Integrand size = 20, antiderivative size = 82 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=-\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\frac {5 c^2 \text {Chi}(\text {arccosh}(a x))}{8 a}-\frac {15 c^2 \text {Chi}(3 \text {arccosh}(a x))}{16 a}+\frac {5 c^2 \text {Chi}(5 \text {arccosh}(a x))}{16 a} \]
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Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5904, 5953, 5556, 3382} \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=\frac {5 c^2 \text {Chi}(\text {arccosh}(a x))}{8 a}-\frac {15 c^2 \text {Chi}(3 \text {arccosh}(a x))}{16 a}+\frac {5 c^2 \text {Chi}(5 \text {arccosh}(a x))}{16 a}-\frac {c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \text {arccosh}(a x)} \]
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Rule 3382
Rule 5556
Rule 5904
Rule 5953
Rubi steps \begin{align*} \text {integral}& = -\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\left (5 a c^2\right ) \int \frac {x (-1+a x)^{3/2} (1+a x)^{3/2}}{\text {arccosh}(a x)} \, dx \\ & = -\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\frac {\left (5 c^2\right ) \text {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,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{8 a}-\frac {\left (15 c^2\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{16 a} \\ & = -\frac {c^2 (-1+a x)^{5/2} (1+a x)^{5/2}}{a \text {arccosh}(a x)}+\frac {5 c^2 \text {Chi}(\text {arccosh}(a x))}{8 a}-\frac {15 c^2 \text {Chi}(3 \text {arccosh}(a x))}{16 a}+\frac {5 c^2 \text {Chi}(5 \text {arccosh}(a x))}{16 a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(82)=164\).
Time = 0.34 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.37 \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=-\frac {c^2 \left (16 \sqrt {\frac {-1+a x}{1+a x}}+16 a x \sqrt {\frac {-1+a x}{1+a x}}-32 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}}-32 a^3 x^3 \sqrt {\frac {-1+a x}{1+a x}}+16 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}+16 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}-10 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))+15 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))-5 \text {arccosh}(a x) \text {Chi}(5 \text {arccosh}(a x))\right )}{16 a \text {arccosh}(a x)} \]
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Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-15 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-10 \sqrt {a x -1}\, \sqrt {a x +1}+5 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{16 a \,\operatorname {arccosh}\left (a x \right )}\) | \(87\) |
default | \(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-15 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-10 \sqrt {a x -1}\, \sqrt {a x +1}+5 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{16 a \,\operatorname {arccosh}\left (a x \right )}\) | \(87\) |
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx\right ) \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^2}{\text {arccosh}(a x)^2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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