Optimal. Leaf size=291 \[ -\frac {\sqrt {1-c x} \text {Int}\left (\frac {c^2 x^2-1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt {c x-1}}-\frac {9 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 \sqrt {c x-1}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c x \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.87, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \int \frac {-1+c^2 x^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c \sqrt {1-c^2 x^2}\right ) \int \frac {-1+c^2 x^2}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {-1+c^2 x^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (3 i \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 (a+b x)}-\frac {i \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {-1+c^2 x^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (9 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {-1+c^2 x^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \int \frac {-1+c^2 x^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (9 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (9 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {9 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {9 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {-1+c^2 x^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 34.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b^{2} x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x \operatorname {arcosh}\left (c x\right ) + a^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left ({\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{5} x^{5} - 2 \, c^{3} x^{3} + c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{3} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x^{2} - a b c x + {\left (b^{2} c^{3} x^{3} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x^{2} - b^{2} c x\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} - \int \frac {{\left ({\left (3 \, c^{5} x^{5} - c^{3} x^{3} - 2 \, c x\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + {\left (6 \, c^{6} x^{6} - 7 \, c^{4} x^{4} + 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + 3 \, {\left (c^{7} x^{7} - 2 \, c^{5} x^{5} + c^{3} x^{3}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{6} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{4} - 2 \, a b c^{3} x^{4} + a b c x^{2} + 2 \, {\left (a b c^{4} x^{5} - a b c^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{6} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{4} - 2 \, b^{2} c^{3} x^{4} + b^{2} c x^{2} + 2 \, {\left (b^{2} c^{4} x^{5} - b^{2} c^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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