Integrand size = 18, antiderivative size = 398 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d^3 x+\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2+\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3}-\frac {3 b d^2 e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c}-\frac {3 b e^3 x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c}-\frac {d^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {3 d^2 e (a+b \text {arccosh}(c x))^2}{4 c^2}-\frac {3 e^3 (a+b \text {arccosh}(c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))^2}{4 e} \] Output:
2*b^2*d^3*x+4/3*b^2*d*e^2*x/c^2+3/4*b^2*d^2*e*x^2+3/32*b^2*e^3*x^2/c^2+2/9 *b^2*d*e^2*x^3+1/32*b^2*e^3*x^4-2*b*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*a rccosh(c*x))/c-4/3*b*d*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/ c^3-3/2*b*d^2*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c-3/16*b* e^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c^3-2/3*b*d*e^2*x^2*( c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c-1/8*b*e^3*x^3*(c*x-1)^(1/2 )*(c*x+1)^(1/2)*(a+b*arccosh(c*x))/c-1/4*d^4*(a+b*arccosh(c*x))^2/e-3/4*d^ 2*e*(a+b*arccosh(c*x))^2/c^2-3/32*e^3*(a+b*arccosh(c*x))^2/c^4+1/4*(e*x+d) ^4*(a+b*arccosh(c*x))^2/e
Time = 0.38 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.97 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+b^2 c x \left (3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )-6 b c \left (-24 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \text {arccosh}(c x)+9 b^2 \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arccosh}(c x)^2-54 a b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{288 c^4} \] Input:
Integrate[(d + e*x)^3*(a + b*ArcCosh[c*x])^2,x]
Output:
(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 6*a*b*Sqrt[ -1 + c*x]*Sqrt[1 + c*x]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 3 2*d*e^2*x^2 + 6*e^3*x^3)) + b^2*c*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) - 6*b*c*(-24*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2*( 64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)))*Arc Cosh[c*x] + 9*b^2*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4* d*e^2*x^2 + e^3*x^3))*ArcCosh[c*x]^2 - 54*a*b*e*(8*c^2*d^2 + e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(288*c^4)
Time = 2.56 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6378, 6390, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {b c \int \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 e}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {b c \int \left (\frac {(a+b \text {arccosh}(c x)) d^4}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {4 e x (a+b \text {arccosh}(c x)) d^3}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {6 e^2 x^2 (a+b \text {arccosh}(c x)) d^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {4 e^3 x^3 (a+b \text {arccosh}(c x)) d}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {e^4 x^4 (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))^2}{4 e}-\frac {b c \left (\frac {3 e^4 (a+b \text {arccosh}(c x))^2}{16 b c^5}+\frac {8 d e^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^4}+\frac {3 e^4 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c^4}+\frac {3 d^2 e^2 (a+b \text {arccosh}(c x))^2}{2 b c^3}+\frac {4 d^3 e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}+\frac {3 d^2 e^2 x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}+\frac {4 d e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^2}+\frac {e^4 x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{4 c^2}+\frac {d^4 (a+b \text {arccosh}(c x))^2}{2 b c}-\frac {8 b d e^3 x}{3 c^3}-\frac {3 b e^4 x^2}{16 c^3}-\frac {4 b d^3 e x}{c}-\frac {3 b d^2 e^2 x^2}{2 c}-\frac {4 b d e^3 x^3}{9 c}-\frac {b e^4 x^4}{16 c}\right )}{2 e}\) |
Input:
Int[(d + e*x)^3*(a + b*ArcCosh[c*x])^2,x]
Output:
((d + e*x)^4*(a + b*ArcCosh[c*x])^2)/(4*e) - (b*c*((-4*b*d^3*e*x)/c - (8*b *d*e^3*x)/(3*c^3) - (3*b*d^2*e^2*x^2)/(2*c) - (3*b*e^4*x^2)/(16*c^3) - (4* b*d*e^3*x^3)/(9*c) - (b*e^4*x^4)/(16*c) + (4*d^3*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^2 + (8*d*e^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^4) + (3*d^2*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^2 + (3*e^4*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*Arc Cosh[c*x]))/(8*c^4) + (4*d*e^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*Arc Cosh[c*x]))/(3*c^2) + (e^4*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh [c*x]))/(4*c^2) + (d^4*(a + b*ArcCosh[c*x])^2)/(2*b*c) + (3*d^2*e^2*(a + b *ArcCosh[c*x])^2)/(2*b*c^3) + (3*e^4*(a + b*ArcCosh[c*x])^2)/(16*b*c^5)))/ (2*e)
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 ] || (EqQ[m, 2] && LtQ[p, -2]))
Time = 0.68 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.52
| method | result | size |
| orering | \(\frac {\left (111 e^{5} c^{4} x^{6}+699 e^{4} c^{4} x^{5} d +1928 e^{3} c^{4} x^{4} d^{2}+3480 e^{2} c^{4} x^{3} d^{3}+672 c^{4} d^{4} e \,x^{2}+63 e^{5} c^{2} x^{4}+192 c^{4} d^{5} x +1079 e^{4} c^{2} x^{3} d -1632 e^{3} c^{2} x^{2} d^{2}-3600 e^{2} c^{2} d^{3} x -720 c^{2} d^{4} e -180 e^{5} x^{2}-2010 e^{4} d x -402 e^{3} d^{2}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{192 \left (e x +d \right )^{2} c^{4}}-\frac {\left (81 e^{4} c^{4} x^{6}+539 e^{3} c^{4} x^{5} d +1640 e^{2} c^{4} x^{4} d^{2}+3672 e \,c^{4} x^{3} d^{3}+99 e^{4} c^{2} x^{4}+1719 e^{3} c^{2} x^{3} d -1920 e^{2} c^{2} x^{2} d^{2}-4464 e \,c^{2} d^{3} x -576 c^{2} d^{4}-216 e^{4} x^{2}-2742 e^{3} d x -384 e^{2} d^{2}\right ) \left (3 \left (e x +d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} e +\frac {2 \left (e x +d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{576 \left (e x +d \right )^{4} c^{4}}+\frac {x \left (9 e^{3} c^{2} x^{3}+64 e^{2} c^{2} x^{2} d +216 e \,c^{2} d^{2} x +576 c^{2} d^{3}+27 e^{3} x +384 d \,e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (6 \left (e x +d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} e^{2}+\frac {12 \left (e x +d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 b^{2} c^{2} \left (e x +d \right )^{3}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (e x +d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (e x +d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{576 c^{4} \left (e x +d \right )^{3}}\) | \(603\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (c^{3} d^{3} \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {3 d^{2} e \,c^{2} \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}+\frac {c d \,e^{2} \left (9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+\frac {e^{3} \left (8 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x +c^{4} x^{4}-3 \operatorname {arccosh}\left (c x \right )^{2}+3 c^{2} x^{2}\right )}{32}\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\) | \(616\) |
| default | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (c^{3} d^{3} \left (\operatorname {arccosh}\left (c x \right )^{2} c x -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {3 d^{2} e \,c^{2} \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}+\frac {c d \,e^{2} \left (9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+\frac {e^{3} \left (8 \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c x +c^{4} x^{4}-3 \operatorname {arccosh}\left (c x \right )^{2}+3 c^{2} x^{2}\right )}{32}\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\) | \(616\) |
| parts | \(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{3} x +432 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{2} e \,x^{2}+288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d \,e^{2} x^{3}+72 \operatorname {arccosh}\left (c x \right )^{2} e^{3} c^{4} x^{4}-576 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d^{3}-432 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d^{2} e x -192 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d \,e^{2} x^{2}-36 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{3} c^{3} x^{3}-216 \operatorname {arccosh}\left (c x \right )^{2} c^{2} d^{2} e -384 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c d \,e^{2}-54 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{3} c x +576 c^{4} d^{3} x +216 c^{4} d^{2} e \,x^{2}+64 c^{4} d \,e^{2} x^{3}+9 c^{4} e^{3} x^{4}-27 \operatorname {arccosh}\left (c x \right )^{2} e^{3}+384 c^{2} d \,e^{2} x +27 c^{2} e^{3} x^{2}\right )}{288 c^{4}}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+c \,e^{2} \operatorname {arccosh}\left (c x \right ) d \,x^{3}+\frac {c \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 c^{3} e \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(649\) |
Input:
int((e*x+d)^3*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/192*(111*c^4*e^5*x^6+699*c^4*d*e^4*x^5+1928*c^4*d^2*e^3*x^4+3480*c^4*d^3 *e^2*x^3+672*c^4*d^4*e*x^2+63*c^2*e^5*x^4+192*c^4*d^5*x+1079*c^2*d*e^4*x^3 -1632*c^2*d^2*e^3*x^2-3600*c^2*d^3*e^2*x-720*c^2*d^4*e-180*e^5*x^2-2010*d* e^4*x-402*d^2*e^3)/(e*x+d)^2/c^4*(a+b*arccosh(c*x))^2-1/576*(81*c^4*e^4*x^ 6+539*c^4*d*e^3*x^5+1640*c^4*d^2*e^2*x^4+3672*c^4*d^3*e*x^3+99*c^2*e^4*x^4 +1719*c^2*d*e^3*x^3-1920*c^2*d^2*e^2*x^2-4464*c^2*d^3*e*x-576*c^2*d^4-216* e^4*x^2-2742*d*e^3*x-384*d^2*e^2)/(e*x+d)^4/c^4*(3*(e*x+d)^2*(a+b*arccosh( c*x))^2*e+2*(e*x+d)^3*(a+b*arccosh(c*x))*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2))+ 1/576*x*(9*c^2*e^3*x^3+64*c^2*d*e^2*x^2+216*c^2*d^2*e*x+576*c^2*d^3+27*e^3 *x+384*d*e^2)/c^4*(c*x-1)*(c*x+1)/(e*x+d)^3*(6*(e*x+d)*(a+b*arccosh(c*x))^ 2*e^2+12*(e*x+d)^2*(a+b*arccosh(c*x))*e*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2* b^2*c^2*(e*x+d)^3/(c*x-1)/(c*x+1)-(e*x+d)^3*(a+b*arccosh(c*x))*b*c^2/(c*x- 1)^(3/2)/(c*x+1)^(1/2)-(e*x+d)^3*(a+b*arccosh(c*x))*b*c^2/(c*x-1)^(1/2)/(c *x+1)^(3/2))
Time = 0.12 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {9 \, {\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e + b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} + 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \, {\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x - 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} - {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{4}} \] Input:
integrate((e*x+d)^3*(a+b*arccosh(c*x))^2,x, algorithm="fricas")
Output:
1/288*(9*(8*a^2 + b^2)*c^4*e^3*x^4 + 32*(9*a^2 + 2*b^2)*c^4*d*e^2*x^3 + 27 *(8*(2*a^2 + b^2)*c^4*d^2*e + b^2*c^2*e^3)*x^2 + 9*(8*b^2*c^4*e^3*x^4 + 32 *b^2*c^4*d*e^2*x^3 + 48*b^2*c^4*d^2*e*x^2 + 32*b^2*c^4*d^3*x - 24*b^2*c^2* d^2*e - 3*b^2*e^3)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 96*(3*(a^2 + 2*b^2)*c^ 4*d^3 + 4*b^2*c^2*d*e^2)*x + 6*(24*a*b*c^4*e^3*x^4 + 96*a*b*c^4*d*e^2*x^3 + 144*a*b*c^4*d^2*e*x^2 + 96*a*b*c^4*d^3*x - 72*a*b*c^2*d^2*e - 9*a*b*e^3 - (6*b^2*c^3*e^3*x^3 + 32*b^2*c^3*d*e^2*x^2 + 96*b^2*c^3*d^3 + 64*b^2*c*d* e^2 + 9*(8*b^2*c^3*d^2*e + b^2*c*e^3)*x)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt (c^2*x^2 - 1)) - 6*(6*a*b*c^3*e^3*x^3 + 32*a*b*c^3*d*e^2*x^2 + 96*a*b*c^3* d^3 + 64*a*b*c*d*e^2 + 9*(8*a*b*c^3*d^2*e + a*b*c*e^3)*x)*sqrt(c^2*x^2 - 1 ))/c^4
\[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{3}\, dx \] Input:
integrate((e*x+d)**3*(a+b*acosh(c*x))**2,x)
Output:
Integral((a + b*acosh(c*x))**2*(d + e*x)**3, x)
\[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^3*(a+b*arccosh(c*x))^2,x, algorithm="maxima")
Output:
1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + b^2*d^3*x*arccosh(c*x)^2 + 3/2*a^2*d^2*e *x^2 + 3/2*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*a*b*d^2*e + 2/3*(3*x^3*arccosh(c*x) - c*(sq rt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a*b*d*e^2 + 1/16*(8*x^ 4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*a*b*e^3 + 2*b^2*d^3*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d^3*x + 2*(c*x*arccosh(c*x) - sqrt (c^2*x^2 - 1))*a*b*d^3/c + 1/4*(b^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2* e*x^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - integrate(1/2*(b^2*c^3*e ^3*x^6 + 4*b^2*c^3*d*e^2*x^5 - 4*b^2*c*d*e^2*x^3 - 6*b^2*c*d^2*e*x^2 + (6* c^3*d^2*e - c*e^3)*b^2*x^4 + (b^2*c^2*e^3*x^5 + 4*b^2*c^2*d*e^2*x^4 + 6*b^ 2*c^2*d^2*e*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt (c*x - 1))/(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)
Exception generated. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x+d)^3*(a+b*arccosh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int((a + b*acosh(c*x))^2*(d + e*x)^3,x)
Output:
int((a + b*acosh(c*x))^2*(d + e*x)^3, x)
\[ \int (d+e x)^3 (a+b \text {arccosh}(c x))^2 \, dx=\frac {96 \mathit {acosh} \left (c x \right ) a b \,c^{4} d^{3} x +144 \mathit {acosh} \left (c x \right ) a b \,c^{4} d^{2} e \,x^{2}+96 \mathit {acosh} \left (c x \right ) a b \,c^{4} d \,e^{2} x^{3}+24 \mathit {acosh} \left (c x \right ) a b \,c^{4} e^{3} x^{4}-72 \sqrt {c^{2} x^{2}-1}\, a b \,c^{3} d^{2} e x -32 \sqrt {c^{2} x^{2}-1}\, a b \,c^{3} d \,e^{2} x^{2}-6 \sqrt {c^{2} x^{2}-1}\, a b \,c^{3} e^{3} x^{3}-64 \sqrt {c^{2} x^{2}-1}\, a b c d \,e^{2}-9 \sqrt {c^{2} x^{2}-1}\, a b c \,e^{3} x -96 \sqrt {c x +1}\, \sqrt {c x -1}\, a b \,c^{3} d^{3}+48 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c^{4} d^{3}+48 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4} e^{3}+144 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{4} d \,e^{2}+144 \left (\int \mathit {acosh} \left (c x \right )^{2} x d x \right ) b^{2} c^{4} d^{2} e -72 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) a b \,c^{2} d^{2} e -9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) a b \,e^{3}+48 a^{2} c^{4} d^{3} x +72 a^{2} c^{4} d^{2} e \,x^{2}+48 a^{2} c^{4} d \,e^{2} x^{3}+12 a^{2} c^{4} e^{3} x^{4}}{48 c^{4}} \] Input:
int((e*x+d)^3*(a+b*acosh(c*x))^2,x)
Output:
(96*acosh(c*x)*a*b*c**4*d**3*x + 144*acosh(c*x)*a*b*c**4*d**2*e*x**2 + 96* acosh(c*x)*a*b*c**4*d*e**2*x**3 + 24*acosh(c*x)*a*b*c**4*e**3*x**4 - 72*sq rt(c**2*x**2 - 1)*a*b*c**3*d**2*e*x - 32*sqrt(c**2*x**2 - 1)*a*b*c**3*d*e* *2*x**2 - 6*sqrt(c**2*x**2 - 1)*a*b*c**3*e**3*x**3 - 64*sqrt(c**2*x**2 - 1 )*a*b*c*d*e**2 - 9*sqrt(c**2*x**2 - 1)*a*b*c*e**3*x - 96*sqrt(c*x + 1)*sqr t(c*x - 1)*a*b*c**3*d**3 + 48*int(acosh(c*x)**2,x)*b**2*c**4*d**3 + 48*int (acosh(c*x)**2*x**3,x)*b**2*c**4*e**3 + 144*int(acosh(c*x)**2*x**2,x)*b**2 *c**4*d*e**2 + 144*int(acosh(c*x)**2*x,x)*b**2*c**4*d**2*e - 72*log(sqrt(c **2*x**2 - 1) + c*x)*a*b*c**2*d**2*e - 9*log(sqrt(c**2*x**2 - 1) + c*x)*a* b*e**3 + 48*a**2*c**4*d**3*x + 72*a**2*c**4*d**2*e*x**2 + 48*a**2*c**4*d*e **2*x**3 + 12*a**2*c**4*e**3*x**4)/(48*c**4)