Optimal. Leaf size=144 \[ \frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d}+\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.198872, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4716, 4676, 3717, 2190, 2279, 2391, 321, 216} \[ \frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d}+\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4716
Rule 4676
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 321
Rule 216
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{\int \frac{x \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2}-\frac{b \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2 c d}\\ &=\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}-\frac{\operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d}-\frac{b \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 c^3 d}\\ &=\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d}\\ &=\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}-\frac{\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d}\\ &=\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}-\frac{\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d}\\ &=\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d}+\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}-\frac{\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d}+\frac{i b \text{Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d}\\ \end{align*}
Mathematica [A] time = 0.224683, size = 161, normalized size = 1.12 \[ -\frac{-4 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )-4 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )+2 a c^2 x^2+2 a \log \left (1-c^2 x^2\right )-b c x \sqrt{1-c^2 x^2}+2 b c^2 x^2 \cos ^{-1}(c x)+b \sin ^{-1}(c x)-2 i b \cos ^{-1}(c x)^2+4 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+4 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{4 c^4 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.196, size = 243, normalized size = 1.7 \begin{align*} -{\frac{a{x}^{2}}{2\,{c}^{2}d}}-{\frac{a\ln \left ( cx-1 \right ) }{2\,d{c}^{4}}}-{\frac{a\ln \left ( cx+1 \right ) }{2\,d{c}^{4}}}+{\frac{{\frac{i}{2}}b \left ( \arccos \left ( cx \right ) \right ) ^{2}}{d{c}^{4}}}+{\frac{bx}{4\,d{c}^{3}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arccos \left ( cx \right ){x}^{2}}{2\,{c}^{2}d}}+{\frac{b\arccos \left ( cx \right ) }{4\,d{c}^{4}}}-{\frac{b\arccos \left ( cx \right ) }{d{c}^{4}}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{b\arccos \left ( cx \right ) }{d{c}^{4}}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{ib}{d{c}^{4}}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{ib}{d{c}^{4}}{\it polylog} \left ( 2,cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{x^{2}}{c^{2} d} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4} d}\right )} + \frac{{\left (c^{4} d \int \frac{c^{2} x^{2} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} + e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) + e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{c^{7} d x^{4} - c^{5} d x^{2} -{\left (c^{5} d x^{2} - c^{3} d\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x} -{\left (c^{2} x^{2} + \log \left (c x + 1\right ) + \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )\right )} b}{2 \, c^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b x^{3} \arccos \left (c x\right ) + a x^{3}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a x^{3}}{c^{2} x^{2} - 1}\, dx + \int \frac{b x^{3} \operatorname{acos}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b \arccos \left (c x\right ) + a\right )} x^{3}}{c^{2} d x^{2} - d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]