Integrand size = 20, antiderivative size = 98 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}+\frac {35 c^3 \text {Chi}(\text {arccosh}(a x))}{64 a}-\frac {63 c^3 \text {Chi}(3 \text {arccosh}(a x))}{64 a}+\frac {35 c^3 \text {Chi}(5 \text {arccosh}(a x))}{64 a}-\frac {7 c^3 \text {Chi}(7 \text {arccosh}(a x))}{64 a} \]
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Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5904, 5953, 5556, 3382} \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\frac {35 c^3 \text {Chi}(\text {arccosh}(a x))}{64 a}-\frac {63 c^3 \text {Chi}(3 \text {arccosh}(a x))}{64 a}+\frac {35 c^3 \text {Chi}(5 \text {arccosh}(a x))}{64 a}-\frac {7 c^3 \text {Chi}(7 \text {arccosh}(a x))}{64 a}+\frac {c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \text {arccosh}(a x)} \]
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Rule 3382
Rule 5556
Rule 5904
Rule 5953
Rubi steps \begin{align*} \text {integral}& = \frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}-\left (7 a c^3\right ) \int \frac {x (-1+a x)^{5/2} (1+a x)^{5/2}}{\text {arccosh}(a x)} \, dx \\ & = \frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^6(x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = \frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \left (-\frac {5 \cosh (x)}{64 x}+\frac {9 \cosh (3 x)}{64 x}-\frac {5 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = \frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a}-\frac {\left (63 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a} \\ & = \frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \text {arccosh}(a x)}+\frac {35 c^3 \text {Chi}(\text {arccosh}(a x))}{64 a}-\frac {63 c^3 \text {Chi}(3 \text {arccosh}(a x))}{64 a}+\frac {35 c^3 \text {Chi}(5 \text {arccosh}(a x))}{64 a}-\frac {7 c^3 \text {Chi}(7 \text {arccosh}(a x))}{64 a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(257\) vs. \(2(98)=196\).
Time = 0.53 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.62 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\frac {c^3 \left (-64 \sqrt {\frac {-1+a x}{1+a x}}-64 a x \sqrt {\frac {-1+a x}{1+a x}}+192 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}}+192 a^3 x^3 \sqrt {\frac {-1+a x}{1+a x}}-192 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-192 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}+64 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}}+64 a^7 x^7 \sqrt {\frac {-1+a x}{1+a x}}+35 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))-63 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))+35 \text {arccosh}(a x) \text {Chi}(5 \text {arccosh}(a x))-7 \text {arccosh}(a x) \text {Chi}(7 \text {arccosh}(a x))\right )}{64 a \text {arccosh}(a x)} \]
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Time = 0.85 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-63 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+35 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-7 \,\operatorname {Chi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-35 \sqrt {a x -1}\, \sqrt {a x +1}+21 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-7 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )+\sinh \left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arccosh}\left (a x \right )}\) | \(107\) |
default | \(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-63 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )+35 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-7 \,\operatorname {Chi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-35 \sqrt {a x -1}\, \sqrt {a x +1}+21 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )-7 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )+\sinh \left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a \,\operatorname {arccosh}\left (a x \right )}\) | \(107\) |
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)^2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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