Integrand size = 26, antiderivative size = 126 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {3 b x^2}{16 c^3 \sqrt {\pi }}-\frac {b x^4}{16 c \sqrt {\pi }}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{4 c^2 \pi }+\frac {3 (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {\pi }} \]
3/16*b*x^2/c^3/Pi^(1/2)-1/16*b*x^4/c/Pi^(1/2)+3/16*(a+b*arcsinh(c*x))^2/b/ c^5/Pi^(1/2)-3/8*x*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)/c^4/Pi+1/4*x^3 *(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)/c^2/Pi
Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {-48 a c x \sqrt {1+c^2 x^2}+32 a c^3 x^3 \sqrt {1+c^2 x^2}+24 b \text {arcsinh}(c x)^2+16 b \cosh (2 \text {arcsinh}(c x))-b \cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (12 a-8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))}{128 c^5 \sqrt {\pi }} \]
(-48*a*c*x*Sqrt[1 + c^2*x^2] + 32*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 24*b*ArcSi nh[c*x]^2 + 16*b*Cosh[2*ArcSinh[c*x]] - b*Cosh[4*ArcSinh[c*x]] + 4*ArcSinh [c*x]*(12*a - 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]))/(128*c^5 *Sqrt[Pi])
Time = 0.58 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6227, 15, 6227, 15, 6198}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi c^2 x^2+\pi }} \, dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 \pi x^2+\pi }}dx}{4 c^2}-\frac {b \int x^3dx}{4 \sqrt {\pi } c}+\frac {x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{4 \pi c^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 \pi x^2+\pi }}dx}{4 c^2}+\frac {x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{4 \pi c^2}-\frac {b x^4}{16 \sqrt {\pi } c}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{2 c^2}-\frac {b \int xdx}{2 \sqrt {\pi } c}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{4 \pi c^2}-\frac {b x^4}{16 \sqrt {\pi } c}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 \pi x^2+\pi }}dx}{2 c^2}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}-\frac {b x^2}{4 \sqrt {\pi } c}\right )}{4 c^2}+\frac {x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{4 \pi c^2}-\frac {b x^4}{16 \sqrt {\pi } c}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{4 \pi c^2}-\frac {3 \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 \sqrt {\pi } b c^3}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}-\frac {b x^2}{4 \sqrt {\pi } c}\right )}{4 c^2}-\frac {b x^4}{16 \sqrt {\pi } c}\) |
-1/16*(b*x^4)/(c*Sqrt[Pi]) + (x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x ]))/(4*c^2*Pi) - (3*(-1/4*(b*x^2)/(c*Sqrt[Pi]) + (x*Sqrt[Pi + c^2*Pi*x^2]* (a + b*ArcSinh[c*x]))/(2*c^2*Pi) - (a + b*ArcSinh[c*x])^2/(4*b*c^3*Sqrt[Pi ])))/(4*c^2)
3.1.81.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {a \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}{4 \pi \,c^{2}}-\frac {3 a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8 c^{4} \pi }+\frac {3 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+3\right )}{16 c^{5} \sqrt {\pi }}\) | \(165\) |
| parts | \(\frac {a \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}{4 \pi \,c^{2}}-\frac {3 a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8 c^{4} \pi }+\frac {3 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+3\right )}{16 c^{5} \sqrt {\pi }}\) | \(165\) |
1/4*a*x^3/Pi/c^2*(Pi*c^2*x^2+Pi)^(1/2)-3/8*a/c^4*x/Pi*(Pi*c^2*x^2+Pi)^(1/2 )+3/8*a/c^4*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/ 2)+1/16*b*(4*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3-c^4*x^4-6*arcsinh(c*x) *c*x*(c^2*x^2+1)^(1/2)+3*c^2*x^2+3*arcsinh(c*x)^2+3)/c^5/Pi^(1/2)
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {\pi + \pi c^{2} x^{2}}} \,d x } \]
Time = 2.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.47 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {x^{3} \sqrt {c^{2} x^{2} + 1}}{4 c^{2}} - \frac {3 x \sqrt {c^{2} x^{2} + 1}}{8 c^{4}} + \frac {3 \log {\left (2 c^{2} x + 2 \sqrt {c^{2} x^{2} + 1} \sqrt {c^{2}} \right )}}{8 c^{4} \sqrt {c^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{4}}{16 c} + \frac {x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} + \frac {3 x^{2}}{16 c^{3}} - \frac {3 x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c^{4}} + \frac {3 \operatorname {asinh}^{2}{\left (c x \right )}}{16 c^{5}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \]
a*Piecewise((x**3*sqrt(c**2*x**2 + 1)/(4*c**2) - 3*x*sqrt(c**2*x**2 + 1)/( 8*c**4) + 3*log(2*c**2*x + 2*sqrt(c**2*x**2 + 1)*sqrt(c**2))/(8*c**4*sqrt( c**2)), Ne(c**2, 0)), (x**5/5, True))/sqrt(pi) + b*Piecewise((-x**4/(16*c) + x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(4*c**2) + 3*x**2/(16*c**3) - 3*x*s qrt(c**2*x**2 + 1)*asinh(c*x)/(8*c**4) + 3*asinh(c*x)**2/(16*c**5), Ne(c, 0)), (0, True))/sqrt(pi)
Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {\pi + \pi c^{2} x^{2}}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]