Optimal. Leaf size=101 \[ \frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {i (c+d x)^2}{a f}-\frac {4 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3} \]
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Rubi [A] time = 0.20, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3318, 4184, 3719, 2190, 2279, 2391} \[ \frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {i (c+d x)^2}{a f}-\frac {4 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3719
Rule 4184
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{a+a \cos (e+f x)} \, dx &=\frac {\int (c+d x)^2 \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{2 a}\\ &=\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(2 d) \int (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f}\\ &=-\frac {i (c+d x)^2}{a f}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(4 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f}\\ &=-\frac {i (c+d x)^2}{a f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {\left (4 d^2\right ) \int \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)^2}{a f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (4 i d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^3}\\ &=-\frac {i (c+d x)^2}{a f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a f^2}-\frac {4 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{a f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 125, normalized size = 1.24 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \left (f (c+d x) \sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right ) \left (4 d \log \left (1+e^{i (e+f x)}\right )-i f (c+d x)\right )\right )-4 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{a f^3 (\cos (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 222, normalized size = 2.20 \[ \frac {{\left (2 i \, d^{2} \cos \left (f x + e\right ) + 2 i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + {\left (-2 i \, d^{2} \cos \left (f x + e\right ) - 2 i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) + {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} \sin \left (f x + e\right )}{a f^{3} \cos \left (f x + e\right ) + a f^{3}} \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{a \cos \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 197, normalized size = 1.95 \[ \frac {2 i \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}+\frac {4 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a \,f^{2}}-\frac {4 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{2}}-\frac {2 i d^{2} x^{2}}{a f}-\frac {4 i d^{2} e x}{a \,f^{2}}-\frac {2 i d^{2} e^{2}}{a \,f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{a \,f^{2}}-\frac {4 i d^{2} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}+\frac {4 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 286, normalized size = 2.83 \[ \frac {2 \, c^{2} f^{2} + {\left (4 \, d^{2} f x + 4 \, c d f + 4 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right ) + {\left (4 i \, d^{2} f x + 4 i \, c d f\right )} \sin \left (f x + e\right )\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x\right )} \cos \left (f x + e\right ) - {\left (4 \, d^{2} \cos \left (f x + e\right ) + 4 i \, d^{2} \sin \left (f x + e\right ) + 4 \, d^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) + {\left (-2 i \, d^{2} f x - 2 i \, c d f + {\left (-2 i \, d^{2} f x - 2 i \, c d f\right )} \cos \left (f x + e\right ) + 2 \, {\left (d^{2} f x + c d f\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + {\left (-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x\right )} \sin \left (f x + e\right )}{-i \, a f^{3} \cos \left (f x + e\right ) + a f^{3} \sin \left (f x + e\right ) - i \, a f^{3}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^2}{a+a\,\cos \left (e+f\,x\right )} \,d x \]
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 \[ \frac {\int \frac {c^{2}}{\cos {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cos {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cos {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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