Optimal. Leaf size=58 \[ \frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}+\frac {c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.24, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5695, 5781, 5448, 3301} \[ \frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}+\frac {c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5448
Rule 5695
Rule 5781
Rubi steps
\begin {align*} \int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-(3 a c) \int \frac {x \sqrt {-1+a x} \sqrt {1+a x}}{\cosh ^{-1}(a x)} \, dx\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac {(3 c) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end {align*}
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Mathematica [B] time = 1.78, size = 217, normalized size = 3.74 \[ \frac {c \sqrt {a x-1} \left (-3 \left (a^2 x^2-1\right ) \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )-(a x-1) \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right ) \left (a x-4 \sqrt {a x-1} \sqrt {a x+1} \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )+1\right )+4 \left (\frac {a x-1}{a x+1}\right )^{5/2} (a x+1)^5+4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \log \left (\cosh ^{-1}(a x)\right ) \left (\sqrt {\frac {a x-1}{a x+1}} (a x+1)+(1-a x) \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )\right )}{4 a \left (\frac {a x-1}{a x+1}\right )^{3/2} (a x+1)^{5/2} \cosh ^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 63, normalized size = 1.09 \[ -\frac {c \left (3 \sqrt {a x -1}\, \sqrt {a x +1}+3 \Chi \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \Chi \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a \,\mathrm {arccosh}\left (a x \right )} \]
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 \[ \frac {a^{5} c x^{5} - 2 \, a^{3} c x^{3} + a c x + {\left (a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )} - \int \frac {3 \, a^{6} c x^{6} - 7 \, a^{4} c x^{4} + 5 \, a^{2} c x^{2} + {\left (3 \, a^{4} c x^{4} - 2 \, a^{2} c x^{2} - c\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 3 \, {\left (2 \, a^{5} c x^{5} - 3 \, a^{3} c x^{3} + a c x\right )} \sqrt {a x + 1} \sqrt {a x - 1} - c}{{\left (a^{4} x^{4} + {\left (a x + 1\right )} {\left (a x - 1\right )} a^{2} x^{2} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 1\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {c-a^2\,c\,x^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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 \[ - c \left (\int \frac {a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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