Integrand size = 18, antiderivative size = 58 \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \text {arccosh}(a x)}+\frac {3 c \text {Chi}(\text {arccosh}(a x))}{4 a}-\frac {3 c \text {Chi}(3 \text {arccosh}(a x))}{4 a} \]
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Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5904, 5953, 5556, 3382} \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\frac {3 c \text {Chi}(\text {arccosh}(a x))}{4 a}-\frac {3 c \text {Chi}(3 \text {arccosh}(a x))}{4 a}+\frac {c (a x-1)^{3/2} (a x+1)^{3/2}}{a \text {arccosh}(a x)} \]
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Rule 3382
Rule 5556
Rule 5904
Rule 5953
Rubi steps \begin{align*} \text {integral}& = \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \text {arccosh}(a x)}-(3 a c) \int \frac {x \sqrt {-1+a x} \sqrt {1+a x}}{\text {arccosh}(a x)} \, dx \\ & = \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \text {arccosh}(a x)}-\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \text {arccosh}(a x)}-\frac {(3 c) \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \text {arccosh}(a x)}+\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{4 a}-\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{4 a} \\ & = \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \text {arccosh}(a x)}+\frac {3 c \text {Chi}(\text {arccosh}(a x))}{4 a}-\frac {3 c \text {Chi}(3 \text {arccosh}(a x))}{4 a} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.12 \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\frac {c \left (4 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3+3 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))-3 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))\right )}{4 a \text {arccosh}(a x)} \]
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Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {c \left (3 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-3 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-3 \sqrt {a x -1}\, \sqrt {a x +1}+\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{4 a \,\operatorname {arccosh}\left (a x \right )}\) | \(61\) |
default | \(\frac {c \left (3 \,\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-3 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right ) \operatorname {arccosh}\left (a x \right )-3 \sqrt {a x -1}\, \sqrt {a x +1}+\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{4 a \,\operatorname {arccosh}\left (a x \right )}\) | \(61\) |
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=- c \left (\int \frac {a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)^2} \, dx=\int \frac {c-a^2\,c\,x^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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