Optimal. Leaf size=83 \[ -\frac{i d^2 \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac{2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac{(c+d x)^2 \cot (a+b x)}{b}-\frac{i (c+d x)^2}{b} \]
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Rubi [A] time = 0.135961, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4184, 3717, 2190, 2279, 2391} \[ -\frac{i d^2 \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac{2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac{(c+d x)^2 \cot (a+b x)}{b}-\frac{i (c+d x)^2}{b} \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (c+d x)^2 \csc ^2(a+b x) \, dx &=-\frac{(c+d x)^2 \cot (a+b x)}{b}+\frac{(2 d) \int (c+d x) \cot (a+b x) \, dx}{b}\\ &=-\frac{i (c+d x)^2}{b}-\frac{(c+d x)^2 \cot (a+b x)}{b}-\frac{(4 i d) \int \frac{e^{2 i (a+b x)} (c+d x)}{1-e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac{i (c+d x)^2}{b}-\frac{(c+d x)^2 \cot (a+b x)}{b}+\frac{2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac{\left (2 d^2\right ) \int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{i (c+d x)^2}{b}-\frac{(c+d x)^2 \cot (a+b x)}{b}+\frac{2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}+\frac{\left (i d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^3}\\ &=-\frac{i (c+d x)^2}{b}-\frac{(c+d x)^2 \cot (a+b x)}{b}+\frac{2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac{i d^2 \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}\\ \end{align*}
Mathematica [B] time = 4.82682, size = 181, normalized size = 2.18 \[ \frac{\csc (a) \left (d^2 \left (-\sin (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )-i b x \left (\pi -2 \tan ^{-1}(\tan (a))\right )-2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )+2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )-b^2 x^2 \cos (a) e^{i \tan ^{-1}(\tan (a))} \sqrt{\sec ^2(a)}\right )+b^2 \sin (b x) (c+d x)^2 \csc (a+b x)-2 b c d (b x \cos (a)-\sin (a) \log (\sin (a+b x)))\right )}{b^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.043, size = 276, normalized size = 3.3 \begin{align*}{\frac{-2\,i \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2} \right ) }{b \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) }}-4\,{\frac{cd\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+2\,{\frac{cd\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-1 \right ) }{{b}^{2}}}+2\,{\frac{cd\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) }{{b}^{2}}}-{\frac{2\,i{d}^{2}{x}^{2}}{b}}-{\frac{4\,i{d}^{2}ax}{{b}^{2}}}-{\frac{2\,i{d}^{2}{a}^{2}}{{b}^{3}}}+2\,{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}+2\,{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{3}}}-{\frac{2\,i{d}^{2}{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+2\,{\frac{{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) x}{{b}^{2}}}-{\frac{2\,i{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+4\,{\frac{{d}^{2}a\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-2\,{\frac{{d}^{2}a\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-1 \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46171, size = 749, normalized size = 9.02 \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 = 1.79532, size = 1026, normalized size = 12.36 \begin{align*} \frac{-i \, d^{2}{\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d^{2}{\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d^{2}{\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d^{2}{\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) +{\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) +{\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) +{\left (b c d - a d^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2} i \, \sin \left (b x + a\right ) + \frac{1}{2}\right ) \sin \left (b x + a\right ) +{\left (b c d - a d^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) - \frac{1}{2} i \, \sin \left (b x + a\right ) + \frac{1}{2}\right ) \sin \left (b x + a\right ) +{\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) +{\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (b x + a\right )}{b^{3} \sin \left (b x + a\right )} \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*} \int \left (c + d x\right )^{2} \csc ^{2}{\left (a + b x \right )}\, dx \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{\left (d x + c\right )}^{2} \csc \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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