Optimal. Leaf size=330 \[ \frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6321, 5468, 4191, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac {6 b \text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}+\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3320
Rule 4191
Rule 5468
Rule 6321
Rule 6589
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)^3}{x^2} \, dx &=-\left (b \operatorname {Subst}\left (\int \frac {x^3 \text {sech}(x) \tanh (x)}{(-a+\text {sech}(x))^2} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )\\ &=-\frac {\text {sech}^{-1}(a+b x)^3}{x}+(3 b) \operatorname {Subst}\left (\int \frac {x^2}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\frac {\text {sech}^{-1}(a+b x)^3}{x}+(3 b) \operatorname {Subst}\left (\int \left (-\frac {x^2}{a}+\frac {x^2}{a (1-a \cosh (x))}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^2}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {e^x x^2}{-a+2 e^x-a e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}-\frac {(6 b) \operatorname {Subst}\left (\int \frac {e^x x^2}{2-2 \sqrt {1-a^2}-2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a^2}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {e^x x^2}{2+2 \sqrt {1-a^2}-2 a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}+\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {(6 b) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 a e^x}{2-2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}+\frac {(6 b) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 a e^x}{2+2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}+\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {(6 b) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 a e^x}{2-2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}+\frac {(6 b) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 a e^x}{2+2 \sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}+\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {(6 b) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}\\ &=-\frac {b \text {sech}^{-1}(a+b x)^3}{a}-\frac {\text {sech}^{-1}(a+b x)^3}{x}+\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {3 b \text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 46.92, size = 1849, normalized size = 5.60 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.90, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arcsech}\left (b x +a \right )^{3}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{3}}{x} - \int \frac {8 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{3} + 8 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{3} - 3 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} b - b\right )} x - 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) - {\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \log \left (b x + a\right ) - {\left (2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + {\left (2 \, a^{2} b - b\right )} x - {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )\right )} \sqrt {b x + a + 1}\right )} \sqrt {-b x - a + 1}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )^{2} - 12 \, {\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{2}\right )} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right )}{b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2} + {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} b - b\right )} x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^3}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asech}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________