Integrand size = 20, antiderivative size = 67 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\frac {35 c^3 \text {Shi}(\text {arccosh}(a x))}{64 a}-\frac {21 c^3 \text {Shi}(3 \text {arccosh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(5 \text {arccosh}(a x))}{64 a}-\frac {c^3 \text {Shi}(7 \text {arccosh}(a x))}{64 a} \]
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Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5906, 3393, 3379} \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\frac {35 c^3 \text {Shi}(\text {arccosh}(a x))}{64 a}-\frac {21 c^3 \text {Shi}(3 \text {arccosh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(5 \text {arccosh}(a x))}{64 a}-\frac {c^3 \text {Shi}(7 \text {arccosh}(a x))}{64 a} \]
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Rule 3379
Rule 3393
Rule 5906
Rubi steps \begin{align*} \text {integral}& = -\frac {c^3 \text {Subst}\left (\int \frac {\sinh ^7(x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {\left (i c^3\right ) \text {Subst}\left (\int \left (\frac {35 i \sinh (x)}{64 x}-\frac {21 i \sinh (3 x)}{64 x}+\frac {7 i \sinh (5 x)}{64 x}-\frac {i \sinh (7 x)}{64 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {c^3 \text {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a}+\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a}-\frac {\left (21 c^3\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a} \\ & = \frac {35 c^3 \text {Shi}(\text {arccosh}(a x))}{64 a}-\frac {21 c^3 \text {Shi}(3 \text {arccosh}(a x))}{64 a}+\frac {7 c^3 \text {Shi}(5 \text {arccosh}(a x))}{64 a}-\frac {c^3 \text {Shi}(7 \text {arccosh}(a x))}{64 a} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\frac {c^3 (35 \text {Shi}(\text {arccosh}(a x))-21 \text {Shi}(3 \text {arccosh}(a x))+7 \text {Shi}(5 \text {arccosh}(a x))-\text {Shi}(7 \text {arccosh}(a x)))}{64 a} \]
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Time = 0.54 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-21 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+7 \,\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )-\operatorname {Shi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a}\) | \(44\) |
default | \(\frac {c^{3} \left (35 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )-21 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+7 \,\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )-\operatorname {Shi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right )\right )}{64 a}\) | \(44\) |
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx\right ) \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\text {arccosh}(a x)} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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