Optimal. Leaf size=75 \[ -\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {(a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{2 b^2}-\frac {a \tanh ^{-1}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )}{b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x) \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6322, 5469, 3773, 3770, 3767, 8} \[ -\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {(a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{2 b^2}-\frac {a \tanh ^{-1}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )}{b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3773
Rule 5469
Rule 6322
Rubi steps
\begin {align*} \int x \text {csch}^{-1}(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x \coth (x) \text {csch}(x) (-a+\text {csch}(x)) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\text {csch}(x))^2 \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 b^2}+\frac {a \operatorname {Subst}\left (\int \text {csch}(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {a \tanh ^{-1}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{b^2}+\frac {i \operatorname {Subst}\left (\int 1 \, dx,x,-i (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}\right )}{2 b^2}\\ &=\frac {(a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{2 b^2}-\frac {a^2 \text {csch}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {csch}^{-1}(a+b x)-\frac {a \tanh ^{-1}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 110, normalized size = 1.47 \[ \frac {(a+b x) \sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}-2 a \log \left ((a+b x) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+1\right )\right )+a^2 \left (-\sinh ^{-1}\left (\frac {1}{a+b x}\right )\right )+b^2 x^2 \text {csch}^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 285, normalized size = 3.80 \[ \frac {b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - a^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) + a^{2} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + 2 \, a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, b^{2}} \]
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 \[ \int x \operatorname {arcsch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 97, normalized size = 1.29 \[ \frac {\frac {\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\mathrm {arccsch}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {1+\left (b x +a \right )^{2}}\, \left (2 a \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{2}} \]
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 \[ \frac {i \, a {\left (\log \left (\frac {i \, {\left (b^{2} x + a b\right )}}{b} + 1\right ) - \log \left (-\frac {i \, {\left (b^{2} x + a b\right )}}{b} + 1\right )\right )}}{2 \, b^{2}} + \frac {2 \, b^{2} x^{2} \log \left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) - {\left (a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x + a\right )}{4 \, b^{2}} + \int \frac {b^{2} x^{3} + a b x^{2}}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + 1\right )}}\,{d x} \]
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 \[ \int x\,\mathrm {asinh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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 \[ \int x \operatorname {acsch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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