Integrand size = 29, antiderivative size = 341 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
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Time = 0.46 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {5932, 5899, 5913, 3797, 2221, 2317, 2438, 5912, 5916, 5569, 4267} \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {4 b c \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 4267
Rule 5569
Rule 5899
Rule 5912
Rule 5913
Rule 5916
Rule 5932
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\left (2 c^2\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x (-1+c x) (1+c x)} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 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 )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 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 )}{d \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 \text {arccosh}(c x)}\right )}{d \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 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 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 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {4 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 )}{d \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 )}{d \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 )}{d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.37 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a b \left (-1+2 c^2 x^2\right ) \text {arccosh}(c x)+b^2 \text {arccosh}(c x)^2+a \left (-a+2 a c^2 x^2-2 b c x \sqrt {-1+c x} \sqrt {1+c x} \log (c x)-b c x \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )\right )}{d x \sqrt {d-c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1270\) vs. \(2(359)=718\).
Time = 1.48 (sec) , antiderivative size = 1271, normalized size of antiderivative = 3.73
method | result | size |
default | \(\text {Expression too large to display}\) | \(1271\) |
parts | \(\text {Expression too large to display}\) | \(1271\) |
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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