Optimal. Leaf size=243 \[ -\frac{4 i d^2 \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{3 a^2 f^3}+\frac{4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{6 a^2 f}-\frac{i (c+d x)^2}{3 a^2 f}-\frac{2 d^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f^3} \]
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Rubi [A] time = 0.287215, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3318, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ -\frac{4 i d^2 \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{3 a^2 f^3}+\frac{4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{6 a^2 f}-\frac{i (c+d x)^2}{3 a^2 f}-\frac{2 d^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f^3} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 4186
Rule 3767
Rule 8
Rule 4184
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a+a \sin (e+f x))^2} \, dx &=\frac{\int (c+d x)^2 \csc ^4\left (\frac{1}{2} \left (e+\frac{\pi }{2}\right )+\frac{f x}{2}\right ) \, dx}{4 a^2}\\ &=-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\int (c+d x)^2 \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{6 a^2}+\frac{d^2 \int \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{3 a^2 f^2}\\ &=-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right )}{3 a^2 f^3}+\frac{(2 d) \int (c+d x) \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac{i (c+d x)^2}{3 a^2 f}-\frac{2 d^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{(4 d) \int \frac{e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)}{1-i e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=-\frac{i (c+d x)^2}{3 a^2 f}-\frac{2 d^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{\left (4 d^2\right ) \int \log \left (1-i e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=-\frac{i (c+d x)^2}{3 a^2 f}-\frac{2 d^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac{\left (4 i d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=-\frac{i (c+d x)^2}{3 a^2 f}-\frac{2 d^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}-\frac{d (c+d x) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \csc ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{4 i d^2 \text{Li}_2\left (i e^{i (e+f x)}\right )}{3 a^2 f^3}\\ \end{align*}
Mathematica [A] time = 2.20246, size = 175, normalized size = 0.72 \[ \frac{-8 i d^2 \text{PolyLog}\left (2,i e^{i (e+f x)}\right )+2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right )-2 i f (c+d x) \left (f (c+d x)+4 i d \log \left (1-i e^{i (e+f x)}\right )\right )+f (c+d x) \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ) \left (f (c+d x) \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right )-2 d\right )}{6 a^2 f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.495, size = 421, normalized size = 1.7 \begin{align*}{\frac{-{\frac{2\,i}{3}} \left ( i{d}^{2}{f}^{2}{x}^{2}+3\,{d}^{2}{f}^{2}{x}^{2}{{\rm e}^{i \left ( fx+e \right ) }}+2\,icd{f}^{2}x+2\,if{d}^{2}x{{\rm e}^{i \left ( fx+e \right ) }}+6\,cd{f}^{2}x{{\rm e}^{i \left ( fx+e \right ) }}+2\,f{d}^{2}x{{\rm e}^{2\,i \left ( fx+e \right ) }}+i{c}^{2}{f}^{2}+2\,ifcd{{\rm e}^{i \left ( fx+e \right ) }}-2\,i{d}^{2}{{\rm e}^{2\,i \left ( fx+e \right ) }}+3\,{c}^{2}{f}^{2}{{\rm e}^{i \left ( fx+e \right ) }}+2\,fcd{{\rm e}^{2\,i \left ( fx+e \right ) }}+2\,i{d}^{2}+4\,{d}^{2}{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{ \left ({{\rm e}^{i \left ( fx+e \right ) }}+i \right ) ^{3}{f}^{3}{a}^{2}}}+{\frac{4\,\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+i \right ) cd}{3\,{a}^{2}{f}^{2}}}-{\frac{4\,\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) cd}{3\,{a}^{2}{f}^{2}}}-{\frac{{\frac{2\,i}{3}}{d}^{2}{x}^{2}}{{a}^{2}f}}-{\frac{{\frac{4\,i}{3}}{d}^{2}ex}{{a}^{2}{f}^{2}}}-{\frac{{\frac{2\,i}{3}}{d}^{2}{e}^{2}}{{a}^{2}{f}^{3}}}+{\frac{4\,{d}^{2}\ln \left ( 1-i{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{3\,{a}^{2}{f}^{2}}}+{\frac{4\,{d}^{2}\ln \left ( 1-i{{\rm e}^{i \left ( fx+e \right ) }} \right ) e}{3\,{a}^{2}{f}^{3}}}-{\frac{{\frac{4\,i}{3}}{d}^{2}{\it polylog} \left ( 2,i{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{a}^{2}{f}^{3}}}-{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+i \right ) }{3\,{a}^{2}{f}^{3}}}+{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{3\,{a}^{2}{f}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.21389, size = 1123, normalized size = 4.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06389, size = 2074, normalized size = 8.53 \begin{align*} \text{result too large to display} \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}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin{\left (e + f x \right )} + 1}\, dx + \int \frac{d^{2} x^{2}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin{\left (e + f x \right )} + 1}\, dx + \int \frac{2 c d x}{\sin ^{2}{\left (e + f x \right )} + 2 \sin{\left (e + f x \right )} + 1}\, dx}{a^{2}} \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}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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