Integrand size = 26, antiderivative size = 131 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {b x^2}{4 c^3 \pi ^{3/2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^4 \pi ^2}-\frac {3 (a+b \text {arcsinh}(c x))^2}{4 b c^5 \pi ^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c^5 \pi ^{3/2}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5810, 5812, 5783, 30, 272, 45} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {3 (a+b \text {arcsinh}(c x))^2}{4 \pi ^{3/2} b c^5}-\frac {x^3 (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {3 x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi ^2 c^4}-\frac {b x^2}{4 \pi ^{3/2} c^3}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^5} \]
[In]
[Out]
Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5810
Rule 5812
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \int \frac {x^3}{1+c^2 x^2} \, dx}{c \pi ^{3/2}}+\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^2 \pi } \\ & = -\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^4 \pi ^2}-\frac {(3 b) \int x \, dx}{2 c^3 \pi ^{3/2}}+\frac {b \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{2 c \pi ^{3/2}}-\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{2 c^4 \pi } \\ & = -\frac {3 b x^2}{4 c^3 \pi ^{3/2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^4 \pi ^2}-\frac {3 (a+b \text {arcsinh}(c x))^2}{4 b c^5 \pi ^{3/2}}+\frac {b \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{2 c \pi ^{3/2}} \\ & = -\frac {b x^2}{4 c^3 \pi ^{3/2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^4 \pi ^2}-\frac {3 (a+b \text {arcsinh}(c x))^2}{4 b c^5 \pi ^{3/2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c^5 \pi ^{3/2}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {12 a c x+4 a c^3 x^3-6 b \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-b \sqrt {1+c^2 x^2} \cosh (2 \text {arcsinh}(c x))-4 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+\text {arcsinh}(c x) \left (9 b c x-12 a \sqrt {1+c^2 x^2}+b \sinh (3 \text {arcsinh}(c x))\right )}{8 c^5 \pi ^{3/2} \sqrt {1+c^2 x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(260\) vs. \(2(113)=226\).
Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.99
method | result | size |
default | \(\frac {a \,x^{3}}{2 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {3 a x}{2 c^{4} \pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {3 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{4} \pi \sqrt {\pi \,c^{2}}}-\frac {b \left (-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{8 \pi ^{\frac {3}{2}} c^{5} \left (c^{2} x^{2}+1\right )}\) | \(261\) |
parts | \(\frac {a \,x^{3}}{2 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {3 a x}{2 c^{4} \pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {3 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{4} \pi \sqrt {\pi \,c^{2}}}-\frac {b \left (-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{8 \pi ^{\frac {3}{2}} c^{5} \left (c^{2} x^{2}+1\right )}\) | \(261\) |
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{4}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
[In]
[Out]
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
[In]
[Out]