Optimal. Leaf size=167 \[ -\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 d x^2+d}}+\frac {2 b c \sqrt {c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 d x^2+d}}-\frac {b^2 c \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.23, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5723, 5659, 3716, 2190, 2279, 2391} \[ \frac {b^2 c \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac {c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 d x^2+d}}+\frac {2 b c \sqrt {c^2 x^2+1} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5723
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}-\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {\left (b^2 c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 168, normalized size = 1.01 \[ \frac {-a \left (a c^2 x^2+a-2 b c x \sqrt {c^2 x^2+1} \log (c x)\right )-2 b \sinh ^{-1}(c x) \left (a c^2 x^2+a-b c x \sqrt {c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-b^2 c x \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )+b^2 \left (-c^2 x^2+c x \sqrt {c^2 x^2+1}-1\right ) \sinh ^{-1}(c x)^2}{x \sqrt {c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{c^{2} d x^{4} + d x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 526, normalized size = 3.15 \[ -\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{d x}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x \,c^{2}}{\left (c^{2} x^{2}+1\right ) d}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{\left (c^{2} x^{2}+1\right ) d x}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}+1\right ) d}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{\left (c^{2} x^{2}+1\right ) d x}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}\, d} \]
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 \[ -\frac {{\left (\left (-1\right )^{2 \, c^{2} d x^{2} + 2 \, d} \sqrt {d} \log \left (2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) - \sqrt {d} \log \left (x^{2} + \frac {1}{c^{2}}\right )\right )} a b c}{d} + b^{2} \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{\sqrt {c^{2} d x^{2} + d} x^{2}}\,{d x} - \frac {2 \, \sqrt {c^{2} d x^{2} + d} a b \operatorname {arsinh}\left (c x\right )}{d x} - \frac {\sqrt {c^{2} d x^{2} + d} a^{2}}{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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