Integrand size = 25, antiderivative size = 210 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {15}{64} b^2 \pi ^{3/2} x \sqrt {1+c^2 x^2}+\frac {1}{32} b^2 \pi ^{3/2} x \left (1+c^2 x^2\right )^{3/2}-\frac {9 b^2 \pi ^{3/2} \text {arcsinh}(c x)}{64 c}-\frac {3}{8} b c \pi ^{3/2} x^2 (a+b \text {arcsinh}(c x))-\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c} \]
1/32*b^2*Pi^(3/2)*x*(c^2*x^2+1)^(3/2)-9/64*b^2*Pi^(3/2)*arcsinh(c*x)/c-3/8 *b*c*Pi^(3/2)*x^2*(a+b*arcsinh(c*x))-1/8*b*Pi^(3/2)*(c^2*x^2+1)^2*(a+b*arc sinh(c*x))/c+1/4*x*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))^2+1/8*Pi^(3/2) *(a+b*arcsinh(c*x))^3/b/c+15/64*b^2*Pi^(3/2)*x*(c^2*x^2+1)^(1/2)+3/8*Pi*x* (a+b*arcsinh(c*x))^2*(Pi*c^2*x^2+Pi)^(1/2)
Time = 1.00 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.96 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\pi ^{3/2} \left (160 a^2 c x \sqrt {1+c^2 x^2}+64 a^2 c^3 x^3 \sqrt {1+c^2 x^2}+32 b^2 \text {arcsinh}(c x)^3-64 a b \cosh (2 \text {arcsinh}(c x))-4 a b \cosh (4 \text {arcsinh}(c x))+32 b^2 \sinh (2 \text {arcsinh}(c x))+b^2 \sinh (4 \text {arcsinh}(c x))+8 b \text {arcsinh}(c x)^2 (12 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))+4 \text {arcsinh}(c x) \left (-16 b^2 \cosh (2 \text {arcsinh}(c x))-b^2 \cosh (4 \text {arcsinh}(c x))+4 a (6 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))\right )\right )}{256 c} \]
(Pi^(3/2)*(160*a^2*c*x*Sqrt[1 + c^2*x^2] + 64*a^2*c^3*x^3*Sqrt[1 + c^2*x^2 ] + 32*b^2*ArcSinh[c*x]^3 - 64*a*b*Cosh[2*ArcSinh[c*x]] - 4*a*b*Cosh[4*Arc Sinh[c*x]] + 32*b^2*Sinh[2*ArcSinh[c*x]] + b^2*Sinh[4*ArcSinh[c*x]] + 8*b* ArcSinh[c*x]^2*(12*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]) + 4*ArcSinh[c*x]*(-16*b^2*Cosh[2*ArcSinh[c*x]] - b^2*Cosh[4*ArcSinh[c*x]] + 4*a*(6*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]))))/(256*c)
Time = 0.95 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 222}
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.
| \(\displaystyle \int \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \pi \int \sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))^2dx+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \pi \left (\frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx-\sqrt {\pi } b c \int x (a+b \text {arcsinh}(c x))dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \pi \left (-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \pi \left (-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )+\frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \pi \left (\frac {1}{2} \sqrt {\pi } \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{3/2}dx}{4 c}\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {1}{2} \pi ^{3/2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c}\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {1}{2} \pi ^{3/2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {3}{4} \pi \left (\frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {\sqrt {\pi } (a+b \text {arcsinh}(c x))^3}{6 b c}\right )\) |
(x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*Pi*((x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[Pi]*(a + b*ArcSinh[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)))/4 - (b*c*Pi^(3/2)*(((1 + c ^2*x^2)^2*(a + b*ArcSinh[c*x]))/(4*c^2) - (b*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/(4*c)))/2
3.3.53.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
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]
Time = 0.18 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {3}{2}} \left (16 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+2 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+40 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x -40 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+17 c x \sqrt {c^{2} x^{2}+1}+8 \operatorname {arcsinh}\left (c x \right )^{3}-17 \,\operatorname {arcsinh}\left (c x \right )\right )}{64 c}+\frac {a b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}+10 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-5 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{8 c}\) | \(289\) |
| parts | \(\frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {3}{2}} \left (16 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+2 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+40 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x -40 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+17 c x \sqrt {c^{2} x^{2}+1}+8 \operatorname {arcsinh}\left (c x \right )^{3}-17 \,\operatorname {arcsinh}\left (c x \right )\right )}{64 c}+\frac {a b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}+10 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-5 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{8 c}\) | \(289\) |
1/4*a^2*x*(Pi*c^2*x^2+Pi)^(3/2)+3/8*a^2*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)+3/8*a^2 *Pi^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/6 4*b^2*Pi^(3/2)*(16*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*x^3*c^3-8*arcsinh(c*x) *c^4*x^4+2*c^3*x^3*(c^2*x^2+1)^(1/2)+40*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*c *x-40*arcsinh(c*x)*c^2*x^2+17*c*x*(c^2*x^2+1)^(1/2)+8*arcsinh(c*x)^3-17*ar csinh(c*x))/c+1/8*a*b*Pi^(3/2)*(4*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3-c ^4*x^4+10*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)-5*c^2*x^2+3*arcsinh(c*x)^2-4) /c
\[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
integral(sqrt(pi + pi*c^2*x^2)*(pi*a^2*c^2*x^2 + pi*a^2 + (pi*b^2*c^2*x^2 + pi*b^2)*arcsinh(c*x)^2 + 2*(pi*a*b*c^2*x^2 + pi*a*b)*arcsinh(c*x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (197) = 394\).
Time = 2.91 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.93 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a^{2} x \sqrt {c^{2} x^{2} + 1}}{8} + \frac {3 \pi ^{\frac {3}{2}} a^{2} \operatorname {asinh}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} a b c^{3} x^{4}}{8} + \frac {\pi ^{\frac {3}{2}} a b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {5 \pi ^{\frac {3}{2}} a b c x^{2}}{8} + \frac {5 \pi ^{\frac {3}{2}} a b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4} + \frac {3 \pi ^{\frac {3}{2}} a b \operatorname {asinh}^{2}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b^{2} c^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{32} - \frac {5 \pi ^{\frac {3}{2}} b^{2} c x^{2} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {5 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {17 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1}}{64} + \frac {\pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}^{3}{\left (c x \right )}}{8 c} - \frac {17 \pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}{\left (c x \right )}}{64 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} a^{2} x & \text {otherwise} \end {cases} \]
Piecewise((pi**(3/2)*a**2*c**2*x**3*sqrt(c**2*x**2 + 1)/4 + 5*pi**(3/2)*a* *2*x*sqrt(c**2*x**2 + 1)/8 + 3*pi**(3/2)*a**2*asinh(c*x)/(8*c) - pi**(3/2) *a*b*c**3*x**4/8 + pi**(3/2)*a*b*c**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/ 2 - 5*pi**(3/2)*a*b*c*x**2/8 + 5*pi**(3/2)*a*b*x*sqrt(c**2*x**2 + 1)*asinh (c*x)/4 + 3*pi**(3/2)*a*b*asinh(c*x)**2/(8*c) - pi**(3/2)*b**2*c**3*x**4*a sinh(c*x)/8 + pi**(3/2)*b**2*c**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)**2/4 + pi**(3/2)*b**2*c**2*x**3*sqrt(c**2*x**2 + 1)/32 - 5*pi**(3/2)*b**2*c*x* *2*asinh(c*x)/8 + 5*pi**(3/2)*b**2*x*sqrt(c**2*x**2 + 1)*asinh(c*x)**2/8 + 17*pi**(3/2)*b**2*x*sqrt(c**2*x**2 + 1)/64 + pi**(3/2)*b**2*asinh(c*x)**3 /(8*c) - 17*pi**(3/2)*b**2*asinh(c*x)/(64*c), Ne(c, 0)), (pi**(3/2)*a**2*x , True))
Exception generated. \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \]