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 = 0.10, 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 = {5860, 5791,
3393, 3382} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 5791
Rule 5860
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 {\text {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\text {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 {\text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b}+\frac {\text {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 = 0.18, size = 37, normalized size = 0.79 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.96, size = 36, normalized size = 0.77
method | result | size |
default | \(\frac {3 \ln \left (\arcsinh \left (b x +a \right )\right )+4 \hyperbolicCosineIntegral \left (2 \arcsinh \left (b x +a \right )\right )+\hyperbolicCosineIntegral \left (4 \arcsinh \left (b x +a \right )\right )}{8 b}\) | \(36\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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