Optimal. Leaf size=15 \[ \frac {\sinh ^{-1}(a+b x)^3}{3 b} \]
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Rubi [A] time = 0.07, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {5867, 5675} \[ \frac {\sinh ^{-1}(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 5675
Rule 5867
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)^2}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sinh ^{-1}(a+b x)^3}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \[ \frac {\sinh ^{-1}(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 32, normalized size = 2.13 \[ \frac {\log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3}}{3 \, b} \]
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 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 14, normalized size = 0.93 \[ \frac {\arcsinh \left (b x +a \right )^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.78, size = 132, normalized size = 8.80 \[ \frac {\operatorname {arsinh}\left (b x + a\right )^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \frac {\operatorname {arsinh}\left (b x + a\right ) \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{2}}{b} + \frac {\operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 13, normalized size = 0.87 \[ \frac {{\mathrm {asinh}\left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 26, normalized size = 1.73 \[ \begin {cases} \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\\frac {x \operatorname {asinh}^{2}{\relax (a )}}{\sqrt {a^{2} + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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