Optimal. Leaf size=147 \[ -\frac {\left (2-17 a^2\right ) (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^4}+\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)+\frac {a \left (1-2 a^2\right ) \tanh ^{-1}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{2 b^4} \]
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Rubi [A]
time = 0.11, 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 = {6457, 5577,
3867, 4133, 3855, 3852, 8} \begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3867
Rule 4133
Rule 5577
Rule 6457
Rubi steps
\begin {align*} \int x^3 \text {csch}^{-1}(a+b x) \, dx &=-\frac {\text {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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 = 0.19, size = 149, normalized size = 1.01 \begin {gather*} \frac {\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (-2 a+13 a^3-2 b x+9 a^2 b x-3 a b^2 x^2+b^3 x^3\right )+3 b^4 x^4 \text {csch}^{-1}(a+b x)-3 a^4 \sinh ^{-1}\left (\frac {1}{a+b x}\right )+6 a \left (1-2 a^2\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 227, normalized size = 1.54
method | result | size |
derivativedivides | \(\frac {\frac {\mathrm {arccsch}\left (b x +a \right ) a^{4}}{4}-\mathrm {arccsch}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\mathrm {arccsch}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\mathrm {arccsch}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (3 a^{4} \arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right )+12 a^{3} \arcsinh \left (b x +a \right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}+1}+6 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}+1}-\left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}+1}-6 a \arcsinh \left (b x +a \right )+2 \sqrt {\left (b x +a \right )^{2}+1}\right )}{12 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}}\) | \(227\) |
default | \(\frac {\frac {\mathrm {arccsch}\left (b x +a \right ) a^{4}}{4}-\mathrm {arccsch}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\mathrm {arccsch}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\mathrm {arccsch}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\mathrm {arccsch}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (3 a^{4} \arctanh \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right )+12 a^{3} \arcsinh \left (b x +a \right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}+1}+6 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}+1}-\left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}+1}-6 a \arcsinh \left (b x +a \right )+2 \sqrt {\left (b x +a \right )^{2}+1}\right )}{12 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}}\) | \(227\) |
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. 325 vs.
\(2 (127) = 254\).
time = 0.38, size = 325, normalized size = 2.21 \begin {gather*} \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}} \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 x^{3} \operatorname {acsch}{\left (a + b x \right )}\, 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 x^3\,\mathrm {asinh}\left (\frac {1}{a+b\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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