Optimal. Leaf size=123 \[ \frac{3}{2} b^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{4} i b^3 \text{PolyLog}\left (4,e^{2 i \sin ^{-1}(c x)}\right )-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.146619, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4625, 3717, 2190, 2531, 6609, 2282, 6589} \[ \frac{3}{2} b^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{3}{2} i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{4} i b^3 \text{PolyLog}\left (4,e^{2 i \sin ^{-1}(c x)}\right )-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{x} \, dx &=\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}-2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (3 i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{3}{2} b^2 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} \left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{3}{2} b^2 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{4} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{3}{2} b^2 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{3}{4} i b^3 \text{Li}_4\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.238776, size = 244, normalized size = 1.98 \[ 3 a^2 b \left (\sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} i \left (\sin ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )\right )\right )+\frac{1}{8} a b^2 \left (24 i \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(c x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(c x)}\right )+8 i \sin ^{-1}(c x)^3+24 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )-i \pi ^3\right )-\frac{1}{64} i b^3 \left (-96 \sin ^{-1}(c x)^2 \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(c x)}\right )+96 i \sin ^{-1}(c x) \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(c x)}\right )+48 \text{PolyLog}\left (4,e^{-2 i \sin ^{-1}(c x)}\right )-16 \sin ^{-1}(c x)^4+64 i \sin ^{-1}(c x)^3 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+\pi ^4\right )+a^3 \log (c x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.026, size = 592, normalized size = 4.8 \begin{align*}{a}^{3}\ln \left ( cx \right ) -3\,i{a}^{2}b{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{3}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,i{b}^{3}{\it polylog} \left ( 4,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,{b}^{3}\arcsin \left ( cx \right ){\it polylog} \left ( 3,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,i{b}^{3}{\it polylog} \left ( 4,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{3}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -6\,ia{b}^{2}\arcsin \left ( cx \right ){\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,{b}^{3}\arcsin \left ( cx \right ){\it polylog} \left ( 3,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -6\,ia{b}^{2}\arcsin \left ( cx \right ){\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{i}{4}}{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{4}-3\,i{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{3\,i}{2}}{a}^{2}b \left ( \arcsin \left ( cx \right ) \right ) ^{2}+3\,a{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,a{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,a{b}^{2}{\it polylog} \left ( 3,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,a{b}^{2}{\it polylog} \left ( 3,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -3\,i{a}^{2}b{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}b\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}b\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -3\,i{b}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -ia{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{3} \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*} a^{3} \log \left (x\right ) + \int \frac{b^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{3} + 3 \, a b^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 3 \, a^{2} b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x}\,{d x} \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^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}}{x}, 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*} \int \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{3}}{x}\, dx \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 )}^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]