Optimal. Leaf size=209 \[ \frac {c \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {c^2 x^2+1}}+\frac {c \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {2 b c \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {b^2 c \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.25, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5737, 5659, 3716, 2190, 2279, 2391, 5675} \[ \frac {b^2 c \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {c \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {c^2 x^2+1}}-\frac {c \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {2 b c \sqrt {c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5675
Rule 5737
Rubi steps
\begin {align*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {\left (2 b c \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}+\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}-\frac {\left (4 b c \sqrt {d+c^2 d 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 {1+c^2 x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}+\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {2 b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}+\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {2 b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}+\frac {c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {2 b c \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.17, size = 232, normalized size = 1.11 \[ -\frac {a^2 \sqrt {c^2 d x^2+d}}{x}+a^2 c \sqrt {d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+\frac {a b \sqrt {c^2 d x^2+d} \left (-2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right )}{x \sqrt {c^2 x^2+1}}+\frac {b^2 c \sqrt {c^2 d x^2+d} \left (\sinh ^{-1}(c x) \left (\left (3-\frac {3 \sqrt {c^2 x^2+1}}{c x}\right ) \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2+6 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )\right )}{3 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 625, normalized size = 2.99 \[ -\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a^{2} c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a^{2} c^{2} d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3} c}{3 \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x \,c^{2}}{c^{2} x^{2}+1}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{x \left (c^{2} x^{2}+1\right )}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x \,c^{2}}{c^{2} x^{2}+1}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}} \]
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 \[ {\left (c \sqrt {d} \operatorname {arsinh}\left (c x\right ) - \frac {\sqrt {c^{2} d x^{2} + d}}{x}\right )} a^{2} + \int \frac {\sqrt {c^{2} d x^{2} + d} b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x^{2}} + \frac {2 \, \sqrt {c^{2} d x^{2} + d} a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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