Integrand size = 16, antiderivative size = 361 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3} \]
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Time = 1.12 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5886, 5951, 5885, 3388, 2211, 2236, 2235, 5880, 5953} \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {16 \sqrt {c x-1} \sqrt {c x+1}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5880
Rule 5885
Rule 5886
Rule 5951
Rule 5953
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}-\frac {4 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{5/2}} \, dx}{5 b c}+\frac {(6 c) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{5/2}} \, dx}{5 b} \\ & = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {12 \int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx}{5 b^2}-\frac {8 \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx}{15 b^2 c^2} \\ & = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 \text {Subst}\left (\int \left (-\frac {3 \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 )}{5 b^4 c^3}-\frac {16 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx}{15 b^3 c} \\ & = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}-\frac {16 \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 )}{15 b^4 c^3}+\frac {6 \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 )}{5 b^4 c^3}+\frac {18 \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 )}{5 b^4 c^3} \\ & = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}-\frac {8 \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 )}{15 b^4 c^3}-\frac {8 \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 )}{15 b^4 c^3}+\frac {3 \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 )}{5 b^4 c^3}+\frac {3 \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 )}{5 b^4 c^3}+\frac {9 \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 )}{5 b^4 c^3}+\frac {9 \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 )}{5 b^4 c^3} \\ & = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}-\frac {16 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{15 b^4 c^3}-\frac {16 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{15 b^4 c^3}+\frac {6 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{5 b^4 c^3}+\frac {6 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{5 b^4 c^3}+\frac {18 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{5 b^4 c^3}+\frac {18 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{5 b^4 c^3} \\ & = -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {8 x}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \text {arccosh}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3} \\ \end{align*}
Time = 1.69 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\frac {-6 b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-2 e^{-\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \left (-2 a+b-2 b \text {arccosh}(c x)+2 e^{\frac {a}{b}+\text {arccosh}(c x)} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 e^{-\frac {a}{b}} (a+b \text {arccosh}(c x)) \left (e^{\frac {a}{b}+\text {arccosh}(c x)} (2 a+b+2 b \text {arccosh}(c x))+2 b \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )-3 (a+b \text {arccosh}(c x)) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 e^{-3 \text {arccosh}(c x)} \left (b+6 a \left (-1+e^{6 \text {arccosh}(c x)}\right )-6 b \text {arccosh}(c x)+b e^{6 \text {arccosh}(c x)} (1+6 \text {arccosh}(c x))+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} (a+b \text {arccosh}(c x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )-6 b^2 \sinh (3 \text {arccosh}(c x))}{60 b^3 c^3 (a+b \text {arccosh}(c x))^{5/2}} \]
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\[\int \frac {x^{2}}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {7}{2}}}d x\]
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Exception generated. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{7/2}} \,d x \]
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