Integrand size = 26, antiderivative size = 108 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {1}{4} b c^3 \pi ^{3/2} x^2+\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {3 c \pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{4 b}+b c \pi ^{3/2} \log (x) \]
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Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5807, 5785, 5783, 30, 14} \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {3}{2} \pi c^2 x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {3 \pi ^{3/2} c (a+b \text {arcsinh}(c x))^2}{4 b}-\frac {1}{4} \pi ^{3/2} b c^3 x^2+\pi ^{3/2} b c \log (x) \]
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Rule 14
Rule 30
Rule 5783
Rule 5785
Rule 5807
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\left (3 c^2 \pi \right ) \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx+\left (b c \pi ^{3/2}\right ) \int \frac {1+c^2 x^2}{x} \, dx \\ & = \frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\left (b c \pi ^{3/2}\right ) \int \left (\frac {1}{x}+c^2 x\right ) \, dx+\frac {1}{2} \left (3 c^2 \pi ^{3/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (3 b c^3 \pi ^{3/2}\right ) \int x \, dx \\ & = -\frac {1}{4} b c^3 \pi ^{3/2} x^2+\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {3 c \pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{4 b}+b c \pi ^{3/2} \log (x) \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {\pi ^{3/2} \left (-8 a \sqrt {1+c^2 x^2}+4 a c^2 x^2 \sqrt {1+c^2 x^2}+6 b c x \text {arcsinh}(c x)^2-b c x \cosh (2 \text {arcsinh}(c x))+8 b c x \log (c x)+2 \text {arcsinh}(c x) \left (6 a c x-4 b \sqrt {1+c^2 x^2}+b c x \sinh (2 \text {arcsinh}(c x))\right )\right )}{8 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(92)=184\).
Time = 0.17 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.93
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right )}{8 x}\) | \(208\) |
parts | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right )}{8 x}\) | \(208\) |
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\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\pi ^{\frac {3}{2}} \left (\int a c^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
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Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^2} \,d x \]
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