Optimal. Leaf size=47 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b} \]
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Rubi [A] time = 0.14, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5867, 5699, 3312, 3301} \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5699
Rule 5867
Rubi steps
\begin {align*} \int \frac {\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 {\left (1+x^2\right )^{3/2}}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\text {Chi}\left (4 \sinh ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 37, normalized size = 0.79 \[ \frac {4 \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )+\text {Chi}\left (4 \sinh ^{-1}(a+b x)\right )+3 \log \left (\sinh ^{-1}(a+b x)\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 42, normalized size = 0.89 \[ \frac {\Chi \left (2 \arcsinh \left (b x +a \right )\right )}{2 b}+\frac {\Chi \left (4 \arcsinh \left (b x +a \right )\right )}{8 b}+\frac {3 \ln \left (\arcsinh \left (b x +a \right )\right )}{8 b} \]
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 \[ \int \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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