Integrand size = 23, antiderivative size = 291 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {4322 b^2 d^3 x}{3675}+\frac {1514 b^2 c^2 d^3 x^3}{11025}+\frac {234 b^2 c^4 d^3 x^5}{6125}+\frac {2}{343} b^2 c^6 d^3 x^7-\frac {32 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{35 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{105 c}-\frac {12 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{175 c}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{49 c}+\frac {16}{35} d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{35} d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {6}{35} d^3 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \] Output:
4322/3675*b^2*d^3*x+1514/11025*b^2*c^2*d^3*x^3+234/6125*b^2*c^4*d^3*x^5+2/ 343*b^2*c^6*d^3*x^7-32/35*b*d^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c-16/ 105*b*d^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c-12/175*b*d^3*(c^2*x^2+1)^ (5/2)*(a+b*arcsinh(c*x))/c-2/49*b*d^3*(c^2*x^2+1)^(7/2)*(a+b*arcsinh(c*x)) /c+16/35*d^3*x*(a+b*arcsinh(c*x))^2+8/35*d^3*x*(c^2*x^2+1)*(a+b*arcsinh(c* x))^2+6/35*d^3*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/7*d^3*x*(c^2*x^2+1)^ 3*(a+b*arcsinh(c*x))^2
Time = 1.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.82 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (11025 a^2 c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )-210 a b \sqrt {1+c^2 x^2} \left (2161+757 c^2 x^2+351 c^4 x^4+75 c^6 x^6\right )+2 b^2 c x \left (226905+26495 c^2 x^2+7371 c^4 x^4+1125 c^6 x^6\right )-210 b \left (-105 a c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {1+c^2 x^2} \left (2161+757 c^2 x^2+351 c^4 x^4+75 c^6 x^6\right )\right ) \text {arcsinh}(c x)+11025 b^2 c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{385875 c} \] Input:
Integrate[(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*(11025*a^2*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) - 210*a*b*S qrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*c^6*x^6) + 2*b^2*c *x*(226905 + 26495*c^2*x^2 + 7371*c^4*x^4 + 1125*c^6*x^6) - 210*b*(-105*a* c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*c^6*x^6))*ArcSinh[c*x] + 11025*b^2*c*x*( 35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6)*ArcSinh[c*x]^2))/(385875*c)
Time = 1.38 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6201, 27, 6201, 6201, 6187, 6213, 24, 210, 2009}
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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {2}{7} b c d^3 \int x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {6}{7} d \int d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{7} b c d^3 \int x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {6}{7} d^3 \int \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {2}{7} b c d^3 \int x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {6}{7} d^3 \left (-\frac {2}{5} b c \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {2}{7} b c d^3 \int x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {6}{7} d^3 \left (-\frac {2}{5} b c \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} \left (-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{3} \int (a+b \text {arcsinh}(c x))^2dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{5} b c \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{7} b c d^3 \int x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^2 x^2+1\right )^2dx}{5 c}\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{7} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {b \int \left (c^2 x^2+1\right )^3dx}{7 c}\right )+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^2 x^2+1\right )^2dx}{5 c}\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{7} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {b \int \left (c^2 x^2+1\right )^3dx}{7 c}\right )+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^4 x^4+2 c^2 x^2+1\right )dx}{5 c}\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{7} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {b \int \left (c^6 x^6+3 c^4 x^4+3 c^2 x^2+1\right )dx}{7 c}\right )+\frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )-\frac {2}{7} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {b \left (\frac {c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+c^2 x^3+x\right )}{7 c}\right )\) |
Input:
Int[(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*x*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/7 - (2*b*c*d^3*(-1/7*(b*(x + c^2*x^3 + (3*c^4*x^5)/5 + (c^6*x^7)/7))/c + ((1 + c^2*x^2)^(7/2)*(a + b* ArcSinh[c*x]))/(7*c^2)))/7 + (6*d^3*((x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x ])^2)/5 - (2*b*c*(-1/5*(b*(x + (2*c^2*x^3)/3 + (c^4*x^5)/5))/c + ((1 + c^2 *x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2)))/5 + (4*((x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3 - (2*b*c*(-1/3*(b*(x + (c^2*x^3)/3))/c + ((1 + c^2*x^ 2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2)))/3 + (2*(x*(a + b*ArcSinh[c*x])^2 - 2*b*c*(-((b*x)/c) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2)))/3))/ 5))/7
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)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && 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 = 2.71 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+x^{3} c^{3}+x c \right )+d^{3} b^{2} \left (\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{35}+\frac {\left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2} x c}{7}+\frac {6 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{35}-\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{35}+\frac {413312 x c}{385875}-\frac {2 \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \operatorname {arcsinh}\left (x c \right )}{49}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {888 x c \left (c^{2} x^{2}+1\right )^{2}}{42875}+\frac {30256 x c \left (c^{2} x^{2}+1\right )}{385875}-\frac {12 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {757 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {117 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c}\) | \(354\) |
| default | \(\frac {d^{3} a^{2} \left (\frac {1}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+x^{3} c^{3}+x c \right )+d^{3} b^{2} \left (\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{35}+\frac {\left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2} x c}{7}+\frac {6 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{35}-\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{35}+\frac {413312 x c}{385875}-\frac {2 \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \operatorname {arcsinh}\left (x c \right )}{49}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {888 x c \left (c^{2} x^{2}+1\right )^{2}}{42875}+\frac {30256 x c \left (c^{2} x^{2}+1\right )}{385875}-\frac {12 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {757 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {117 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c}\) | \(354\) |
| parts | \(d^{3} a^{2} \left (\frac {1}{7} c^{6} x^{7}+\frac {3}{5} c^{4} x^{5}+x^{3} c^{2}+x \right )+\frac {d^{3} b^{2} \left (\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{35}+\frac {\left (c^{2} x^{2}+1\right )^{3} \operatorname {arcsinh}\left (x c \right )^{2} x c}{7}+\frac {6 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{35}-\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{35}+\frac {413312 x c}{385875}-\frac {2 \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \operatorname {arcsinh}\left (x c \right )}{49}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {888 x c \left (c^{2} x^{2}+1\right )^{2}}{42875}+\frac {30256 x c \left (c^{2} x^{2}+1\right )}{385875}-\frac {12 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )}{c}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {757 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {117 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c}\) | \(354\) |
| orering | \(\frac {x \left (47625 c^{8} x^{8}+271212 c^{6} x^{6}+741678 c^{4} x^{4}+3539900 c^{2} x^{2}+128625\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{128625 \left (c^{2} x^{2}+1\right )^{4}}-\frac {\left (20250 c^{8} x^{8}+125811 c^{6} x^{6}+407785 c^{4} x^{4}+2802345 c^{2} x^{2}+226905\right ) \left (6 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d x +\frac {2 \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{385875 c^{2} \left (c^{2} x^{2}+1\right )^{3}}+\frac {x \left (1125 c^{6} x^{6}+7371 c^{4} x^{4}+26495 c^{2} x^{2}+226905\right ) \left (24 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2} x^{2}+\frac {24 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d x b}{\sqrt {c^{2} x^{2}+1}}+6 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 \left (c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3} x}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{385875 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\) | \(398\) |
Input:
int((c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(d^3*a^2*(1/7*x^7*c^7+3/5*x^5*c^5+x^3*c^3+x*c)+d^3*b^2*(16/35*arcsinh( x*c)^2*x*c+1/7*(c^2*x^2+1)^3*arcsinh(x*c)^2*x*c+6/35*arcsinh(x*c)^2*x*c*(c ^2*x^2+1)^2+8/35*arcsinh(x*c)^2*x*c*(c^2*x^2+1)-32/35*arcsinh(x*c)*(c^2*x^ 2+1)^(1/2)+413312/385875*x*c-2/49*(c^2*x^2+1)^(7/2)*arcsinh(x*c)+2/343*x*c *(c^2*x^2+1)^3+888/42875*x*c*(c^2*x^2+1)^2+30256/385875*x*c*(c^2*x^2+1)-12 /175*arcsinh(x*c)*(c^2*x^2+1)^(5/2)-16/105*arcsinh(x*c)*(c^2*x^2+1)^(3/2)) +2*d^3*a*b*(1/7*arcsinh(x*c)*x^7*c^7+3/5*arcsinh(x*c)*x^5*c^5+arcsinh(x*c) *x^3*c^3+x*c*arcsinh(x*c)-2161/3675*(c^2*x^2+1)^(1/2)-757/3675*x^2*c^2*(c^ 2*x^2+1)^(1/2)-117/1225*x^4*c^4*(c^2*x^2+1)^(1/2)-1/49*x^6*c^6*(c^2*x^2+1) ^(1/2)))
Time = 0.10 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.22 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (1225 \, a^{2} + 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \, {\left (11025 \, a^{2} + 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} + 105 \, {\left (3675 \, a^{2} + 4322 \, b^{2}\right )} c d^{3} x + 11025 \, {\left (5 \, b^{2} c^{7} d^{3} x^{7} + 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} + 35 \, b^{2} c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (525 \, a b c^{7} d^{3} x^{7} + 2205 \, a b c^{5} d^{3} x^{5} + 3675 \, a b c^{3} d^{3} x^{3} + 3675 \, a b c d^{3} x - {\left (75 \, b^{2} c^{6} d^{3} x^{6} + 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} + 2161 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 210 \, {\left (75 \, a b c^{6} d^{3} x^{6} + 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} + 2161 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{385875 \, c} \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/385875*(1125*(49*a^2 + 2*b^2)*c^7*d^3*x^7 + 189*(1225*a^2 + 78*b^2)*c^5* d^3*x^5 + 35*(11025*a^2 + 1514*b^2)*c^3*d^3*x^3 + 105*(3675*a^2 + 4322*b^2 )*c*d^3*x + 11025*(5*b^2*c^7*d^3*x^7 + 21*b^2*c^5*d^3*x^5 + 35*b^2*c^3*d^3 *x^3 + 35*b^2*c*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 210*(525*a*b*c^7*d ^3*x^7 + 2205*a*b*c^5*d^3*x^5 + 3675*a*b*c^3*d^3*x^3 + 3675*a*b*c*d^3*x - (75*b^2*c^6*d^3*x^6 + 351*b^2*c^4*d^3*x^4 + 757*b^2*c^2*d^3*x^2 + 2161*b^2 *d^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 210*(75*a*b*c^6*d^ 3*x^6 + 351*a*b*c^4*d^3*x^4 + 757*a*b*c^2*d^3*x^2 + 2161*a*b*d^3)*sqrt(c^2 *x^2 + 1))/c
Time = 0.91 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.80 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{6} d^{3} x^{7}}{7} + \frac {3 a^{2} c^{4} d^{3} x^{5}}{5} + a^{2} c^{2} d^{3} x^{3} + a^{2} d^{3} x + \frac {2 a b c^{6} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {2 a b c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {6 a b c^{4} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {234 a b c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + 2 a b c^{2} d^{3} x^{3} \operatorname {asinh}{\left (c x \right )} - \frac {1514 a b c d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{3675} + 2 a b d^{3} x \operatorname {asinh}{\left (c x \right )} - \frac {4322 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{3675 c} + \frac {b^{2} c^{6} d^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{6} d^{3} x^{7}}{343} - \frac {2 b^{2} c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{49} + \frac {3 b^{2} c^{4} d^{3} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {234 b^{2} c^{4} d^{3} x^{5}}{6125} - \frac {234 b^{2} c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1225} + b^{2} c^{2} d^{3} x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {1514 b^{2} c^{2} d^{3} x^{3}}{11025} - \frac {1514 b^{2} c d^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {4322 b^{2} d^{3} x}{3675} - \frac {4322 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675 c} & \text {for}\: c \neq 0 \\a^{2} d^{3} x & \text {otherwise} \end {cases} \] Input:
integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**6*d**3*x**7/7 + 3*a**2*c**4*d**3*x**5/5 + a**2*c**2*d** 3*x**3 + a**2*d**3*x + 2*a*b*c**6*d**3*x**7*asinh(c*x)/7 - 2*a*b*c**5*d**3 *x**6*sqrt(c**2*x**2 + 1)/49 + 6*a*b*c**4*d**3*x**5*asinh(c*x)/5 - 234*a*b *c**3*d**3*x**4*sqrt(c**2*x**2 + 1)/1225 + 2*a*b*c**2*d**3*x**3*asinh(c*x) - 1514*a*b*c*d**3*x**2*sqrt(c**2*x**2 + 1)/3675 + 2*a*b*d**3*x*asinh(c*x) - 4322*a*b*d**3*sqrt(c**2*x**2 + 1)/(3675*c) + b**2*c**6*d**3*x**7*asinh( c*x)**2/7 + 2*b**2*c**6*d**3*x**7/343 - 2*b**2*c**5*d**3*x**6*sqrt(c**2*x* *2 + 1)*asinh(c*x)/49 + 3*b**2*c**4*d**3*x**5*asinh(c*x)**2/5 + 234*b**2*c **4*d**3*x**5/6125 - 234*b**2*c**3*d**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x )/1225 + b**2*c**2*d**3*x**3*asinh(c*x)**2 + 1514*b**2*c**2*d**3*x**3/1102 5 - 1514*b**2*c*d**3*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/3675 + b**2*d**3* x*asinh(c*x)**2 + 4322*b**2*d**3*x/3675 - 4322*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3675*c), Ne(c, 0)), (a**2*d**3*x, True))
Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (259) = 518\).
Time = 0.06 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.45 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/7*b^2*c^6*d^3*x^7*arcsinh(c*x)^2 + 1/7*a^2*c^6*d^3*x^7 + 3/5*b^2*c^4*d^3 *x^5*arcsinh(c*x)^2 + 3/5*a^2*c^4*d^3*x^5 + 2/245*(35*x^7*arcsinh(c*x) - ( 5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*a*b*c^6*d^3 - 2/25725*(105*(5 *sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c*arcsinh(c*x) - (75*c^6*x^7 - 12 6*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^6)*b^2*c^6*d^3 + b^2*c^2*d^3*x^3*arcsi nh(c*x)^2 + 2/25*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*s qrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^4*d^3 - 2/375 *(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c ^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^ 2*c^4*d^3 + a^2*c^2*d^3*x^3 + 2/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c^2*d^3 - 2/9*(3*c*(sqrt(c^2*x^ 2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c ^2)*b^2*c^2*d^3 + b^2*d^3*x*arcsinh(c*x)^2 + 2*b^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^3*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1)) *a*b*d^3/c
Exception generated. \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((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. \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3,x)
Output:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3, x)
\[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{3} \left (3675 \mathit {asinh} \left (c x \right )^{2} b^{2} c x -7350 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2}+1050 \mathit {asinh} \left (c x \right ) a b \,c^{7} x^{7}+4410 \mathit {asinh} \left (c x \right ) a b \,c^{5} x^{5}+7350 \mathit {asinh} \left (c x \right ) a b \,c^{3} x^{3}+7350 \mathit {asinh} \left (c x \right ) a b c x -150 \sqrt {c^{2} x^{2}+1}\, a b \,c^{6} x^{6}-702 \sqrt {c^{2} x^{2}+1}\, a b \,c^{4} x^{4}-1514 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-4322 \sqrt {c^{2} x^{2}+1}\, a b +3675 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+11025 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+11025 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+525 a^{2} c^{7} x^{7}+2205 a^{2} c^{5} x^{5}+3675 a^{2} c^{3} x^{3}+3675 a^{2} c x +7350 b^{2} c x \right )}{3675 c} \] Input:
int((c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2,x)
Output:
(d**3*(3675*asinh(c*x)**2*b**2*c*x - 7350*sqrt(c**2*x**2 + 1)*asinh(c*x)*b **2 + 1050*asinh(c*x)*a*b*c**7*x**7 + 4410*asinh(c*x)*a*b*c**5*x**5 + 7350 *asinh(c*x)*a*b*c**3*x**3 + 7350*asinh(c*x)*a*b*c*x - 150*sqrt(c**2*x**2 + 1)*a*b*c**6*x**6 - 702*sqrt(c**2*x**2 + 1)*a*b*c**4*x**4 - 1514*sqrt(c**2 *x**2 + 1)*a*b*c**2*x**2 - 4322*sqrt(c**2*x**2 + 1)*a*b + 3675*int(asinh(c *x)**2*x**6,x)*b**2*c**7 + 11025*int(asinh(c*x)**2*x**4,x)*b**2*c**5 + 110 25*int(asinh(c*x)**2*x**2,x)*b**2*c**3 + 525*a**2*c**7*x**7 + 2205*a**2*c* *5*x**5 + 3675*a**2*c**3*x**3 + 3675*a**2*c*x + 7350*b**2*c*x))/(3675*c)