Optimal. Leaf size=114 \[ \frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\left (2 a^2+1\right ) b^2 \tanh ^{-1}\left (\frac {\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )+a}{\sqrt {a^2+1}}\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac {b (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{2 a \left (a^2+1\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.21, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6322, 5469, 3785, 3919, 3831, 2660, 618, 206} \[ \frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\left (2 a^2+1\right ) b^2 \tanh ^{-1}\left (\frac {\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )+a}{\sqrt {a^2+1}}\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac {b (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{2 a \left (a^2+1\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 3785
Rule 3831
Rule 3919
Rule 5469
Rule 6322
Rubi steps
\begin {align*} \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx &=-\left (b^2 \operatorname {Subst}\left (\int \frac {x \coth (x) \text {csch}(x)}{(-a+\text {csch}(x))^3} \, dx,x,\text {csch}^{-1}(a+b x)\right )\right )\\ &=-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{(-a+\text {csch}(x))^2} \, dx,x,\text {csch}^{-1}(a+b x)\right )\\ &=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {-1-a^2-a \text {csch}(x)}{-a+\text {csch}(x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 a \left (1+a^2\right )}\\ &=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1+2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {\text {csch}(x)}{-a+\text {csch}(x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 a^2 \left (1+a^2\right )}\\ &=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1+2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a \sinh (x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 a^2 \left (1+a^2\right )}\\ &=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1+2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x-x^2} \, dx,x,\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a^2 \left (1+a^2\right )}\\ &=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}+\frac {\left (2 \left (1+2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+a^2\right )-x^2} \, dx,x,-2 a-2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )\right )}{a^2 \left (1+a^2\right )}\\ &=\frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (1+2 a^2\right ) b^2 \tanh ^{-1}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 220, normalized size = 1.93 \[ \frac {1}{2} \left (\frac {b (a+b x) \sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}}{a \left (a^2+1\right ) x}-\frac {\left (2 a^2+1\right ) b^2 \log \left (\sqrt {a^2+1} a \sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+\sqrt {a^2+1} b x \sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+a^2+a b x+1\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac {\left (2 a^2+1\right ) b^2 \log (x)}{a^2 \left (a^2+1\right )^{3/2}}+\frac {b^2 \sinh ^{-1}\left (\frac {1}{a+b x}\right )}{a^2}-\frac {\text {csch}^{-1}(a+b x)}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 461, normalized size = 4.04 \[ \frac {{\left (2 \, a^{2} + 1\right )} \sqrt {a^{2} + 1} b^{2} x^{2} \log \left (-\frac {a^{2} b x + a^{3} - {\left (a b x + a^{2} + {\left (a b x + a^{2}\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1\right )} \sqrt {a^{2} + 1} + {\left (a^{3} + {\left (a^{2} + 1\right )} b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + a}{x}\right ) + {\left (a^{4} + 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - {\left (a^{4} + 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + {\left (a^{3} + a\right )} b^{2} x^{2} - {\left (a^{6} + 2 \, a^{4} + a^{2}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + {\left ({\left (a^{3} + a\right )} b^{2} x^{2} + {\left (a^{4} + a^{2}\right )} b x\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, {\left (a^{6} + 2 \, a^{4} + a^{2}\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcsch}\left (b x + a\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 453, normalized size = 3.97 \[ -\frac {\mathrm {arccsch}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \arctanh \left (\frac {1}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}{2 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}+1\right )}-\frac {b^{2} \sqrt {1+\left (b x +a \right )^{2}}\, a^{2} \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )+2}{b x}\right )}{\sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {b^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \arctanh \left (\frac {1}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}{2 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )}+\frac {b \left (1+\left (b x +a \right )^{2}\right )}{2 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \left (a^{2}+1\right ) x}-\frac {3 b^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )+2}{b x}\right )}{2 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {b^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )+2}{b x}\right )}{2 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {i \, a b^{2} {\left (\log \left (\frac {i \, {\left (b^{2} x + a b\right )}}{b} + 1\right ) - \log \left (-\frac {i \, {\left (b^{2} x + a b\right )}}{b} + 1\right )\right )}}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )}} + \frac {{\left (3 \, a^{2} b^{2} + b^{2}\right )} \log \relax (x)}{2 \, {\left (a^{6} + 2 \, a^{4} + a^{2}\right )}} + \frac {{\left (a^{4} b^{2} - a^{2} b^{2}\right )} x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) + 2 \, {\left (a^{3} b + a b\right )} x + 2 \, {\left (a^{6} + 2 \, a^{4} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{2} + b^{2}\right )} x^{2} + a^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (a^{6} + 2 \, a^{4} + a^{2}\right )} \log \left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right )}{4 \, {\left (a^{6} + 2 \, a^{4} + a^{2}\right )} x^{2}} - \int \frac {b^{2} x + a b}{2 \, {\left (b^{2} x^{4} + 2 \, a b x^{3} + {\left (a^{2} + 1\right )} x^{2} + {\left (b^{2} x^{4} + 2 \, a b x^{3} + {\left (a^{2} + 1\right )} x^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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