Optimal. Leaf size=78 \[ -\frac {6 \sqrt {1+(a+b x)^2}}{b}+\frac {6 (a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^3}{b} \]
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Rubi [A]
time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5772,
5798, 267} \begin {gather*} -\frac {6 \sqrt {(a+b x)^2+1}}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^3}{b}-\frac {3 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b}+\frac {6 (a+b x) \sinh ^{-1}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rule 5798
Rule 5858
Rubi steps
\begin {align*} \int \sinh ^{-1}(a+b x)^3 \, dx &=\frac {\text {Subst}\left (\int \sinh ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)^3}{b}-\frac {3 \text {Subst}\left (\int \frac {x \sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^3}{b}+\frac {6 \text {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {6 (a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^3}{b}-\frac {6 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {6 \sqrt {1+(a+b x)^2}}{b}+\frac {6 (a+b x) \sinh ^{-1}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^3}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 0.90 \begin {gather*} \frac {-6 \sqrt {1+(a+b x)^2}+6 (a+b x) \sinh ^{-1}(a+b x)-3 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2+(a+b x) \sinh ^{-1}(a+b x)^3}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.00, size = 67, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )-3 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \arcsinh \left (b x +a \right ) \left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}}{b}\) | \(67\) |
default | \(\frac {\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )-3 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \arcsinh \left (b x +a \right ) \left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}}{b}\) | \(67\) |
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 [A]
time = 0.48, size = 139, normalized size = 1.78 \begin {gather*} \frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 6 \, {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 109, normalized size = 1.40 \begin {gather*} \begin {cases} \frac {a \operatorname {asinh}^{3}{\left (a + b x \right )}}{b} + \frac {6 a \operatorname {asinh}{\left (a + b x \right )}}{b} + x \operatorname {asinh}^{3}{\left (a + b x \right )} + 6 x \operatorname {asinh}{\left (a + b x \right )} - \frac {3 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{b} - \frac {6 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asinh}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \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 {\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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