Optimal. Leaf size=137 \[ 6 i b^2 c \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 i b^2 c \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-6 b^3 c \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )+6 b^3 c \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4627, 4709, 4183, 2531, 2282, 6589} \[ 6 i b^2 c \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 i b^2 c \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-6 b^3 c \text {PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )+6 b^3 c \text {PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 4183
Rule 4627
Rule 4709
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x^2} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}+(3 b c) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (6 b^2 c\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (6 b^2 c\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\left (6 i b^3 c\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (6 i b^3 c\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\left (6 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\left (6 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x}-6 b c \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i b^2 c \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-6 b^3 c \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )+6 b^3 c \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.32, size = 283, normalized size = 2.07 \[ -\frac {a^3}{x}-3 a^2 b c \log \left (\sqrt {1-c^2 x^2}+1\right )+3 a^2 b c \log (x)-\frac {3 a^2 b \sin ^{-1}(c x)}{x}+3 a b^2 c \left (2 i \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-\sin ^{-1}(c x) \left (\frac {\sin ^{-1}(c x)}{c x}-2 \log \left (1-e^{i \sin ^{-1}(c x)}\right )+2 \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )+b^3 c \left (6 i \sin ^{-1}(c x) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-6 i \sin ^{-1}(c x) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-6 \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )+6 \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )-\frac {\sin ^{-1}(c x)^3}{c x}+3 \sin ^{-1}(c x)^2 \log \left (1-e^{i \sin ^{-1}(c x)}\right )-3 \sin ^{-1}(c x)^2 \log \left (1+e^{i \sin ^{-1}(c x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.10, size = 378, normalized size = 2.76 \[ -\frac {a^{3}}{x}-\frac {b^{3} \arcsin \left (c x \right )^{3}}{x}-3 c \,b^{3} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i c \,b^{3} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 b^{3} c \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 c \,b^{3} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i c \,b^{3} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 b^{3} c \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 a \,b^{2} \arcsin \left (c x \right )^{2}}{x}+6 c a \,b^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 c a \,b^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i c a \,b^{2} \dilog \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 i c a \,b^{2} \dilog \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 a^{2} b \arcsin \left (c x \right )}{x}-3 c \,a^{2} b \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -3 \, {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} a^{2} b - \frac {a^{3}}{x} - \frac {b^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{3} + \frac {3}{2} \, {\left (a b^{2} c^{2} {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} \arctan \left (\frac {c x}{\sqrt {c x + 1} \sqrt {-c x + 1}}\right )^{2} - {\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} a b^{2} \arctan \left (\frac {c x}{\sqrt {c x + 1} \sqrt {-c x + 1}}\right )^{2} + 8 \, b^{3} c \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}}{4 \, {\left (c^{2} x^{3} - x\right )}}\,{d x}\right )} x}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________