Optimal. Leaf size=187 \[ -\frac{3 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{3 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{3 i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2}+\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}-\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.236777, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4703, 4715, 4657, 4181, 2279, 2391, 261, 266, 43} \[ -\frac{3 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{3 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{3 i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^5 d^2}+\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}-\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4703
Rule 4715
Rule 4657
Rule 4181
Rule 2279
Rule 2391
Rule 261
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b \int \frac{x^3}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac{3 \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{(3 b) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{2 c^3 d^2}-\frac{b \operatorname{Subst}\left (\int \frac{x}{\left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d}\\ &=\frac{3 b \sqrt{1-c^2 x^2}}{2 c^5 d^2}+\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^5 d^2}-\frac{b \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \left (1-c^2 x\right )^{3/2}}-\frac{1}{c^2 \sqrt{1-c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2}\\ &=-\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}+\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d^2}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^5 d^2}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=-\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}+\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d^2}-\frac{(3 i b) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{(3 i b) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=-\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}+\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \sin ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^5 d^2}-\frac{3 i b \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{3 i b \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end{align*}
Mathematica [A] time = 0.452945, size = 332, normalized size = 1.78 \[ \frac{-6 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+6 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-\frac{2 a c x}{c^2 x^2-1}+4 a c x+3 a \log (1-c x)-3 a \log (c x+1)+\frac{b \sqrt{1-c^2 x^2}}{c x-1}-\frac{b \sqrt{1-c^2 x^2}}{c x+1}+4 b \sqrt{1-c^2 x^2}+4 b c x \sin ^{-1}(c x)+\frac{b \sin ^{-1}(c x)}{1-c x}-\frac{b \sin ^{-1}(c x)}{c x+1}+3 i \pi b \sin ^{-1}(c x)-6 b \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-3 \pi b \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+6 b \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-3 \pi b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+3 \pi b \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+3 \pi b \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{4 c^5 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.267, size = 305, normalized size = 1.6 \begin{align*}{\frac{ax}{{c}^{4}{d}^{2}}}-{\frac{a}{4\,{c}^{5}{d}^{2} \left ( cx-1 \right ) }}+{\frac{3\,a\ln \left ( cx-1 \right ) }{4\,{c}^{5}{d}^{2}}}-{\frac{a}{4\,{c}^{5}{d}^{2} \left ( cx+1 \right ) }}-{\frac{3\,a\ln \left ( cx+1 \right ) }{4\,{c}^{5}{d}^{2}}}+{\frac{b}{{c}^{5}{d}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arcsin \left ( cx \right ) x}{{c}^{4}{d}^{2}}}-{\frac{b\arcsin \left ( cx \right ) x}{2\,{c}^{4}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{2\,{c}^{5}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,b\arcsin \left ( cx \right ) }{2\,{c}^{5}{d}^{2}}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{3\,b\arcsin \left ( cx \right ) }{2\,{c}^{5}{d}^{2}}\ln \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{{\frac{3\,i}{2}}b}{{c}^{5}{d}^{2}}{\it dilog} \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{{\frac{3\,i}{2}}b}{{c}^{5}{d}^{2}}{\it dilog} \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \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}{4} \, a{\left (\frac{2 \, x}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac{4 \, x}{c^{4} d^{2}} + \frac{3 \, \log \left (c x + 1\right )}{c^{5} d^{2}} - \frac{3 \, \log \left (c x - 1\right )}{c^{5} d^{2}}\right )} - \frac{{\left (3 \,{\left (c^{2} x^{2} - 1\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 3 \,{\left (c^{2} x^{2} - 1\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (2 \, c^{3} x^{3} - 3 \, c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (c^{7} d^{2} x^{2} - c^{5} d^{2}\right )} \int \frac{{\left (4 \, c^{3} x^{3} - 6 \, c x - 3 \,{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \,{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}}\,{d x}\right )} b}{4 \,{\left (c^{7} d^{2} x^{2} - c^{5} d^{2}\right )}} \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^{4} \arcsin \left (c x\right ) + a x^{4}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, 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^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{4} \operatorname{asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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 \arcsin \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]