Optimal. Leaf size=82 \[ \frac {5 c^2 \text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {5 c^2 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a}-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \cosh ^{-1}(a x)} \]
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Rubi [A] time = 0.31, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5695, 5781, 5448, 3301} \[ \frac {5 c^2 \text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {5 c^2 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a}-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \cosh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5448
Rule 5695
Rule 5781
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^2}{\cosh ^{-1}(a x)^2} \, dx &=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \cosh ^{-1}(a x)}+\left (5 a c^2\right ) \int \frac {x (-1+a x)^{3/2} (1+a x)^{3/2}}{\cosh ^{-1}(a x)} \, dx\\ &=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \cosh ^{-1}(a x)}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \cosh ^{-1}(a x)}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {\cosh (x)}{8 x}-\frac {3 \cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \cosh ^{-1}(a x)}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a}-\frac {\left (15 c^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}\\ &=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \cosh ^{-1}(a x)}+\frac {5 c^2 \text {Chi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {5 c^2 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [B] time = 0.32, size = 194, normalized size = 2.37 \[ -\frac {c^2 \left (16 a^5 x^5 \sqrt {\frac {a x-1}{a x+1}}+16 a^4 x^4 \sqrt {\frac {a x-1}{a x+1}}-32 a^3 x^3 \sqrt {\frac {a x-1}{a x+1}}-32 a^2 x^2 \sqrt {\frac {a x-1}{a x+1}}-10 \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )+15 \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )-5 \cosh ^{-1}(a x) \text {Chi}\left (5 \cosh ^{-1}(a x)\right )+16 a x \sqrt {\frac {a x-1}{a x+1}}+16 \sqrt {\frac {a x-1}{a x+1}}\right )}{16 a \cosh ^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}{\operatorname {arcosh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 87, normalized size = 1.06 \[ \frac {c^{2} \left (10 \Chi \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-15 \Chi \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )+5 \Chi \left (5 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-10 \sqrt {a x -1}\, \sqrt {a x +1}+5 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )-\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{16 a \,\mathrm {arccosh}\left (a x \right )} \]
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 \[ -\frac {a^{7} c^{2} x^{7} - 3 \, a^{5} c^{2} x^{5} + 3 \, a^{3} c^{2} x^{3} - a c^{2} x + {\left (a^{6} c^{2} x^{6} - 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} - c^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{{\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )} + \int \frac {5 \, a^{8} c^{2} x^{8} - 16 \, a^{6} c^{2} x^{6} + 18 \, a^{4} c^{2} x^{4} - 8 \, a^{2} c^{2} x^{2} + {\left (5 \, a^{6} c^{2} x^{6} - 9 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 5 \, {\left (2 \, a^{7} c^{2} x^{7} - 5 \, a^{5} c^{2} x^{5} + 4 \, a^{3} c^{2} x^{3} - a c^{2} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + c^{2}}{{\left (a^{4} x^{4} + {\left (a x + 1\right )} {\left (a x - 1\right )} a^{2} x^{2} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 1\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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 \[ c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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