Optimal. Leaf size=147 \[ -\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}-\frac {\left (2-17 a^2\right ) (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{12 b^4}+\frac {\left (1-2 a^2\right ) a \tanh ^{-1}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )}{2 b^4}-\frac {a (a+b x)^2 \sqrt {\frac {1}{(a+b x)^2}+1}}{3 b^4}+\frac {x^2 (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{12 b^2}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x) \]
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Rubi [A] time = 0.15, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6322, 5469, 3782, 4048, 3770, 3767, 8} \[ -\frac {\left (2-17 a^2\right ) (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{12 b^4}+\frac {\left (1-2 a^2\right ) a \tanh ^{-1}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )}{2 b^4}-\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}+\frac {x^2 (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {\frac {1}{(a+b x)^2}+1}}{3 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 4048
Rule 5469
Rule 6322
Rubi steps
\begin {align*} \int x^3 \text {csch}^{-1}(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x \coth (x) \text {csch}(x) (-a+\text {csch}(x))^3 \, dx,x,\text {csch}^{-1}(a+b x)\right )}{b^4}\\ &=\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\text {csch}(x))^4 \, dx,x,\text {csch}^{-1}(a+b x)\right )}{4 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\text {csch}(x)) \left (-3 a^3-\left (2-9 a^2\right ) \text {csch}(x)-8 a \text {csch}^2(x)\right ) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{12 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}}}{3 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \left (6 a^4+12 a \left (1-2 a^2\right ) \text {csch}(x)-2 \left (2-17 a^2\right ) \text {csch}^2(x)\right ) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{24 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)+\frac {\left (2-17 a^2\right ) \operatorname {Subst}\left (\int \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{12 b^4}-\frac {\left (a \left (1-2 a^2\right )\right ) \operatorname {Subst}\left (\int \text {csch}(x) \, dx,x,\text {csch}^{-1}(a+b x)\right )}{2 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)+\frac {a \left (1-2 a^2\right ) \tanh ^{-1}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{2 b^4}-\frac {\left (i \left (2-17 a^2\right )\right ) \operatorname {Subst}\left (\int 1 \, dx,x,-i (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}\right )}{12 b^4}\\ &=-\frac {\left (2-17 a^2\right ) (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^4}+\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)+\frac {a \left (1-2 a^2\right ) \tanh ^{-1}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 149, normalized size = 1.01 \[ \frac {-3 a^4 \sinh ^{-1}\left (\frac {1}{a+b x}\right )+6 \left (1-2 a^2\right ) a \log \left ((a+b x) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+1\right )\right )+\sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}} \left (13 a^3+9 a^2 b x-3 a b^2 x^2-2 a+b^3 x^3-2 b x\right )+3 b^4 x^4 \text {csch}^{-1}(a+b x)}{12 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 325, normalized size = 2.21 \[ \frac {3 \, b^{4} x^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 3 \, a^{4} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) + 3 \, a^{4} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + 6 \, {\left (2 \, a^{3} - a\right )} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} + 13 \, a^{3} + {\left (9 \, a^{2} - 2\right )} b x - 2 \, a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{12 \, b^{4}} \]
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 \[ \int x^{3} \operatorname {arcsch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 226, normalized size = 1.54 \[ \frac {\frac {\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right )^{3} a +\frac {3 \,\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right )^{2} a^{2}}{2}-\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right ) a^{3}+\frac {\mathrm {arccsch}\left (b x +a \right ) a^{4}}{4}+\frac {\sqrt {1+\left (b x +a \right )^{2}}\, \left (-3 a^{4} \arctanh \left (\frac {1}{\sqrt {1+\left (b x +a \right )^{2}}}\right )-12 \arcsinh \left (b x +a \right ) a^{3}+\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}-6 a \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+18 a^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 a \arcsinh \left (b x +a \right )-2 \sqrt {1+\left (b x +a \right )^{2}}\right )}{12 \sqrt {\frac {1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}} \]
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 \[ -\frac {{\left (-i \, a^{3} + i \, a\right )} {\left (\log \left (\frac {i \, {\left (b^{2} x + a b\right )}}{b} + 1\right ) - \log \left (-\frac {i \, {\left (b^{2} x + a b\right )}}{b} + 1\right )\right )}}{2 \, b^{4}} + \frac {2 \, b^{4} x^{4} \log \left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) + b^{2} x^{2} - 6 \, a b x - {\left (a^{4} - 6 \, a^{2} + 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, {\left (b^{4} x^{4} - a^{4}\right )} \log \left (b x + a\right )}{8 \, b^{4}} + \int \frac {b^{2} x^{5} + a b x^{4}}{4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + 1\right )}}\,{d x} \]
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 \[ \int x^3\,\mathrm {asinh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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 \[ \int x^{3} \operatorname {acsch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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