Integrand size = 18, antiderivative size = 29 \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\frac {3 c \text {Shi}(\text {arccosh}(a x))}{4 a}-\frac {c \text {Shi}(3 \text {arccosh}(a x))}{4 a} \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5906, 3393, 3379} \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\frac {3 c \text {Shi}(\text {arccosh}(a x))}{4 a}-\frac {c \text {Shi}(3 \text {arccosh}(a x))}{4 a} \]
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Rule 3379
Rule 3393
Rule 5906
Rubi steps \begin{align*} \text {integral}& = -\frac {c \text {Subst}\left (\int \frac {\sinh ^3(x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {(i c) \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 x}-\frac {i \sinh (3 x)}{4 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{4 a}+\frac {(3 c) \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{4 a} \\ & = \frac {3 c \text {Shi}(\text {arccosh}(a x))}{4 a}-\frac {c \text {Shi}(3 \text {arccosh}(a x))}{4 a} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\frac {c (3 \text {Shi}(\text {arccosh}(a x))-\text {Shi}(3 \text {arccosh}(a x)))}{4 a} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {c \left (3 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{4 a}\) | \(24\) |
default | \(\frac {c \left (3 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{4 a}\) | \(24\) |
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=- c \left (\int \frac {a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {c-a^2 c x^2}{\text {arccosh}(a x)} \, dx=\int \frac {c-a^2\,c\,x^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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