Optimal. Leaf size=276 \[ -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}-\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3} \]
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Rubi [A]
time = 0.86, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5886, 5951,
5887, 5556, 3389, 2211, 2236, 2235, 5881} \begin {gather*} -\frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}-\frac {\sqrt {3 \pi } e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {\sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5881
Rule 5886
Rule 5887
Rule 5951
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}+\frac {(2 c) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{b}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {12 \int \frac {x^2}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b^2}-\frac {8 \int \frac {1}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{3 b^2 c^2}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac {12 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {4 \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{3 b^3 c^3}-\frac {4 \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{3 b^3 c^3}+\frac {12 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {\sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {8 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{3 b^3 c^3}-\frac {8 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{3 b^3 c^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b^2 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}-\frac {4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b^2 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}-\frac {4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{3 b^{5/2} c^3}-\frac {3 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}-\frac {3 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}+\frac {3 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}+\frac {3 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 b c \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {8 x}{3 b^2 c^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{b^2 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}-\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{6 b^{5/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{5/2} c^3}\\ \end {align*}
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Mathematica [A]
time = 1.43, size = 340, normalized size = 1.23 \begin {gather*} \frac {e^{-3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )} \left (2 e^{\frac {4 a}{b}+3 \cosh ^{-1}(c x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )-6 \sqrt {3} b e^{3 \cosh ^{-1}(c x)} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-2 b e^{\frac {2 a}{b}+3 \cosh ^{-1}(c x)} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+e^{\frac {3 a}{b}} \left (-\left (\left (1+e^{2 \cosh ^{-1}(c x)}\right ) \left (a \left (6-4 e^{2 \cosh ^{-1}(c x)}+6 e^{4 \cosh ^{-1}(c x)}\right )+b \left (-1+6 \cosh ^{-1}(c x)-4 e^{2 \cosh ^{-1}(c x)} \cosh ^{-1}(c x)+e^{4 \cosh ^{-1}(c x)} \left (1+6 \cosh ^{-1}(c x)\right )\right )\right )\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )\right )}{12 b^2 c^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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