Integrand size = 23, antiderivative size = 116 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]
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Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5859, 12, 5775, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=-\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e}+\frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {\log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5859
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x} \, dx,x,c+d x\right )}{d e} \\ & = -\frac {\text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {2 \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 \text {Subst}\left (\int x \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}+\frac {b \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\frac {-2 (a+b \text {arcsinh}(c+d x))^2 \left (a+b \text {arcsinh}(c+d x)-3 b \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )\right )+6 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )-3 b^3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c+d x)}\right )}{6 b d e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(311\) vs. \(2(136)=272\).
Time = 0.43 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.69
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(312\) |
default | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(312\) |
parts | \(\frac {a^{2} \ln \left (d x +c \right )}{e d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}\) | \(317\) |
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\frac {\int \frac {a^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{c\,e+d\,e\,x} \,d x \]
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