Optimal. Leaf size=78 \[ x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {4 b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c}+\frac {2 i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6279, 5451, 4180, 2279, 2391} \[ \frac {2 i b^2 \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{c}+x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {4 b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 4180
Rule 5451
Rule 6279
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {4 b \left (a+b \text {sech}^{-1}(c x)\right ) \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}-\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {4 b \left (a+b \text {sech}^{-1}(c x)\right ) \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {4 b \left (a+b \text {sech}^{-1}(c x)\right ) \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 126, normalized size = 1.62 \[ a^2 x+\frac {2 a b \left (c x \text {sech}^{-1}(c x)-2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )\right )\right )}{c}+\frac {i b^2 \left (2 \text {Li}_2\left (-i e^{-\text {sech}^{-1}(c x)}\right )-2 \text {Li}_2\left (i e^{-\text {sech}^{-1}(c x)}\right )+\text {sech}^{-1}(c x) \left (-i c x \text {sech}^{-1}(c x)+2 \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )\right )}{c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {arsech}\left (c x\right )^{2} + 2 \, a b \operatorname {arsech}\left (c x\right ) + a^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.30, size = 250, normalized size = 3.21 \[ x \,b^{2} \mathrm {arcsech}\left (c x \right )^{2}-\frac {2 i \mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right ) b^{2}}{c}+\frac {2 i \mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right ) b^{2}}{c}+2 x a b \,\mathrm {arcsech}\left (c x \right )+\frac {2 i \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right ) b^{2}}{c}-\frac {2 i \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right ) b^{2}}{c}+a^{2} x -\frac {2 \arctan \left (\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right ) a b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (x \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )^{2} - \int -\frac {c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} - 1\right )} \log \relax (x)^{2} + {\left (c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} - 1\right )} \log \relax (x)^{2} - \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) - \log \relax (c)\right )} \log \relax (x)\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - 2 \, {\left (c^{2} x^{2} \log \relax (c) + {\left (c^{2} x^{2} {\left (\log \relax (c) + 1\right )} + {\left (c^{2} x^{2} - 1\right )} \log \relax (x) - \log \relax (c)\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + {\left (c^{2} x^{2} - 1\right )} \log \relax (x) - \log \relax (c)\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) - \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) - \log \relax (c)\right )} \log \relax (x)}{c^{2} x^{2} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - 1}\,{d x}\right )} b^{2} + a^{2} x + \frac {2 \, {\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} a b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________