Integrand size = 29, antiderivative size = 186 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5917, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {c \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5917
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx}{\sqrt {d-c^2 d x^2}} \\ & = -\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}+\frac {\left (2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}} \\ & = -\frac {c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}+\frac {\left (4 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}} \\ & = -\frac {c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}} \\ & = -\frac {c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{\sqrt {d-c^2 d x^2}} \\ & = -\frac {c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{d x}-\frac {2 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{\sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {a^2 \sqrt {-d \left (-1+c^2 x^2\right )}}{d x}-2 a b c \left (\frac {\sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{c d x}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\log \left (-1+\sqrt {1+c x}\right )+\log \left (1+\sqrt {1+c x}\right )\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {b^2 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{c x}-2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{\sqrt {-d (-1+c x) (1+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(475\) vs. \(2(192)=384\).
Time = 1.15 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.56
method | result | size |
default | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right ) d}-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} c}{d \left (c^{2} x^{2}-1\right )}+\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\) | \(476\) |
parts | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right ) d}-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2} c}{d \left (c^{2} x^{2}-1\right )}+\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {2 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \operatorname {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\right )\) | \(476\) |
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \]
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