Integrand size = 19, antiderivative size = 67 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\frac {35 c^3 \text {Chi}(\text {arcsinh}(a x))}{64 a}+\frac {21 c^3 \text {Chi}(3 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Chi}(5 \text {arcsinh}(a x))}{64 a}+\frac {c^3 \text {Chi}(7 \text {arcsinh}(a x))}{64 a} \]
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Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5791, 3393, 3382} \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\frac {35 c^3 \text {Chi}(\text {arcsinh}(a x))}{64 a}+\frac {21 c^3 \text {Chi}(3 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Chi}(5 \text {arcsinh}(a x))}{64 a}+\frac {c^3 \text {Chi}(7 \text {arcsinh}(a x))}{64 a} \]
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Rule 3382
Rule 3393
Rule 5791
Rubi steps \begin{align*} \text {integral}& = \frac {c^3 \text {Subst}\left (\int \frac {\cosh ^7(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = \frac {c^3 \text {Subst}\left (\int \left (\frac {35 \cosh (x)}{64 x}+\frac {21 \cosh (3 x)}{64 x}+\frac {7 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a} \\ & = \frac {c^3 \text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a}+\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a}+\frac {\left (21 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a} \\ & = \frac {35 c^3 \text {Chi}(\text {arcsinh}(a x))}{64 a}+\frac {21 c^3 \text {Chi}(3 \text {arcsinh}(a x))}{64 a}+\frac {7 c^3 \text {Chi}(5 \text {arcsinh}(a x))}{64 a}+\frac {c^3 \text {Chi}(7 \text {arcsinh}(a x))}{64 a} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\frac {c^3 (35 \text {Chi}(\text {arcsinh}(a x))+21 \text {Chi}(3 \text {arcsinh}(a x))+7 \text {Chi}(5 \text {arcsinh}(a x))+\text {Chi}(7 \text {arcsinh}(a x)))}{64 a} \]
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Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )+21 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )+7 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )+\operatorname {Chi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{64 a}\) | \(42\) |
default | \(\frac {c^{3} \left (35 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )+21 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )+7 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )+\operatorname {Chi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right )\right )}{64 a}\) | \(42\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}{\left (a x \right )}}\, dx\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\text {arcsinh}(a x)} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{\mathrm {asinh}\left (a\,x\right )} \,d x \]
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