5}{16} a^3 c^4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2748
Rule 2815
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\left (a^3 c^4\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{6} \left (5 a^3 c^4\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{8} \left (5 a^3 c^4\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{16} \left (5 a^3 c^4\right ) \int 1 \, dx\\ &=\frac {5}{16} a^3 c^4 x+\frac {a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 c^4 \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 89, normalized size = 0.79 \begin {gather*} \frac {a^3 c^4 (420 e+420 f x+105 \cos (e+f x)+63 \cos (3 (e+f x))+21 \cos (5 (e+f x))+3 \cos (7 (e+f x))+315 \sin (2 (e+f x))+63 \sin (4 (e+f x))+7 \sin (6 (e+f x)))}{1344 f} \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. \(254\) vs.
\(2(102)=204\).
time = 0.47, size = 255, normalized size = 2.28
method | result | size |
risch | \(\frac {5 a^{3} c^{4} x}{16}+\frac {5 c^{4} a^{3} \cos \left (f x +e \right )}{64 f}+\frac {c^{4} a^{3} \cos \left (7 f x +7 e \right )}{448 f}+\frac {c^{4} a^{3} \sin \left (6 f x +6 e \right )}{192 f}+\frac {c^{4} a^{3} \cos \left (5 f x +5 e \right )}{64 f}+\frac {3 c^{4} a^{3} \sin \left (4 f x +4 e \right )}{64 f}+\frac {3 c^{4} a^{3} \cos \left (3 f x +3 e \right )}{64 f}+\frac {15 c^{4} a^{3} \sin \left (2 f x +2 e \right )}{64 f}\) | \(148\) |
derivativedivides | \(\frac {-\frac {c^{4} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-c^{4} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {3 c^{4} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 c^{4} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-c^{4} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 c^{4} a^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+c^{4} a^{3} \cos \left (f x +e \right )+c^{4} a^{3} \left (f x +e \right )}{f}\) | \(255\) |
default | \(\frac {-\frac {c^{4} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-c^{4} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {3 c^{4} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 c^{4} a^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-c^{4} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 c^{4} a^{3} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+c^{4} a^{3} \cos \left (f x +e \right )+c^{4} a^{3} \left (f x +e \right )}{f}\) | \(255\) |
norman | \(\frac {\frac {2 c^{4} a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 c^{4} a^{3}}{7 f}+\frac {5 a^{3} c^{4} x}{16}+\frac {6 c^{4} a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {10 c^{4} a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {35 a^{3} c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {105 a^{3} c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {175 a^{3} c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {175 a^{3} c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {105 a^{3} c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {35 a^{3} c^{4} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {5 a^{3} c^{4} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {11 c^{4} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {7 c^{4} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}+\frac {85 c^{4} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {85 c^{4} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {7 c^{4} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 f}-\frac {11 c^{4} a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{7}}\) | \(374\) |
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. 276 vs.
\(2 (109) = 218\).
time = 0.29, size = 276, normalized size = 2.46 \begin {gather*} \frac {192 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} + 1344 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} + 6720 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{4} - 35 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} + 630 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} - 5040 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{4} + 6720 \, {\left (f x + e\right )} a^{3} c^{4} + 6720 \, a^{3} c^{4} \cos \left (f x + e\right )}{6720 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 92, normalized size = 0.82 \begin {gather*} \frac {48 \, a^{3} c^{4} \cos \left (f x + e\right )^{7} + 105 \, a^{3} c^{4} f x + 7 \, {\left (8 \, a^{3} c^{4} \cos \left (f x + e\right )^{5} + 10 \, a^{3} c^{4} \cos \left (f x + e\right )^{3} + 15 \, a^{3} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{336 \, 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. 631 vs.
\(2 (107) = 214\).
time = 0.73, size = 631, normalized size = 5.63 \begin {gather*} \begin {cases} - \frac {5 a^{3} c^{4} x \sin ^{6}{\left (e + f x \right )}}{16} - \frac {15 a^{3} c^{4} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {9 a^{3} c^{4} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {15 a^{3} c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {9 a^{3} c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 a^{3} c^{4} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {5 a^{3} c^{4} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {9 a^{3} c^{4} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {3 a^{3} c^{4} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{3} c^{4} x - \frac {a^{3} c^{4} \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {11 a^{3} c^{4} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {2 a^{3} c^{4} \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 a^{3} c^{4} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{3} c^{4} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {15 a^{3} c^{4} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {8 a^{3} c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {4 a^{3} c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {3 a^{3} c^{4} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{3} c^{4} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {9 a^{3} c^{4} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {3 a^{3} c^{4} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {16 a^{3} c^{4} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {8 a^{3} c^{4} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {2 a^{3} c^{4} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {a^{3} c^{4} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\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 [A]
time = 0.43, size = 154, normalized size = 1.38 \begin {gather*} \frac {5}{16} \, a^{3} c^{4} x + \frac {a^{3} c^{4} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {a^{3} c^{4} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {3 \, a^{3} c^{4} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {5 \, a^{3} c^{4} \cos \left (f x + e\right )}{64 \, f} + \frac {a^{3} c^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {3 \, a^{3} c^{4} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {15 \, a^{3} c^{4} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.45, size = 301, normalized size = 2.69 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (\frac {a^3\,c^4\,\left (735\,e+735\,f\,x+672\right )}{336}-\frac {35\,a^3\,c^4\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^3\,c^4\,\left (2205\,e+2205\,f\,x+2016\right )}{336}-\frac {105\,a^3\,c^4\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^3\,c^4\,\left (3675\,e+3675\,f\,x+3360\right )}{336}-\frac {175\,a^3\,c^4\,\left (e+f\,x\right )}{16}\right )+\frac {7\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6}+\frac {85\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{24}-\frac {85\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}-\frac {7\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{6}-\frac {11\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{8}+\frac {a^3\,c^4\,\left (105\,e+105\,f\,x+96\right )}{336}+\frac {11\,a^3\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}-\frac {5\,a^3\,c^4\,\left (e+f\,x\right )}{16}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^7}+\frac {5\,a^3\,c^4\,x}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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