Optimal. Leaf size=361 \[ \frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {16 \sqrt {c x-1} \sqrt {c x+1}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \]
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Rubi [A] time = 1.63, antiderivative size = 361, 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 = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5656, 5781} \[ \frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {c x-1} \sqrt {c x+1}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {24 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5656
Rule 5666
Rule 5668
Rule 5775
Rule 5781
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{7/2}} \, dx &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}-\frac {4 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b c}+\frac {(6 c) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {12 \int \frac {x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{5 b^2}-\frac {8 \int \frac {1}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{15 b^2 c^2}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}-\frac {16 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{15 b^3 c}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {18 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^3}-\frac {8 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac {3 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {3 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {9 \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {9 \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {16 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{15 b^4 c^3}-\frac {16 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{15 b^4 c^3}+\frac {6 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {6 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {18 \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {18 \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \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 )}{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 \cosh ^{-1}(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 \cosh ^{-1}(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 \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}\\ \end {align*}
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Mathematica [A] time = 2.59, size = 394, normalized size = 1.09 \[ \frac {-2 e^{-\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \left (2 e^{\frac {a}{b}+\cosh ^{-1}(c x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )-2 a-2 b \cosh ^{-1}(c x)+b\right )-2 e^{-\frac {a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (e^{\frac {a}{b}+\cosh ^{-1}(c x)} \left (2 a+2 b \cosh ^{-1}(c x)+b\right )+2 b \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )\right )-3 \left (a+b \cosh ^{-1}(c x)\right ) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2 e^{-3 \cosh ^{-1}(c x)} \left (6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+6 a \left (e^{6 \cosh ^{-1}(c x)}-1\right )-6 b \cosh ^{-1}(c x)+b e^{6 \cosh ^{-1}(c x)} \left (6 \cosh ^{-1}(c x)+1\right )+b\right )\right )-6 b^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1)-6 b^2 \sinh \left (3 \cosh ^{-1}(c x)\right )}{60 b^3 c^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{7/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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