Integrand size = 12, antiderivative size = 218 \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=-\frac {\left (76 c^3 d^3+26 c^2 d^2 e+64 c d e^2+9 e^3\right ) \sqrt {-1+c x} \sqrt {1+c x}}{96 c^4}-\frac {e \left (26 c^2 d^2+9 e^2\right ) (-1+c x)^{3/2} \sqrt {1+c x}}{96 c^4}-\frac {7 d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {\left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \text {arccosh}(c x)}{32 c^4 e}+\frac {(d+e x)^4 \text {arccosh}(c x)}{4 e} \] Output:
-1/96*(76*c^3*d^3+26*c^2*d^2*e+64*c*d*e^2+9*e^3)*(c*x-1)^(1/2)*(c*x+1)^(1/ 2)/c^4-1/96*e*(26*c^2*d^2+9*e^2)*(c*x-1)^(3/2)*(c*x+1)^(1/2)/c^4-7/48*d*(c *x-1)^(1/2)*(c*x+1)^(1/2)*(e*x+d)^2/c-1/16*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(e* x+d)^3/c-1/32*(8*c^4*d^4+24*c^2*d^2*e^2+3*e^4)*arccosh(c*x)/c^4/e+1/4*(e*x +d)^4*arccosh(c*x)/e
Time = 0.15 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.70 \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=-\frac {c \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )-24 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {arccosh}(c x)+9 \left (8 c^2 d^2 e+e^3\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{96 c^4} \] Input:
Integrate[(d + e*x)^3*ArcCosh[c*x],x]
Output:
-1/96*(c*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)) - 24*c^4*x*(4*d^3 + 6*d^2*e*x + 4* d*e^2*x^2 + e^3*x^3)*ArcCosh[c*x] + 9*(8*c^2*d^2*e + e^3)*Log[c*x + Sqrt[- 1 + c*x]*Sqrt[1 + c*x]])/c^4
Time = 0.41 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6378, 111, 170, 27, 164, 43}
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.
| \(\displaystyle \int \text {arccosh}(c x) (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^4}{4 e}-\frac {c \int \frac {(d+e x)^4}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 e}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^4}{4 e}-\frac {c \left (\frac {\int \frac {(d+e x)^2 \left (4 d^2 c^2+7 d e x c^2+3 e^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^4}{4 e}-\frac {c \left (\frac {\frac {\int \frac {c^2 (d+e x) \left (d \left (12 c^2 d^2+23 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^4}{4 e}-\frac {c \left (\frac {\frac {1}{3} \int \frac {(d+e x) \left (d \left (12 c^2 d^2+23 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^4}{4 e}-\frac {c \left (\frac {\frac {1}{3} \left (\frac {3 \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{2 c^2}\right )+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^4}{4 e}-\frac {c \left (\frac {\frac {1}{3} \left (\frac {3 \text {arccosh}(c x) \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right )}{2 c^3}+\frac {e \sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{2 c^2}\right )+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
Input:
Int[(d + e*x)^3*ArcCosh[c*x],x]
Output:
((d + e*x)^4*ArcCosh[c*x])/(4*e) - (c*((e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^3)/(4*c^2) + ((7*d*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/3 + ((e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^ 2 + 9*e^2)*x))/(2*c^2) + (3*(8*c^4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcCosh[c *x])/(2*c^3))/3)/(4*c^2)))/(4*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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]
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13
| method | result | size |
| orering | \(\frac {\left (14 c^{4} e^{4} x^{5}+72 c^{4} d \,e^{3} x^{4}+152 c^{4} d^{2} e^{2} x^{3}+176 c^{4} d^{3} e \,x^{2}+32 c^{4} d^{4} x +3 c^{2} e^{4} x^{3}+32 c^{2} d \,e^{3} x^{2}-96 c^{2} d^{2} e^{2} x -120 c^{2} d^{3} e -12 e^{4} x -67 d \,e^{3}\right ) \operatorname {arccosh}\left (c x \right )}{32 c^{4} \left (e x +d \right )}-\frac {\left (6 e^{3} c^{2} x^{3}+32 e^{2} c^{2} x^{2} d +72 e \,c^{2} d^{2} x +96 c^{2} d^{3}+9 e^{3} x +64 d \,e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (3 \left (e x +d \right )^{2} \operatorname {arccosh}\left (c x \right ) e +\frac {\left (e x +d \right )^{3} c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{96 c^{4} \left (e x +d \right )^{3}}\) | \(247\) |
| derivativedivides | \(\frac {\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+c \,e^{2} \operatorname {arccosh}\left (c x \right ) d \,x^{3}+\frac {c \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 c^{3} e \sqrt {c^{2} x^{2}-1}}}{c}\) | \(299\) |
| default | \(\frac {\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+c \,e^{2} \operatorname {arccosh}\left (c x \right ) d \,x^{3}+\frac {c \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 c^{3} e \sqrt {c^{2} x^{2}-1}}}{c}\) | \(299\) |
| parts | \(\frac {\operatorname {arccosh}\left (c x \right ) e^{3} x^{4}}{4}+\operatorname {arccosh}\left (c x \right ) e^{2} d \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) e \,d^{2} x^{2}}{2}+\operatorname {arccosh}\left (c x \right ) d^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \,\operatorname {csgn}\left (c \right ) c^{3} e^{4} x^{3} \sqrt {c^{2} x^{2}-1}+32 \,\operatorname {csgn}\left (c \right ) c^{3} d \,e^{3} x^{2} \sqrt {c^{2} x^{2}-1}+72 \sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right ) c^{3} d^{2} e^{2} x +96 \,\operatorname {csgn}\left (c \right ) c^{3} \sqrt {c^{2} x^{2}-1}\, d^{3} e +24 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) c^{4} d^{4}+9 \sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right ) c \,e^{4} x +64 \,\operatorname {csgn}\left (c \right ) c \sqrt {c^{2} x^{2}-1}\, d \,e^{3}+72 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) c^{2} d^{2} e^{2}+9 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) e^{4}\right ) \operatorname {csgn}\left (c \right )}{96 e \,c^{4} \sqrt {c^{2} x^{2}-1}}\) | \(322\) |
Input:
int((e*x+d)^3*arccosh(c*x),x,method=_RETURNVERBOSE)
Output:
1/32*(14*c^4*e^4*x^5+72*c^4*d*e^3*x^4+152*c^4*d^2*e^2*x^3+176*c^4*d^3*e*x^ 2+32*c^4*d^4*x+3*c^2*e^4*x^3+32*c^2*d*e^3*x^2-96*c^2*d^2*e^2*x-120*c^2*d^3 *e-12*e^4*x-67*d*e^3)/c^4/(e*x+d)*arccosh(c*x)-1/96*(6*c^2*e^3*x^3+32*c^2* d*e^2*x^2+72*c^2*d^2*e*x+96*c^2*d^3+9*e^3*x+64*d*e^2)/c^4*(c*x-1)*(c*x+1)/ (e*x+d)^3*(3*(e*x+d)^2*arccosh(c*x)*e+(e*x+d)^3*c/(c*x-1)^(1/2)/(c*x+1)^(1 /2))
Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.70 \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=\frac {3 \, {\left (8 \, c^{4} e^{3} x^{4} + 32 \, c^{4} d e^{2} x^{3} + 48 \, c^{4} d^{2} e x^{2} + 32 \, c^{4} d^{3} x - 24 \, c^{2} d^{2} e - 3 \, e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, c^{3} e^{3} x^{3} + 32 \, c^{3} d e^{2} x^{2} + 96 \, c^{3} d^{3} + 64 \, c d e^{2} + 9 \, {\left (8 \, c^{3} d^{2} e + c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{96 \, c^{4}} \] Input:
integrate((e*x+d)^3*arccosh(c*x),x, algorithm="fricas")
Output:
1/96*(3*(8*c^4*e^3*x^4 + 32*c^4*d*e^2*x^3 + 48*c^4*d^2*e*x^2 + 32*c^4*d^3* x - 24*c^2*d^2*e - 3*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (6*c^3*e^3*x^3 + 32*c^3*d*e^2*x^2 + 96*c^3*d^3 + 64*c*d*e^2 + 9*(8*c^3*d^2*e + c*e^3)*x)*sq rt(c^2*x^2 - 1))/c^4
\[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=\int \left (d + e x\right )^{3} \operatorname {acosh}{\left (c x \right )}\, dx \] Input:
integrate((e*x+d)**3*acosh(c*x),x)
Output:
Integral((d + e*x)**3*acosh(c*x), x)
Time = 0.03 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.05 \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=-\frac {1}{96} \, {\left (\frac {6 \, \sqrt {c^{2} x^{2} - 1} e^{3} x^{3}}{c^{2}} + \frac {32 \, \sqrt {c^{2} x^{2} - 1} d e^{2} x^{2}}{c^{2}} + \frac {72 \, \sqrt {c^{2} x^{2} - 1} d^{2} e x}{c^{2}} + \frac {96 \, \sqrt {c^{2} x^{2} - 1} d^{3}}{c^{2}} + \frac {72 \, d^{2} e \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}} + \frac {9 \, \sqrt {c^{2} x^{2} - 1} e^{3} x}{c^{4}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} d e^{2}}{c^{4}} + \frac {9 \, e^{3} \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c + \frac {1}{4} \, {\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \operatorname {arcosh}\left (c x\right ) \] Input:
integrate((e*x+d)^3*arccosh(c*x),x, algorithm="maxima")
Output:
-1/96*(6*sqrt(c^2*x^2 - 1)*e^3*x^3/c^2 + 32*sqrt(c^2*x^2 - 1)*d*e^2*x^2/c^ 2 + 72*sqrt(c^2*x^2 - 1)*d^2*e*x/c^2 + 96*sqrt(c^2*x^2 - 1)*d^3/c^2 + 72*d ^2*e*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3 + 9*sqrt(c^2*x^2 - 1)*e^3*x/ c^4 + 64*sqrt(c^2*x^2 - 1)*d*e^2/c^4 + 9*e^3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c + 1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)*arcco sh(c*x)
Time = 0.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.80 \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=\frac {{\left (e x + d\right )}^{4} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, e} - \frac {\sqrt {c^{2} x^{2} - 1} {\left ({\left (2 \, {\left (\frac {3 \, e^{4} x}{c} + \frac {16 \, d e^{3}}{c}\right )} x + \frac {9 \, {\left (8 \, c^{5} d^{2} e^{2} + c^{3} e^{4}\right )}}{c^{6}}\right )} x + \frac {32 \, {\left (3 \, c^{5} d^{3} e + 2 \, c^{3} d e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (8 \, c^{4} d^{4} + 24 \, c^{2} d^{2} e^{2} + 3 \, e^{4}\right )} \log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{c^{3} {\left | c \right |}}}{96 \, e} \] Input:
integrate((e*x+d)^3*arccosh(c*x),x, algorithm="giac")
Output:
1/4*(e*x + d)^4*log(c*x + sqrt(c^2*x^2 - 1))/e - 1/96*(sqrt(c^2*x^2 - 1)*( (2*(3*e^4*x/c + 16*d*e^3/c)*x + 9*(8*c^5*d^2*e^2 + c^3*e^4)/c^6)*x + 32*(3 *c^5*d^3*e + 2*c^3*d*e^3)/c^6) - 3*(8*c^4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*lo g(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^3*abs(c)))/e
Timed out. \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=\int \mathrm {acosh}\left (c\,x\right )\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int(acosh(c*x)*(d + e*x)^3,x)
Output:
int(acosh(c*x)*(d + e*x)^3, x)
Time = 0.19 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.04 \[ \int (d+e x)^3 \text {arccosh}(c x) \, dx=\frac {96 \mathit {acosh} \left (c x \right ) c^{4} d^{3} x +144 \mathit {acosh} \left (c x \right ) c^{4} d^{2} e \,x^{2}+96 \mathit {acosh} \left (c x \right ) c^{4} d \,e^{2} x^{3}+24 \mathit {acosh} \left (c x \right ) c^{4} e^{3} x^{4}-72 \sqrt {c^{2} x^{2}-1}\, c^{3} d^{2} e x -32 \sqrt {c^{2} x^{2}-1}\, c^{3} d \,e^{2} x^{2}-6 \sqrt {c^{2} x^{2}-1}\, c^{3} e^{3} x^{3}-64 \sqrt {c^{2} x^{2}-1}\, c d \,e^{2}-9 \sqrt {c^{2} x^{2}-1}\, c \,e^{3} x -96 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} d^{3}-72 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) c^{2} d^{2} e -9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) e^{3}}{96 c^{4}} \] Input:
int((e*x+d)^3*acosh(c*x),x)
Output:
(96*acosh(c*x)*c**4*d**3*x + 144*acosh(c*x)*c**4*d**2*e*x**2 + 96*acosh(c* x)*c**4*d*e**2*x**3 + 24*acosh(c*x)*c**4*e**3*x**4 - 72*sqrt(c**2*x**2 - 1 )*c**3*d**2*e*x - 32*sqrt(c**2*x**2 - 1)*c**3*d*e**2*x**2 - 6*sqrt(c**2*x* *2 - 1)*c**3*e**3*x**3 - 64*sqrt(c**2*x**2 - 1)*c*d*e**2 - 9*sqrt(c**2*x** 2 - 1)*c*e**3*x - 96*sqrt(c*x + 1)*sqrt(c*x - 1)*c**3*d**3 - 72*log(sqrt(c **2*x**2 - 1) + c*x)*c**2*d**2*e - 9*log(sqrt(c**2*x**2 - 1) + c*x)*e**3)/ (96*c**4)