Integrand size = 26, antiderivative size = 62 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c \sqrt {\pi }}{6 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} b c^3 \sqrt {\pi } \log (x) \]
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Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5800, 14} \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} \sqrt {\pi } b c^3 \log (x)-\frac {\sqrt {\pi } b c}{6 x^2} \]
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Rule 14
Rule 5800
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} \left (b c \sqrt {\pi }\right ) \int \frac {1+c^2 x^2}{x^3} \, dx \\ & = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} \left (b c \sqrt {\pi }\right ) \int \left (\frac {1}{x^3}+\frac {c^2}{x}\right ) \, dx \\ & = -\frac {b c \sqrt {\pi }}{6 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} b c^3 \sqrt {\pi } \log (x) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {\sqrt {\pi } \left (-b c x-3 b c^3 x^3-2 a \sqrt {1+c^2 x^2}-2 a c^2 x^2 \sqrt {1+c^2 x^2}-2 b \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)+2 b c^3 x^3 \log (x)\right )}{6 x^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(500\) vs. \(2(50)=100\).
Time = 0.16 (sec) , antiderivative size = 501, normalized size of antiderivative = 8.08
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,x^{3}}-\frac {2 b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{3}+\frac {b \sqrt {\pi }\, x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}+\frac {b \sqrt {\pi }\, x^{4} c^{7}}{18 c^{4} x^{4}+18 c^{2} x^{2}+6}-\frac {b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \sqrt {\pi }\, x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {2 b \sqrt {\pi }\, x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{4}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{9 c^{4} x^{4}+9 c^{2} x^{2}+3}-\frac {4 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \,c^{3} \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\) | \(501\) |
parts | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,x^{3}}-\frac {2 b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{3}+\frac {b \sqrt {\pi }\, x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}+\frac {b \sqrt {\pi }\, x^{4} c^{7}}{18 c^{4} x^{4}+18 c^{2} x^{2}+6}-\frac {b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \sqrt {\pi }\, x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {2 b \sqrt {\pi }\, x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{4}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{9 c^{4} x^{4}+9 c^{2} x^{2}+3}-\frac {4 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \,c^{3} \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\) | \(501\) |
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (50) = 100\).
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \sqrt {\pi } {\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (2 \, a c^{4} x^{4} + 4 \, a c^{2} x^{2} - {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} + 1} + 2 \, a\right )}}{6 \, {\left (c^{2} x^{5} + x^{3}\right )}} \]
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\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\sqrt {\pi } \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (50) = 100\).
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {{\left (\pi ^{\frac {3}{2}} \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} c^{2} \log \left (2 \, \pi c^{2} + \frac {2 \, \pi }{x^{2}}\right ) - \pi ^{\frac {3}{2}} c^{2} \log \left (x^{2} + \frac {1}{c^{2}}\right ) + \frac {\pi \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}}}{x^{2}}\right )} b c}{6 \, \pi } - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, \pi x^{3}} - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} a}{3 \, \pi x^{3}} \]
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Exception generated. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x^4} \,d x \]
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