Integrand size = 14, antiderivative size = 145 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=384 b^4 x-\frac {192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+48 b^2 x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \]
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Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6001, 8} \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=-\frac {192 b^3 \left (d x^4+2 x^2\right ) \left (a+b \text {arccosh}\left (d x^2+1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2+2}}+48 b^2 x \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^2+x \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^4-\frac {8 b \left (d x^4+2 x^2\right ) \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2+2}}+384 b^4 x \]
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Rule 8
Rule 6001
Rubi steps \begin{align*} \text {integral}& = -\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \, dx \\ & = -\frac {192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+48 b^2 x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx \\ & = 384 b^4 x-\frac {192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+48 b^2 x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.82 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=\frac {\left (a^4+48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2+24 b^2\right ) \sqrt {d x^2} \sqrt {2+d x^2}+4 b \left (a^3 d x^2+24 a b^2 d x^2-6 a^2 b \sqrt {d x^2} \sqrt {2+d x^2}-48 b^3 \sqrt {d x^2} \sqrt {2+d x^2}\right ) \text {arccosh}\left (1+d x^2\right )+6 b^2 \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {2+d x^2}\right ) \text {arccosh}\left (1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {2+d x^2}\right ) \text {arccosh}\left (1+d x^2\right )^3+b^4 d x^2 \text {arccosh}\left (1+d x^2\right )^4}{d x} \]
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\[\int {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}+1\right )\right )}^{4}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (137) = 274\).
Time = 0.27 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.06 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{4} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 4 \, {\left (a b^{3} d x^{2} - 2 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{3} - 6 \, {\left (4 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x^{2} - 6 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} {\left (a^{2} b^{2} + 8 \, b^{4}\right )}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right ) - 8 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} {\left (a^{3} b + 24 \, a b^{3}\right )}}{d x} \]
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\[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{4}\, dx \]
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\[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{4} \,d x } \]
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Exception generated. \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^4 \,d x \]
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