Integrand size = 24, antiderivative size = 233 \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c} \]
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Time = 0.45 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5904, 5953, 5556, 3388, 2211, 2236, 2235} \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {3 \sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {\sqrt {3 \pi } d e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 \sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {\sqrt {3 \pi } d e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5904
Rule 5953
Rubi steps \begin{align*} \text {integral}& = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {(6 c d) \int \frac {x \sqrt {-1+c x} \sqrt {1+c x}}{\sqrt {a+b \text {arccosh}(c x)}} \, dx}{b} \\ & = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {(6 d) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c} \\ & = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {(6 d) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c} \\ & = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {(3 d) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c}+\frac {(3 d) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c} \\ & = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {(3 d) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c}+\frac {(3 d) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c}-\frac {(3 d) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c}-\frac {(3 d) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c} \\ & = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {(3 d) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c}+\frac {(3 d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c}+\frac {(3 d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c}-\frac {(3 d) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c} \\ & = \frac {2 d (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c} \\ \end{align*}
Time = 1.10 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.06 \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (-3 d e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-\sqrt {3} d \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+d e^{\frac {2 a}{b}} \left (3 \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+e^{a/b} \left (-6 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+\sqrt {3} e^{\frac {3 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 \sinh (3 \text {arccosh}(c x))\right )\right )\right )}{4 b c \sqrt {a+b \text {arccosh}(c x)}} \]
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\[\int \frac {-c^{2} d \,x^{2}+d}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx\right ) \]
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\[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {d-c^2 d x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {d-c^2\,d\,x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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