Integrand size = 26, antiderivative size = 351 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c} \]
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Time = 0.65 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5904, 5953, 5556, 3388, 2211, 2236, 2235} \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {5 \sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}} \]
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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^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {\left (10 c d^2\right ) \int \frac {x (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}} \, dx}{b} \\ & = -\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {\left (10 d^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\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^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {\left (10 d^2\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}+\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c} \\ & = -\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c}+\frac {\left (5 d^2\right ) \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 )}{4 b^2 c}-\frac {\left (15 d^2\right ) \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 )}{8 b^2 c} \\ & = -\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c}+\frac {\left (5 d^2\right ) \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 )}{8 b^2 c}+\frac {\left (5 d^2\right ) \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 )}{8 b^2 c}-\frac {\left (15 d^2\right ) \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 )}{16 b^2 c}-\frac {\left (15 d^2\right ) \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 )}{16 b^2 c} \\ & = -\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{8 b^2 c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{8 b^2 c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 b^2 c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{4 b^2 c}-\frac {\left (15 d^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{8 b^2 c}-\frac {\left (15 d^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{8 b^2 c} \\ & = -\frac {2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c} \\ \end{align*}
Time = 1.59 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {d^2 e^{-\frac {5 a}{b}} \left (20 e^{\frac {5 a}{b}} \sqrt {\frac {-1+c x}{1+c x}}+20 c e^{\frac {5 a}{b}} x \sqrt {\frac {-1+c x}{1+c x}}+10 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )-\sqrt {5} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+5 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-10 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )-5 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\sqrt {5} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-10 e^{\frac {5 a}{b}} \sinh (3 \text {arccosh}(c x))+2 e^{\frac {5 a}{b}} \sinh (5 \text {arccosh}(c x))\right )}{16 b c \sqrt {a+b \text {arccosh}(c x)}} \]
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\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \frac {1}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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