Integrand size = 40, antiderivative size = 261 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \]
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Time = 0.23 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6813, 5775, 3797, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{2 c}+\frac {3 b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^4}{4 b c}-\frac {\log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{4 c} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 6724
Rule 6744
Rule 6813
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = \frac {\text {Subst}\left (\int x^3 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\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}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\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}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\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}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}-\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}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{2 c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\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}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{b}\right )}\right )}{4 c} \\ & = -\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{b}\right )}\right )}{4 c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{b}-4 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )-6 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )+6 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )-3 b^3 \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1112\) vs. \(2(285)=570\).
Time = 1.40 (sec) , antiderivative size = 1113, normalized size of antiderivative = 4.26
method | result | size |
default | \(\text {Expression too large to display}\) | \(1113\) |
parts | \(\text {Expression too large to display}\) | \(1113\) |
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\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=- \int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{3} \operatorname {asinh}^{3}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a^{2} b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
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\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]
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