Integrand size = 26, antiderivative size = 98 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {2 b x}{3 c^3 \sqrt {\pi }}-\frac {b x^3}{9 c \sqrt {\pi }}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{3 c^4 \pi }+\frac {x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{3 c^2 \pi } \] Output:
2/3*b*x/c^3/Pi^(1/2)-1/9*b*x^3/c/Pi^(1/2)-2/3*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*a rcsinh(c*x))/c^4/Pi+1/3*x^2*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))/c^2/P i
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {3 a \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2}+b \left (6 c x-c^3 x^3\right )+3 b \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{9 c^4 \sqrt {\pi }} \] Input:
Integrate[(x^3*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]
Output:
(3*a*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2] + b*(6*c*x - c^3*x^3) + 3*b*(-2 + c^ 2*x^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(9*c^4*Sqrt[Pi])
Time = 0.46 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6227, 15, 6213, 24}
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.
\(\displaystyle \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi c^2 x^2+\pi }} \, dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle -\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 \pi x^2+\pi }}dx}{3 c^2}-\frac {b \int x^2dx}{3 \sqrt {\pi } c}+\frac {x^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{3 \pi c^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 \pi x^2+\pi }}dx}{3 c^2}+\frac {x^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{3 \pi c^2}-\frac {b x^3}{9 \sqrt {\pi } c}\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle -\frac {2 \left (\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi c^2}-\frac {b \int 1dx}{\sqrt {\pi } c}\right )}{3 c^2}+\frac {x^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{3 \pi c^2}-\frac {b x^3}{9 \sqrt {\pi } c}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {x^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{3 \pi c^2}-\frac {2 \left (\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi c^2}-\frac {b x}{\sqrt {\pi } c}\right )}{3 c^2}-\frac {b x^3}{9 \sqrt {\pi } c}\) |
Input:
Int[(x^3*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]
Output:
-1/9*(b*x^3)/(c*Sqrt[Pi]) + (x^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x] ))/(3*c^2*Pi) - (2*(-((b*x)/(c*Sqrt[Pi])) + (Sqrt[Pi + c^2*Pi*x^2]*(a + b* ArcSinh[c*x]))/(c^2*Pi)))/(3*c^2)
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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -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 = 1.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36
method | result | size |
default | \(a \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )+\frac {b \left (3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-6 \,\operatorname {arcsinh}\left (x c \right )+6 \sqrt {c^{2} x^{2}+1}\, x c \right )}{9 \sqrt {\pi }\, c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(133\) |
parts | \(a \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )+\frac {b \left (3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-6 \,\operatorname {arcsinh}\left (x c \right )+6 \sqrt {c^{2} x^{2}+1}\, x c \right )}{9 \sqrt {\pi }\, c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(133\) |
orering | \(\frac {\left (5 c^{4} x^{4}-12 c^{2} x^{2}-24\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{9 c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\left (c^{2} x^{2}-6\right ) \left (c^{2} x^{2}+1\right ) \left (\frac {3 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{\sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {x^{3} b c}{\sqrt {c^{2} x^{2}+1}\, \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}}{\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}\right )}{9 x^{2} c^{4}}\) | \(155\) |
Input:
int(x^3*(a+b*arcsinh(x*c))/(Pi*c^2*x^2+Pi)^(1/2),x,method=_RETURNVERBOSE)
Output:
a*(1/3*x^2/Pi/c^2*(Pi*c^2*x^2+Pi)^(1/2)-2/3/Pi/c^4*(Pi*c^2*x^2+Pi)^(1/2))+ 1/9*b/Pi^(1/2)/c^4/(c^2*x^2+1)^(1/2)*(3*arcsinh(x*c)*c^4*x^4-3*arcsinh(x*c )*c^2*x^2-(c^2*x^2+1)^(1/2)*c^3*x^3-6*arcsinh(x*c)+6*(c^2*x^2+1)^(1/2)*x*c )
Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.35 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 3 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 6 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 6 \, a\right )}}{9 \, {\left (\pi c^{6} x^{2} + \pi c^{4}\right )}} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(1/2),x, algorithm="frica s")
Output:
1/9*(3*sqrt(pi + pi*c^2*x^2)*(b*c^4*x^4 - b*c^2*x^2 - 2*b)*log(c*x + sqrt( c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2)*(3*a*c^4*x^4 - 3*a*c^2*x^2 - (b*c^3* x^3 - 6*b*c*x)*sqrt(c^2*x^2 + 1) - 6*a))/(pi*c^6*x^2 + pi*c^4)
Time = 3.70 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.27 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{3}}{9 c} + \frac {x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{2}} + \frac {2 x}{3 c^{3}} - \frac {2 \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{4}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \] Input:
integrate(x**3*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(1/2),x)
Output:
a*Piecewise((x**2*sqrt(c**2*x**2 + 1)/(3*c**2) - 2*sqrt(c**2*x**2 + 1)/(3* c**4), Ne(c**2, 0)), (x**4/4, True))/sqrt(pi) + b*Piecewise((-x**3/(9*c) + x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3*c**2) + 2*x/(3*c**3) - 2*sqrt(c**2 *x**2 + 1)*asinh(c*x)/(3*c**4), Ne(c, 0)), (0, True))/sqrt(pi)
Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.19 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {1}{3} \, b {\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{2}} - \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{2}} - \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{4}}\right )} - \frac {{\left (c^{2} x^{3} - 6 \, x\right )} b}{9 \, \sqrt {\pi } c^{3}} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(1/2),x, algorithm="maxim a")
Output:
1/3*b*(sqrt(pi + pi*c^2*x^2)*x^2/(pi*c^2) - 2*sqrt(pi + pi*c^2*x^2)/(pi*c^ 4))*arcsinh(c*x) + 1/3*a*(sqrt(pi + pi*c^2*x^2)*x^2/(pi*c^2) - 2*sqrt(pi + pi*c^2*x^2)/(pi*c^4)) - 1/9*(c^2*x^3 - 6*x)*b/(sqrt(pi)*c^3)
Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(1/2),x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \] Input:
int((x^3*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(1/2),x)
Output:
int((x^3*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(1/2), x)
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a +3 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{4}}{3 \sqrt {\pi }\, c^{4}} \] Input:
int(x^3*(a+b*asinh(c*x))/(Pi*c^2*x^2+Pi)^(1/2),x)
Output:
(sqrt(c**2*x**2 + 1)*a*c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a + 3*int((asinh( c*x)*x**3)/sqrt(c**2*x**2 + 1),x)*b*c**4)/(3*sqrt(pi)*c**4)