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 = 0.19869, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3318
Rule 4184
Rule 3719
Rule 2190
Rule 2279
Rule 2391
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*}
Mathematica [A] time = 0.329933, 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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Maple [B] time = 0.381, size = 197, normalized size = 2. \begin{align*}{\frac{2\,i \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2} \right ) }{af \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) }}+4\,{\frac{cd\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) }{a{f}^{2}}}-4\,{\frac{cd\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{a{f}^{2}}}-{\frac{2\,i{d}^{2}{x}^{2}}{af}}-{\frac{4\,i{d}^{2}ex}{a{f}^{2}}}-{\frac{2\,i{d}^{2}{e}^{2}}{{f}^{3}a}}+4\,{\frac{{d}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) x}{a{f}^{2}}}-{\frac{4\,i{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}a}}+4\,{\frac{{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60632, size = 382, normalized size = 3.78 \begin{align*} \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 ) - 4 \,{\left (d^{2} \cos \left (f x + e\right ) + i \, d^{2} \sin \left (f x + e\right ) + 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68263, size = 564, normalized size = 5.58 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{a \cos \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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