Integrand size = 25, antiderivative size = 122 \[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} b^2 \sqrt {\pi } x \sqrt {1+c^2 x^2}-\frac {b^2 \sqrt {\pi } \text {arcsinh}(c x)}{4 c}-\frac {1}{2} b c \sqrt {\pi } x^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c} \]
-1/4*b^2*arcsinh(c*x)*Pi^(1/2)/c-1/2*b*c*x^2*(a+b*arcsinh(c*x))*Pi^(1/2)+1 /6*(a+b*arcsinh(c*x))^3*Pi^(1/2)/b/c+1/4*b^2*x*Pi^(1/2)*(c^2*x^2+1)^(1/2)+ 1/2*x*(a+b*arcsinh(c*x))^2*(Pi*c^2*x^2+Pi)^(1/2)
Time = 0.51 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {\pi } \left (4 b^2 \text {arcsinh}(c x)^3+6 b \text {arcsinh}(c x)^2 (2 a+b \sinh (2 \text {arcsinh}(c x)))+3 \left (4 a^2 c x \sqrt {1+c^2 x^2}-2 a b \cosh (2 \text {arcsinh}(c x))+b^2 \sinh (2 \text {arcsinh}(c x))\right )+6 \text {arcsinh}(c x) \left (-b^2 \cosh (2 \text {arcsinh}(c x))+2 a (a+b \sinh (2 \text {arcsinh}(c x)))\right )\right )}{24 c} \]
(Sqrt[Pi]*(4*b^2*ArcSinh[c*x]^3 + 6*b*ArcSinh[c*x]^2*(2*a + b*Sinh[2*ArcSi nh[c*x]]) + 3*(4*a^2*c*x*Sqrt[1 + c^2*x^2] - 2*a*b*Cosh[2*ArcSinh[c*x]] + b^2*Sinh[2*ArcSinh[c*x]]) + 6*ArcSinh[c*x]*(-(b^2*Cosh[2*ArcSinh[c*x]]) + 2*a*(a + b*Sinh[2*ArcSinh[c*x]]))))/(24*c)
Time = 0.50 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6200, 6191, 262, 222, 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.
\(\displaystyle \int \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2 \, dx\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle \frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx-\sqrt {\pi } b c \int x (a+b \text {arcsinh}(c x))dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle -\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )+\frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c}\) |
(x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[Pi]*(a + b*ArcS inh[c*x])^3)/(6*b*c) - b*c*Sqrt[Pi]*((x^2*(a + b*ArcSinh[c*x]))/2 - (b*c*( (x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/2)
3.3.54.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && 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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {a^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a^{2} \pi \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \sqrt {\pi }\, \left (6 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x -6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+2 \operatorname {arcsinh}\left (c x \right )^{3}-3 \,\operatorname {arcsinh}\left (c x \right )\right )}{12 c}+\frac {a b \sqrt {\pi }\, \left (2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )^{2}-1\right )}{2 c}\) | \(179\) |
parts | \(\frac {a^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a^{2} \pi \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \sqrt {\pi }\, \left (6 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x -6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+2 \operatorname {arcsinh}\left (c x \right )^{3}-3 \,\operatorname {arcsinh}\left (c x \right )\right )}{12 c}+\frac {a b \sqrt {\pi }\, \left (2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )^{2}-1\right )}{2 c}\) | \(179\) |
1/2*a^2*x*(Pi*c^2*x^2+Pi)^(1/2)+1/2*a^2*Pi*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi* c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/12*b^2*Pi^(1/2)*(6*arcsinh(c*x)^2*(c^2 *x^2+1)^(1/2)*c*x-6*arcsinh(c*x)*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+2*arcsinh (c*x)^3-3*arcsinh(c*x))/c+1/2*a*b*Pi^(1/2)*(2*arcsinh(c*x)*c*x*(c^2*x^2+1) ^(1/2)-c^2*x^2+arcsinh(c*x)^2-1)/c
\[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
\[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\sqrt {\pi } \left (\int a^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int b^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 2 a b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
sqrt(pi)*(Integral(a**2*sqrt(c**2*x**2 + 1), x) + Integral(b**2*sqrt(c**2* x**2 + 1)*asinh(c*x)**2, x) + Integral(2*a*b*sqrt(c**2*x**2 + 1)*asinh(c*x ), x))
Exception generated. \[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \]