Optimal. Leaf size=134 \[ -\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {PolyLog}\left (2,-e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \text {PolyLog}\left (3,-e^{i (e+f x)}\right )}{a f^4}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \]
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Rubi [A]
time = 0.18, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3399, 4269,
3800, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {12 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3}+\frac {6 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {i (c+d x)^3}{a f}+\frac {12 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a f^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3399
Rule 3800
Rule 4269
Rule 6724
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+a \cos (e+f x)} \, dx &=\frac {\int (c+d x)^3 \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{2 a}\\ &=\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(3 d) \int (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f}\\ &=-\frac {i (c+d x)^3}{a f}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(6 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f}\\ &=-\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2}\\ &=-\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 i d^3\right ) \int \text {Li}_2\left (-e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3}\\ &=-\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^4}\\ &=-\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a f^4}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 151, normalized size = 1.13 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {i \cos \left (\frac {1}{2} (e+f x)\right ) \left (f^2 (c+d x)^2 \left (f (c+d x)+6 i d \log \left (1+e^{i (e+f x)}\right )\right )+12 d^2 f (c+d x) \text {PolyLog}\left (2,-e^{i (e+f x)}\right )+12 i d^3 \text {PolyLog}\left (3,-e^{i (e+f x)}\right )\right )}{f^3}+(c+d x)^3 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{a f (1+\cos (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 363 vs. \(2 (122 ) = 244\).
time = 0.14, size = 364, normalized size = 2.72
method | result | size |
risch | \(\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}+\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a \,f^{2}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{2}}-\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {6 i d^{2} c \,e^{2}}{a \,f^{3}}+\frac {4 i d^{3} e^{3}}{a \,f^{4}}+\frac {6 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x^{2}}{a \,f^{2}}-\frac {12 i d^{2} c \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}+\frac {12 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}+\frac {12 d^{2} c e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {2 i d^{3} x^{3}}{a f}-\frac {12 i d^{2} c e x}{a \,f^{2}}-\frac {12 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{3}}-\frac {6 i d^{2} c \,x^{2}}{a f}+\frac {6 i d^{3} e^{2} x}{a \,f^{3}}+\frac {12 d^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{a \,f^{2}}\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 1006 vs. \(2 (123) = 246\).
time = 0.40, size = 1006, normalized size = 7.51 \begin {gather*} -\frac {\frac {6 \, {\left ({\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} c d^{2} e}{a f^{2} \cos \left (f x + e\right )^{2} + a f^{2} \sin \left (f x + e\right )^{2} + 2 \, a f^{2} \cos \left (f x + e\right ) + a f^{2}} - \frac {3 \, {\left ({\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} c^{2} d}{a f \cos \left (f x + e\right )^{2} + a f \sin \left (f x + e\right )^{2} + 2 \, a f \cos \left (f x + e\right ) + a f} - \frac {c^{3} \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {3 \, c^{2} d e \sin \left (f x + e\right )}{a f {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {3 \, c d^{2} e^{2} \sin \left (f x + e\right )}{a f^{2} {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {2 \, d^{3} e^{3} - 6 \, {\left ({\left (f x + e\right )}^{2} d^{3} + d^{3} e^{2} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )} + {\left ({\left (f x + e\right )}^{2} d^{3} + d^{3} e^{2} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}\right )} \cos \left (f x + e\right ) - {\left (-i \, {\left (f x + e\right )}^{2} d^{3} - i \, d^{3} e^{2} + 2 \, {\left (-i \, c d^{2} f + i \, d^{3} e\right )} {\left (f x + e\right )}\right )} \sin \left (f x + e\right )\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) + 2 \, {\left ({\left (f x + e\right )}^{3} d^{3} + 3 \, {\left (f x + e\right )} d^{3} e^{2} + 3 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}^{2}\right )} \cos \left (f x + e\right ) + 12 \, {\left ({\left (f x + e\right )} d^{3} + c d^{2} f - d^{3} e + {\left ({\left (f x + e\right )} d^{3} + c d^{2} f - d^{3} e\right )} \cos \left (f x + e\right ) + {\left (i \, {\left (f x + e\right )} d^{3} + i \, c d^{2} f - i \, d^{3} e\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) + 3 \, {\left (i \, {\left (f x + e\right )}^{2} d^{3} + i \, d^{3} e^{2} + 2 \, {\left (i \, c d^{2} f - i \, d^{3} e\right )} {\left (f x + e\right )} + {\left (i \, {\left (f x + e\right )}^{2} d^{3} + i \, d^{3} e^{2} + 2 \, {\left (i \, c d^{2} f - i \, d^{3} e\right )} {\left (f x + e\right )}\right )} \cos \left (f x + e\right ) - {\left ({\left (f x + e\right )}^{2} d^{3} + d^{3} e^{2} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + 12 \, {\left (i \, d^{3} \cos \left (f x + e\right ) - d^{3} \sin \left (f x + e\right ) + i \, d^{3}\right )} {\rm Li}_{3}(-e^{\left (i \, f x + i \, e\right )}) + 2 \, {\left (i \, {\left (f x + e\right )}^{3} d^{3} + 3 i \, {\left (f x + e\right )} d^{3} e^{2} + 3 \, {\left (i \, c d^{2} f - i \, d^{3} e\right )} {\left (f x + e\right )}^{2}\right )} \sin \left (f x + e\right )}{-i \, a f^{3} \cos \left (f x + e\right ) + a f^{3} \sin \left (f x + e\right ) - i \, a f^{3}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 442 vs. \(2 (123) = 246\).
time = 0.39, size = 442, normalized size = 3.30 \begin {gather*} -\frac {6 \, {\left (-i \, d^{3} f x - i \, c d^{2} f + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 6 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3}\right )} \sin \left (f x + e\right )}{a f^{4} \cos \left (f x + e\right ) + a f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {c^{3}}{\cos {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\cos {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\cos {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\cos {\left (e + f x \right )} + 1}\, dx}{a} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{a+a\,\cos \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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