Optimal. Leaf size=36 \[ \frac {\text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}-\frac {(a+b x)^2+1}{b \sinh ^{-1}(a+b x)} \]
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Rubi [A] time = 0.12, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5867, 5696, 5669, 5448, 12, 3298} \[ \frac {\text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}-\frac {(a+b x)^2+1}{b \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 5448
Rule 5669
Rule 5696
Rule 5867
Rubi steps
\begin {align*} \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {2 \operatorname {Subst}\left (\int \frac {x}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {\text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 1.31 \[ -\frac {a^2-\sinh ^{-1}(a+b x) \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )+2 a b x+b^2 x^2+1}{b \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 44, normalized size = 1.22 \[ \frac {2 \Shi \left (2 \arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\cosh \left (2 \arcsinh \left (b x +a \right )\right )-1}{2 b \arcsinh \left (b x +a \right )} \]
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 \[ -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{2} + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b^{2} x + a b\right )} + b\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} + \int \frac {{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} - 1\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} b + b\right )} x + a\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + {\left (2 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} + 2 \, a^{4} + 3 \, {\left (4 \, a^{2} b^{2} + b^{2}\right )} x^{2} + 3 \, a^{2} + 2 \, {\left (4 \, a^{3} b + 3 \, a b\right )} x + 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + a^{4} + 2 \, {\left (3 \, a^{2} b^{2} + b^{2}\right )} x^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} + 2 \, a^{2} + 4 \, {\left (a^{3} b + a b\right )} x + 2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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