Optimal. Leaf size=55 \[ -2 \text {Int}\left (\frac {a+b x}{\left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)},x\right )-\frac {1}{b \left ((a+b x)^2+1\right ) \sinh ^{-1}(a+b x)} \]
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Rubi [A] time = 0.12, 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 = {} \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^2} \, dx \]
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)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1}{b \left (1+(a+b x)^2\right ) \sinh ^{-1}(a+b x)}-\frac {2 \operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2 \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ \end {align*}
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Mathematica [A] time = 2.92, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \, {\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + a^{4} + 4 \, {\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \operatorname {arsinh}\left (b x + a\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}} \arcsinh \left (b x +a \right )^{2}}\, dx \]
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 \[ -\frac {b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} {\left (b^{2} x + a b\right )} + {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + b\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} - \int \frac {2 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} + 2 \, a^{4} + {\left (12 \, a^{2} b^{2} + b^{2}\right )} x^{2} + {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} + 1\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + a^{2} + 2 \, {\left (4 \, a^{3} b + a b\right )} x + 2 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} b + b\right )} x + a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1}{{\left ({\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + a^{4} + {\left (6 \, a^{2} b^{2} + b^{2}\right )} x^{2} + a^{2} + 2 \, {\left (2 \, a^{3} b + a b\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + a^{5} + 2 \, {\left (5 \, a^{2} b^{3} + b^{3}\right )} x^{3} + 2 \, a^{3} + 2 \, {\left (5 \, a^{3} b^{2} + 3 \, a b^{2}\right )} x^{2} + {\left (5 \, a^{4} b + 6 \, a^{2} b + b\right )} x + a\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + a^{6} + 3 \, {\left (5 \, a^{2} b^{4} + b^{4}\right )} x^{4} + 3 \, a^{4} + 4 \, {\left (5 \, a^{3} b^{3} + 3 \, a b^{3}\right )} x^{3} + 3 \, {\left (5 \, a^{4} b^{2} + 6 \, a^{2} b^{2} + b^{2}\right )} x^{2} + 3 \, a^{2} + 6 \, {\left (a^{5} b + 2 \, a^{3} b + a b\right )} x + 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {asinh}\left (a+b\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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