Optimal. Leaf size=166 \[ \frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-b d \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} b^2 d \text {PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5808, 5775,
3797, 2221, 2611, 2320, 6724, 5785, 5783, 30} \begin {gather*} \frac {1}{2} d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {1}{2} b c d x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-b d \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+d \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b^2 d \text {Li}_3\left (e^{-2 \sinh ^{-1}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5783
Rule 5785
Rule 5808
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+d \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-(b c d) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+d \text {Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{2} (b c d) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{2} \left (b^2 c^2 d\right ) \int x \, dx\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-(2 d) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-(2 b d) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\left (b^2 d\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \left (b^2 d\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} d \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} b^2 d \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 222, normalized size = 1.34 \begin {gather*} \frac {1}{8} d \left (4 a^2 c^2 x^2+8 a b c^2 x^2 \sinh ^{-1}(c x)-4 a b \left (c x \sqrt {1+c^2 x^2}-\tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )+b^2 \left (1+2 \sinh ^{-1}(c x)^2\right ) \cosh \left (2 \sinh ^{-1}(c x)\right )+8 a b \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )+8 a^2 \log (x)-8 a b \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+8 b^2 \left (-\frac {1}{3} \sinh ^{-1}(c x)^3+\sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )\right )-2 b^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs.
\(2(179)=358\).
time = 3.69, size = 425, normalized size = 2.56
method | result | size |
derivativedivides | \(\frac {a^{2} c^{2} d \,x^{2}}{2}+a^{2} d \ln \left (c x \right )-\frac {b^{2} d \arcsinh \left (c x \right )^{3}}{3}+\frac {b^{2} d \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2}-\frac {b^{2} d \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2}+\frac {b^{2} d \arcsinh \left (c x \right )^{2}}{4}+\frac {d \,b^{2} c^{2} x^{2}}{4}+\frac {b^{2} d}{8}+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-b d a \arcsinh \left (c x \right )^{2}+b d a \arcsinh \left (c x \right ) c^{2} x^{2}-\frac {b d a c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {b d a \arcsinh \left (c x \right )}{2}+2 b d a \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 b d a \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )\) | \(425\) |
default | \(\frac {a^{2} c^{2} d \,x^{2}}{2}+a^{2} d \ln \left (c x \right )-\frac {b^{2} d \arcsinh \left (c x \right )^{3}}{3}+\frac {b^{2} d \arcsinh \left (c x \right )^{2} c^{2} x^{2}}{2}-\frac {b^{2} d \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2}+\frac {b^{2} d \arcsinh \left (c x \right )^{2}}{4}+\frac {d \,b^{2} c^{2} x^{2}}{4}+\frac {b^{2} d}{8}+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-b d a \arcsinh \left (c x \right )^{2}+b d a \arcsinh \left (c x \right ) c^{2} x^{2}-\frac {b d a c x \sqrt {c^{2} x^{2}+1}}{2}+\frac {b d a \arcsinh \left (c x \right )}{2}+2 b d a \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 b d a \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 b d a \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )\) | \(425\) |
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*} d \left (\int \frac {a^{2}}{x}\, dx + \int a^{2} c^{2} x\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{2} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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