Integrand size = 23, antiderivative size = 155 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\frac {(a+b \text {arcsinh}(c+d x))^4}{4 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{4 d e} \]
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Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5775, 3797, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=-\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{2 d e}-\frac {3 b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{2 d e}+\frac {(a+b \text {arcsinh}(c+d x))^4}{4 b d e}+\frac {\log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3}{d e}-\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{4 d e} \]
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5859
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x} \, dx,x,c+d x\right )}{d e} \\ & = -\frac {\text {Subst}\left (\int x^3 \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))^4}{4 b d e}+\frac {2 \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^3}{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))^4}{4 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 \text {Subst}\left (\int x^2 \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))^4}{4 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}+\frac {(3 b) \text {Subst}\left (\int x \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))^4}{4 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^4}{4 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{4 d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^4}{4 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e}-\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{4 d e} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{b}+4 (a+b \text {arcsinh}(c+d x))^3 \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )+6 b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )-6 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c+d x)}\right )+3 b^3 \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(c+d x)}\right )}{4 d e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(555\) vs. \(2(179)=358\).
Time = 0.47 (sec) , antiderivative size = 556, normalized size of antiderivative = 3.59
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,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 {3 a^{2} 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}\) | \(556\) |
default | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {3 a \,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 {3 a^{2} 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}\) | \(556\) |
parts | \(\frac {a^{3} \ln \left (d x +c \right )}{e d}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {3 a \,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 {3 a^{2} 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}\) | \(564\) |
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\frac {\int \frac {a^{3}}{c + d x}\, dx + \int \frac {b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a^{2} b \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{c\,e+d\,e\,x} \,d x \]
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