Optimal. Leaf size=114 \[ \frac {b (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac {b^2 \text {csch}^{-1}(a+b x)}{2 a^2}-\frac {\text {csch}^{-1}(a+b x)}{2 x^2}-\frac {\left (1+2 a^2\right ) b^2 \tanh ^{-1}\left (\frac {a+\tanh \left (\frac {1}{2} \text {csch}^{-1}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}} \]
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Rubi [A]
time = 0.15, 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 = {6457, 5577,
3870, 4004, 3916, 2739, 632, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 2739
Rule 3870
Rule 3916
Rule 4004
Rule 5577
Rule 6457
Rubi steps
\begin {align*} \int \frac {\text {csch}^{-1}(a+b x)}{x^3} \, dx &=-\left (b^2 \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.31, size = 220, normalized size = 1.93 \begin {gather*} \frac {1}{2} \left (\frac {b (a+b x) \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}}{a \left (1+a^2\right ) x}-\frac {\text {csch}^{-1}(a+b x)}{x^2}+\frac {b^2 \sinh ^{-1}\left (\frac {1}{a+b x}\right )}{a^2}+\frac {\left (1+2 a^2\right ) b^2 \log (x)}{a^2 \left (1+a^2\right )^{3/2}}-\frac {\left (1+2 a^2\right ) b^2 \log \left (1+a^2+a b x+a \sqrt {1+a^2} \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+\sqrt {1+a^2} b x \sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs.
\(2(100)=200\).
time = 0.46, size = 456, normalized size = 4.00
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\mathrm {arccsch}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (-\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{3}+\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )+2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{5}-2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{4} \left (b x +a \right )-\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a +\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} \left (b x +a \right )+\left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {\left (b x +a \right )^{2}+1}\, a +3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{2} \left (b x +a \right )+a \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right )-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )^{\frac {5}{2}} b x}\right )\) | \(456\) |
default | \(b^{2} \left (-\frac {\mathrm {arccsch}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (-\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{3}+\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )+2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{5}-2 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{4} \left (b x +a \right )-\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} a +\arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right ) \left (a^{2}+1\right )^{\frac {3}{2}} \left (b x +a \right )+\left (a^{2}+1\right )^{\frac {3}{2}} \sqrt {\left (b x +a \right )^{2}+1}\, a +3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{3}-3 \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) a^{2} \left (b x +a \right )+a \ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right )-\ln \left (\frac {2 \sqrt {a^{2}+1}\, \sqrt {\left (b x +a \right )^{2}+1}+2 a \left (b x +a \right )+2}{b x}\right ) \left (b x +a \right )\right )}{2 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}+1\right )^{\frac {5}{2}} b x}\right )\) | \(456\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (100) = 200\).
time = 0.39, size = 461, normalized size = 4.04 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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