Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left ((a+b x)^2+1\right )^{3/2} \sinh ^{-1}(a+b x)},x\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \, {\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + a^{4} + 4 \, {\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \operatorname {arsinh}\left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}} \arcsinh \left (b x +a \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\mathrm {asinh}\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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