Optimal. Leaf size=272 \[ -\frac {b \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{\left (1-a^2\right ) x}-\frac {\text {ArcSin}(a+b x)^2}{2 x^2}-\frac {i a b^2 \text {ArcSin}(a+b x) \log \left (1+\frac {i e^{i \text {ArcSin}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {i a b^2 \text {ArcSin}(a+b x) \log \left (1+\frac {i e^{i \text {ArcSin}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {a b^2 \text {PolyLog}\left (2,-\frac {i e^{i \text {ArcSin}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {a b^2 \text {PolyLog}\left (2,-\frac {i e^{i \text {ArcSin}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.40, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {4889, 4827,
4857, 3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {a b^2 \text {Li}_2\left (-\frac {i e^{i \text {ArcSin}(a+b x)}}{a-\sqrt {a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac {a b^2 \text {Li}_2\left (-\frac {i e^{i \text {ArcSin}(a+b x)}}{a+\sqrt {a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}-\frac {i a b^2 \text {ArcSin}(a+b x) \log \left (1+\frac {i e^{i \text {ArcSin}(a+b x)}}{a-\sqrt {a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac {i a b^2 \text {ArcSin}(a+b x) \log \left (1+\frac {i e^{i \text {ArcSin}(a+b x)}}{\sqrt {a^2-1}+a}\right )}{\left (a^2-1\right )^{3/2}}-\frac {b \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{\left (1-a^2\right ) x}+\frac {b^2 \log (x)}{1-a^2}-\frac {\text {ArcSin}(a+b x)^2}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3404
Rule 3405
Rule 4827
Rule 4857
Rule 4889
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)^2}{x^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}+\text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}+\text {Subst}\left (\int \frac {x}{\left (-\frac {a}{b}+\frac {\sin (x)}{b}\right )^2} \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}+\frac {b \text {Subst}\left (\int \frac {\cos (x)}{-\frac {a}{b}+\frac {\sin (x)}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{1-a^2}+\frac {(a b) \text {Subst}\left (\int \frac {x}{-\frac {a}{b}+\frac {\sin (x)}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{1-a^2}\\ &=-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}+\frac {(2 a b) \text {Subst}\left (\int \frac {e^{i x} x}{\frac {i}{b}-\frac {2 a e^{i x}}{b}-\frac {i e^{2 i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{1-a^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+x} \, dx,x,\frac {a}{b}+x\right )}{1-a^2}\\ &=-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \log (x)}{1-a^2}+\frac {(2 i a b) \text {Subst}\left (\int \frac {e^{i x} x}{-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}-\frac {2 i e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}-\frac {(2 i a b) \text {Subst}\left (\int \frac {e^{i x} x}{-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}-\frac {2 i e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}\\ &=-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}-\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {\left (i a b^2\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x}}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}+\frac {\left (i a b^2\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x}}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}\\ &=-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}-\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {\left (a b^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {\left (a b^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )}{\left (-1+a^2\right )^{3/2}}\\ &=-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2}-\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {a b^2 \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {a b^2 \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 314, normalized size = 1.15 \begin {gather*} \frac {2 \sqrt {-1+a^2} b x \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)+\sqrt {-1+a^2} \text {ArcSin}(a+b x)^2-a^2 \sqrt {-1+a^2} \text {ArcSin}(a+b x)^2-2 i a b^2 x^2 \text {ArcSin}(a+b x) \log \left (\frac {a-\sqrt {-1+a^2}+i e^{i \text {ArcSin}(a+b x)}}{a-\sqrt {-1+a^2}}\right )+2 i a b^2 x^2 \text {ArcSin}(a+b x) \log \left (\frac {a+\sqrt {-1+a^2}+i e^{i \text {ArcSin}(a+b x)}}{a+\sqrt {-1+a^2}}\right )-2 \sqrt {-1+a^2} b^2 x^2 \log (x)-2 a b^2 x^2 \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(a+b x)}}{-a+\sqrt {-1+a^2}}\right )+2 a b^2 x^2 \text {PolyLog}\left (2,-\frac {i e^{i \text {ArcSin}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{2 \left (-1+a^2\right )^{3/2} x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.22, size = 521, normalized size = 1.92
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\arcsin \left (b x +a \right ) \left (2 i \left (b x +a \right )^{2}-\arcsin \left (b x +a \right )-2 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+a^{2} \arcsin \left (b x +a \right )+2 i a^{2}+2 a \sqrt {1-\left (b x +a \right )^{2}}-4 i a \left (b x +a \right )\right )}{2 \left (a^{2}-1\right ) b^{2} x^{2}}+\frac {2 \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}-\frac {\ln \left (i \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+2 a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-i\right )}{a^{2}-1}-\frac {i \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {i \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}\right )\) | \(521\) |
default | \(b^{2} \left (-\frac {\arcsin \left (b x +a \right ) \left (2 i \left (b x +a \right )^{2}-\arcsin \left (b x +a \right )-2 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+a^{2} \arcsin \left (b x +a \right )+2 i a^{2}+2 a \sqrt {1-\left (b x +a \right )^{2}}-4 i a \left (b x +a \right )\right )}{2 \left (a^{2}-1\right ) b^{2} x^{2}}+\frac {2 \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}-\frac {\ln \left (i \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+2 a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-i\right )}{a^{2}-1}-\frac {i \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {i \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}\right )\) | \(521\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________