Integrand size = 26, antiderivative size = 86 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {b x}{c^3 \pi ^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{c^4 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^4 \pi ^2}-\frac {b \arctan (c x)}{c^4 \pi ^{3/2}} \] Output:
-b*x/c^3/Pi^(3/2)+(a+b*arcsinh(c*x))/c^4/Pi/(Pi*c^2*x^2+Pi)^(1/2)+(Pi*c^2* x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))/c^4/Pi^2-b*arctan(c*x)/c^4/Pi^(3/2)
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {2 a+a c^2 x^2-b c x \sqrt {1+c^2 x^2}+b \left (2+c^2 x^2\right ) \text {arcsinh}(c x)-b \sqrt {1+c^2 x^2} \arctan (c x)}{c^4 \pi ^{3/2} \sqrt {1+c^2 x^2}} \] Input:
Integrate[(x^3*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]
Output:
(2*a + a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2] + b*(2 + c^2*x^2)*ArcSinh[c*x] - b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(c^4*Pi^(3/2)*Sqrt[1 + c^2*x^2])
Time = 0.56 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6219, 27, 299, 216}
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.
\(\displaystyle \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6219 |
\(\displaystyle -\sqrt {\pi } b c \int \frac {c^2 x^2+2}{c^4 \pi ^2 \left (c^2 x^2+1\right )}dx+\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^4}+\frac {a+b \text {arcsinh}(c x)}{\pi c^4 \sqrt {\pi c^2 x^2+\pi }}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \int \frac {c^2 x^2+2}{c^2 x^2+1}dx}{\pi ^{3/2} c^3}+\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^4}+\frac {a+b \text {arcsinh}(c x)}{\pi c^4 \sqrt {\pi c^2 x^2+\pi }}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {b \left (\int \frac {1}{c^2 x^2+1}dx+x\right )}{\pi ^{3/2} c^3}+\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^4}+\frac {a+b \text {arcsinh}(c x)}{\pi c^4 \sqrt {\pi c^2 x^2+\pi }}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^4}+\frac {a+b \text {arcsinh}(c x)}{\pi c^4 \sqrt {\pi c^2 x^2+\pi }}-\frac {b \left (\frac {\arctan (c x)}{c}+x\right )}{\pi ^{3/2} c^3}\) |
Input:
Int[(x^3*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]
Output:
(a + b*ArcSinh[c*x])/(c^4*Pi*Sqrt[Pi + c^2*Pi*x^2]) + (Sqrt[Pi + c^2*Pi*x^ 2]*(a + b*ArcSinh[c*x]))/(c^4*Pi^2) - (b*(x + ArcTan[c*x]/c))/(c^3*Pi^(3/2 ))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.90
method | result | size |
default | \(a \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )\right )}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, c^{4}}\) | \(163\) |
parts | \(a \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )-\sqrt {c^{2} x^{2}+1}\, x c +2 \,\operatorname {arcsinh}\left (x c \right )\right )}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, c^{4}}\) | \(163\) |
Input:
int(x^3*(a+b*arcsinh(x*c))/(Pi*c^2*x^2+Pi)^(3/2),x,method=_RETURNVERBOSE)
Output:
a*(x^2/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)+2/Pi/c^4/(Pi*c^2*x^2+Pi)^(1/2))+b/Pi^( 3/2)/(c^2*x^2+1)^(1/2)*(arcsinh(x*c)*c^2*x^2+I*(c^2*x^2+1)^(1/2)*ln(x*c+(c ^2*x^2+1)^(1/2)-I)-I*(c^2*x^2+1)^(1/2)*ln(x*c+(c^2*x^2+1)^(1/2)+I)-(c^2*x^ 2+1)^(1/2)*x*c+2*arcsinh(x*c))/c^4
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (78) = 156\).
Time = 0.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.92 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{2} x^{2} + 2 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (a c^{2} x^{2} - \sqrt {c^{2} x^{2} + 1} b c x + 2 \, a\right )}}{2 \, {\left (\pi ^{2} c^{6} x^{2} + \pi ^{2} c^{4}\right )}} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="frica s")
Output:
1/2*(sqrt(pi)*(b*c^2*x^2 + b)*arctan(-2*sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqr t(c^2*x^2 + 1)*c*x/(pi - pi*c^4*x^4)) + 2*sqrt(pi + pi*c^2*x^2)*(b*c^2*x^2 + 2*b)*log(c*x + sqrt(c^2*x^2 + 1)) + 2*sqrt(pi + pi*c^2*x^2)*(a*c^2*x^2 - sqrt(c^2*x^2 + 1)*b*c*x + 2*a))/(pi^2*c^6*x^2 + pi^2*c^4)
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{3}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{3} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \] Input:
integrate(x**3*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(3/2),x)
Output:
(Integral(a*x**3/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x**3*asinh(c*x)/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x **2 + 1)), x))/pi**(3/2)
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-b c {\left (\frac {x}{\pi ^{\frac {3}{2}} c^{4}} + \frac {\arctan \left (c x\right )}{\pi ^{\frac {3}{2}} c^{5}}\right )} + b {\left (\frac {x^{2}}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{2}} + \frac {2}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) + a {\left (\frac {x^{2}}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{2}} + \frac {2}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{4}}\right )} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="maxim a")
Output:
-b*c*(x/(pi^(3/2)*c^4) + arctan(c*x)/(pi^(3/2)*c^5)) + b*(x^2/(pi*sqrt(pi + pi*c^2*x^2)*c^2) + 2/(pi*sqrt(pi + pi*c^2*x^2)*c^4))*arcsinh(c*x) + a*(x ^2/(pi*sqrt(pi + pi*c^2*x^2)*c^2) + 2/(pi*sqrt(pi + pi*c^2*x^2)*c^4))
Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),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. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \] Input:
int((x^3*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2),x)
Output:
int((x^3*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2), x)
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, a +\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{6} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{4}}{\sqrt {\pi }\, c^{4} \pi \left (c^{2} x^{2}+1\right )} \] Input:
int(x^3*(a+b*asinh(c*x))/(Pi*c^2*x^2+Pi)^(3/2),x)
Output:
(sqrt(c**2*x**2 + 1)*a*c**2*x**2 + 2*sqrt(c**2*x**2 + 1)*a + int((asinh(c* x)*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**6*x **2 + int((asinh(c*x)*x**3)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x** 2 + 1)),x)*b*c**4)/(sqrt(pi)*c**4*pi*(c**2*x**2 + 1))