Optimal. Leaf size=45 \[ \frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}-\frac {2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b}+2 x \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5863, 5653, 5717, 8} \[ \frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}-\frac {2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b}+2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 5653
Rule 5717
Rule 5863
Rubi steps
\begin {align*} \int \sinh ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int \frac {x \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}+\frac {2 \operatorname {Subst}(\int 1 \, dx,x,a+b x)}{b}\\ &=2 x-\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.04 \[ \frac {2 (a+b x)+(a+b x) \sinh ^{-1}(a+b x)^2-2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 88, normalized size = 1.96 \[ \frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 2 \, b x - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{b} \]
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 \[ \int \operatorname {arsinh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 1.02 \[ \frac {\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a}{b} \]
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 \[ x \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - \int \frac {2 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} b + b\right )} x + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b^{2} x^{2} + a b x\right )}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {asinh}\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 63, normalized size = 1.40 \[ \begin {cases} \frac {a \operatorname {asinh}^{2}{\left (a + b x \right )}}{b} + x \operatorname {asinh}^{2}{\left (a + b x \right )} + 2 x - \frac {2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asinh}^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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