Integrand size = 24, antiderivative size = 269 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {35 b^2 d^3 x^2}{1024}+\frac {35 b^2 d^3 \left (1+c^2 x^2\right )^2}{3072 c^2}+\frac {7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac {35 b d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{512 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{768 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{192 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{32 c}-\frac {35 d^3 (a+b \text {arcsinh}(c x))^2}{1024 c^2}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2} \] Output:
35/1024*b^2*d^3*x^2+35/3072*b^2*d^3*(c^2*x^2+1)^2/c^2+7/1152*b^2*d^3*(c^2* x^2+1)^3/c^2+1/256*b^2*d^3*(c^2*x^2+1)^4/c^2-35/512*b*d^3*x*(c^2*x^2+1)^(1 /2)*(a+b*arcsinh(c*x))/c-35/768*b*d^3*x*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x ))/c-7/192*b*d^3*x*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c-1/32*b*d^3*x*(c^ 2*x^2+1)^(7/2)*(a+b*arcsinh(c*x))/c-35/1024*d^3*(a+b*arcsinh(c*x))^2/c^2+1 /8*d^3*(c^2*x^2+1)^4*(a+b*arcsinh(c*x))^2/c^2
Time = 1.35 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.95 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (c x \left (1152 a^2 c x \left (4+6 c^2 x^2+4 c^4 x^4+c^6 x^6\right )+b^2 c x \left (837+489 c^2 x^2+200 c^4 x^4+36 c^6 x^6\right )-6 a b \sqrt {1+c^2 x^2} \left (279+326 c^2 x^2+200 c^4 x^4+48 c^6 x^6\right )\right )+6 b \left (-b c x \sqrt {1+c^2 x^2} \left (279+326 c^2 x^2+200 c^4 x^4+48 c^6 x^6\right )+3 a \left (93+512 c^2 x^2+768 c^4 x^4+512 c^6 x^6+128 c^8 x^8\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (93+512 c^2 x^2+768 c^4 x^4+512 c^6 x^6+128 c^8 x^8\right ) \text {arcsinh}(c x)^2\right )}{9216 c^2} \] Input:
Integrate[x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*(c*x*(1152*a^2*c*x*(4 + 6*c^2*x^2 + 4*c^4*x^4 + c^6*x^6) + b^2*c*x*(8 37 + 489*c^2*x^2 + 200*c^4*x^4 + 36*c^6*x^6) - 6*a*b*Sqrt[1 + c^2*x^2]*(27 9 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x ^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 3*a*(93 + 512*c^2*x^ 2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8))*ArcSinh[c*x] + 9*b^2*(93 + 5 12*c^2*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8)*ArcSinh[c*x]^2))/(92 16*c^2)
Time = 1.06 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6213, 6201, 241, 6201, 241, 6201, 244, 2009, 6200, 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.
| \(\displaystyle \int x \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \int \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))dx}{4 c}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \int \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int x \left (c^2 x^2+1\right )^3dx+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))\right )}{4 c}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \int \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{6} b c \int x \left (c^2 x^2+1\right )^2dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {1}{8} x \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))+\frac {7}{8} \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )-\frac {b \left (c^2 x^2+1\right )^4}{64 c}\right )}{4 c}\) |
Input:
Int[x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x])^2)/(8*c^2) - (b*d^3*(-1/64*(b*(1 + c^2*x^2)^4)/c + (x*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/8 + (7*(-1 /36*(b*(1 + c^2*x^2)^3)/c + (x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/6 + (5*(-1/4*(b*c*(x^2/2 + (c^2*x^4)/4)) + (x*(1 + c^2*x^2)^(3/2)*(a + b*Ar cSinh[c*x]))/4 + (3*(-1/4*(b*c*x^2) + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[ c*x]))/2 + (a + b*ArcSinh[c*x])^2/(4*b*c)))/4))/6))/8))/(4*c)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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 = 1.68 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {\frac {d^{3} a^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}-\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{192}-\frac {35 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{768}-\frac {35 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{512}-\frac {35 \operatorname {arcsinh}\left (x c \right )^{2}}{1024}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}+1\right )^{3}}{1152}+\frac {35 \left (c^{2} x^{2}+1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}+\frac {35}{1024}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (x c \right )}{1024}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 \sqrt {c^{2} x^{2}+1}\, x c}{1024}\right )}{c^{2}}\) | \(306\) |
| default | \(\frac {\frac {d^{3} a^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}-\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{192}-\frac {35 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{768}-\frac {35 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{512}-\frac {35 \operatorname {arcsinh}\left (x c \right )^{2}}{1024}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}+1\right )^{3}}{1152}+\frac {35 \left (c^{2} x^{2}+1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}+\frac {35}{1024}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (x c \right )}{1024}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 \sqrt {c^{2} x^{2}+1}\, x c}{1024}\right )}{c^{2}}\) | \(306\) |
| parts | \(\frac {d^{3} a^{2} \left (c^{2} x^{2}+1\right )^{4}}{8 c^{2}}+\frac {d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}-\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{192}-\frac {35 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{768}-\frac {35 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{512}-\frac {35 \operatorname {arcsinh}\left (x c \right )^{2}}{1024}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}+1\right )^{3}}{1152}+\frac {35 \left (c^{2} x^{2}+1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}+\frac {35}{1024}\right )}{c^{2}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (x c \right )}{1024}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 \sqrt {c^{2} x^{2}+1}\, x c}{1024}\right )}{c^{2}}\) | \(311\) |
| orering | \(\frac {\left (6084 c^{10} x^{10}+32348 c^{8} x^{8}+72453 c^{6} x^{6}+97420 c^{4} x^{4}+34749 c^{2} x^{2}+5022\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{18432 c^{2} \left (c^{2} x^{2}+1\right )^{4}}-\frac {\left (756 c^{8} x^{8}+4160 c^{6} x^{6}+9913 c^{4} x^{4}+15489 c^{2} x^{2}+3348\right ) \left (\left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+6 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{18432 c^{2} \left (c^{2} x^{2}+1\right )^{3}}+\frac {x \left (36 c^{6} x^{6}+200 c^{4} x^{4}+489 c^{2} x^{2}+837\right ) \left (18 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d x +\frac {4 \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+24 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}+\frac {24 x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x \left (c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{18432 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\) | \(475\) |
Input:
int(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(1/8*d^3*a^2*(c^2*x^2+1)^4+d^3*b^2*(1/8*arcsinh(x*c)^2*(c^2*x^2+1)^4 -1/32*arcsinh(x*c)*x*c*(c^2*x^2+1)^(7/2)-7/192*arcsinh(x*c)*x*c*(c^2*x^2+1 )^(5/2)-35/768*arcsinh(x*c)*x*c*(c^2*x^2+1)^(3/2)-35/512*arcsinh(x*c)*(c^2 *x^2+1)^(1/2)*x*c-35/1024*arcsinh(x*c)^2+1/256*(c^2*x^2+1)^4+7/1152*(c^2*x ^2+1)^3+35/3072*(c^2*x^2+1)^2+35/1024*c^2*x^2+35/1024)+2*d^3*a*b*(1/8*arcs inh(x*c)*x^8*c^8+1/2*arcsinh(x*c)*x^6*c^6+3/4*arcsinh(x*c)*c^4*x^4+1/2*arc sinh(x*c)*c^2*x^2+93/1024*arcsinh(x*c)-1/64*x*c*(c^2*x^2+1)^(7/2)-7/384*x* c*(c^2*x^2+1)^(5/2)-35/1536*x*c*(c^2*x^2+1)^(3/2)-35/1024*(c^2*x^2+1)^(1/2 )*x*c))
Time = 0.10 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.42 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {36 \, {\left (32 \, a^{2} + b^{2}\right )} c^{8} d^{3} x^{8} + 8 \, {\left (576 \, a^{2} + 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \, {\left (2304 \, a^{2} + 163 \, b^{2}\right )} c^{4} d^{3} x^{4} + 9 \, {\left (512 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \, {\left (128 \, b^{2} c^{8} d^{3} x^{8} + 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} + 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (384 \, a b c^{8} d^{3} x^{8} + 1536 \, a b c^{6} d^{3} x^{6} + 2304 \, a b c^{4} d^{3} x^{4} + 1536 \, a b c^{2} d^{3} x^{2} + 279 \, a b d^{3} - {\left (48 \, b^{2} c^{7} d^{3} x^{7} + 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} + 279 \, b^{2} c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (48 \, a b c^{7} d^{3} x^{7} + 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} + 279 \, a b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{9216 \, c^{2}} \] Input:
integrate(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/9216*(36*(32*a^2 + b^2)*c^8*d^3*x^8 + 8*(576*a^2 + 25*b^2)*c^6*d^3*x^6 + 3*(2304*a^2 + 163*b^2)*c^4*d^3*x^4 + 9*(512*a^2 + 93*b^2)*c^2*d^3*x^2 + 9 *(128*b^2*c^8*d^3*x^8 + 512*b^2*c^6*d^3*x^6 + 768*b^2*c^4*d^3*x^4 + 512*b^ 2*c^2*d^3*x^2 + 93*b^2*d^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(384*a*b*c^ 8*d^3*x^8 + 1536*a*b*c^6*d^3*x^6 + 2304*a*b*c^4*d^3*x^4 + 1536*a*b*c^2*d^3 *x^2 + 279*a*b*d^3 - (48*b^2*c^7*d^3*x^7 + 200*b^2*c^5*d^3*x^5 + 326*b^2*c ^3*d^3*x^3 + 279*b^2*c*d^3*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(48*a*b*c^7*d^3*x^7 + 200*a*b*c^5*d^3*x^5 + 326*a*b*c^3*d^3*x^3 + 279*a*b*c*d^3*x)*sqrt(c^2*x^2 + 1))/c^2
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (258) = 516\).
Time = 1.39 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.13 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{6} d^{3} x^{8}}{8} + \frac {a^{2} c^{4} d^{3} x^{6}}{2} + \frac {3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{2}}{2} + \frac {a b c^{6} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {a b c^{5} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )} - \frac {25 a b c^{3} d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{192} + \frac {3 a b c^{2} d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {163 a b c d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {93 a b d^{3} x \sqrt {c^{2} x^{2} + 1}}{512 c} + \frac {93 a b d^{3} \operatorname {asinh}{\left (c x \right )}}{512 c^{2}} + \frac {b^{2} c^{6} d^{3} x^{8} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{6} d^{3} x^{8}}{256} - \frac {b^{2} c^{5} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{32} + \frac {b^{2} c^{4} d^{3} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {25 b^{2} c^{4} d^{3} x^{6}}{1152} - \frac {25 b^{2} c^{3} d^{3} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{192} + \frac {3 b^{2} c^{2} d^{3} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {163 b^{2} c^{2} d^{3} x^{4}}{3072} - \frac {163 b^{2} c d^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{768} + \frac {b^{2} d^{3} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {93 b^{2} d^{3} x^{2}}{1024} - \frac {93 b^{2} d^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{512 c} + \frac {93 b^{2} d^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**6*d**3*x**8/8 + a**2*c**4*d**3*x**6/2 + 3*a**2*c**2*d** 3*x**4/4 + a**2*d**3*x**2/2 + a*b*c**6*d**3*x**8*asinh(c*x)/4 - a*b*c**5*d **3*x**7*sqrt(c**2*x**2 + 1)/32 + a*b*c**4*d**3*x**6*asinh(c*x) - 25*a*b*c **3*d**3*x**5*sqrt(c**2*x**2 + 1)/192 + 3*a*b*c**2*d**3*x**4*asinh(c*x)/2 - 163*a*b*c*d**3*x**3*sqrt(c**2*x**2 + 1)/768 + a*b*d**3*x**2*asinh(c*x) - 93*a*b*d**3*x*sqrt(c**2*x**2 + 1)/(512*c) + 93*a*b*d**3*asinh(c*x)/(512*c **2) + b**2*c**6*d**3*x**8*asinh(c*x)**2/8 + b**2*c**6*d**3*x**8/256 - b** 2*c**5*d**3*x**7*sqrt(c**2*x**2 + 1)*asinh(c*x)/32 + b**2*c**4*d**3*x**6*a sinh(c*x)**2/2 + 25*b**2*c**4*d**3*x**6/1152 - 25*b**2*c**3*d**3*x**5*sqrt (c**2*x**2 + 1)*asinh(c*x)/192 + 3*b**2*c**2*d**3*x**4*asinh(c*x)**2/4 + 1 63*b**2*c**2*d**3*x**4/3072 - 163*b**2*c*d**3*x**3*sqrt(c**2*x**2 + 1)*asi nh(c*x)/768 + b**2*d**3*x**2*asinh(c*x)**2/2 + 93*b**2*d**3*x**2/1024 - 93 *b**2*d**3*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(512*c) + 93*b**2*d**3*asinh(c *x)**2/(1024*c**2), Ne(c, 0)), (a**2*d**3*x**2/2, True))
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (241) = 482\).
Time = 0.08 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.44 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Too large to display} \] Input:
integrate(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/8*b^2*c^6*d^3*x^8*arcsinh(c*x)^2 + 1/8*a^2*c^6*d^3*x^8 + 1/2*b^2*c^4*d^3 *x^6*arcsinh(c*x)^2 + 1/2*a^2*c^4*d^3*x^6 + 3/4*b^2*c^2*d^3*x^4*arcsinh(c* x)^2 + 1/1536*(384*x^8*arcsinh(c*x) - (48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*s qrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arcsinh(c*x)/c^9)*c)*a*b*c^6*d^3 + 1/9216*((36*x^8/c^2 - 56*x^6/c^4 + 105*x^4/c^6 - 315*x^2/c^8 + 315*log(c*x + sqrt(c^2*x^2 + 1)) ^2/c^10)*c^2 - 6*(48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/ c^4 + 70*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arc sinh(c*x)/c^9)*c*arcsinh(c*x))*b^2*c^6*d^3 + 3/4*a^2*c^2*d^3*x^4 + 1/48*(4 8*x^6*arcsinh(c*x) - (8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x ^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c)*a*b*c^4*d^3 + 1/288*((8*x^6/c^2 - 15*x^4/c^4 + 45*x^2/c^6 - 45*log(c*x + sqrt(c^2*x^2 + 1))^2/c^8)*c^2 - 6*(8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x ^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c*arcsinh(c*x)) *b^2*c^4*d^3 + 1/2*b^2*d^3*x^2*arcsinh(c*x)^2 + 3/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x) /c^5)*c)*a*b*c^2*d^3 + 3/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + sqrt(c^2*x ^2 + 1))^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1) *x/c^4 + 3*arcsinh(c*x)/c^5)*c*arcsinh(c*x))*b^2*c^2*d^3 + 1/2*a^2*d^3*x^2 + 1/2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/...
Exception generated. \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^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. \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3,x)
Output:
int(x*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3, x)
\[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{3} \left (768 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{2} x^{2}+384 \mathit {asinh} \left (c x \right )^{2} b^{2}-768 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c x +384 \mathit {asinh} \left (c x \right ) a b \,c^{8} x^{8}+1536 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+2304 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}+1536 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-48 \sqrt {c^{2} x^{2}+1}\, a b \,c^{7} x^{7}-200 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-326 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-279 \sqrt {c^{2} x^{2}+1}\, a b c x +1536 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}+4608 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+4608 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+279 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +192 a^{2} c^{8} x^{8}+768 a^{2} c^{6} x^{6}+1152 a^{2} c^{4} x^{4}+768 a^{2} c^{2} x^{2}+384 b^{2} c^{2} x^{2}\right )}{1536 c^{2}} \] Input:
int(x*(c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2,x)
Output:
(d**3*(768*asinh(c*x)**2*b**2*c**2*x**2 + 384*asinh(c*x)**2*b**2 - 768*sqr t(c**2*x**2 + 1)*asinh(c*x)*b**2*c*x + 384*asinh(c*x)*a*b*c**8*x**8 + 1536 *asinh(c*x)*a*b*c**6*x**6 + 2304*asinh(c*x)*a*b*c**4*x**4 + 1536*asinh(c*x )*a*b*c**2*x**2 - 48*sqrt(c**2*x**2 + 1)*a*b*c**7*x**7 - 200*sqrt(c**2*x** 2 + 1)*a*b*c**5*x**5 - 326*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 - 279*sqrt(c* *2*x**2 + 1)*a*b*c*x + 1536*int(asinh(c*x)**2*x**7,x)*b**2*c**8 + 4608*int (asinh(c*x)**2*x**5,x)*b**2*c**6 + 4608*int(asinh(c*x)**2*x**3,x)*b**2*c** 4 + 279*log(sqrt(c**2*x**2 + 1) + c*x)*a*b + 192*a**2*c**8*x**8 + 768*a**2 *c**6*x**6 + 1152*a**2*c**4*x**4 + 768*a**2*c**2*x**2 + 384*b**2*c**2*x**2 ))/(1536*c**2)