Integrand size = 16, antiderivative size = 191 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {7 b d \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2}{48 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3}{16 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}-\frac {b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \text {arccosh}(c x)}{32 c^4 e}+\frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e} \]
-1/32*b*(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* (a+b*arccosh(c*x))/e-7/48*b*d*(e*x+d)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/16 *b*(e*x+d)^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/96*b*(4*d*(19*c^2*d^2+16*e^2) +e*(26*c^2*d^2+9*e^2)*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3
Time = 0.18 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.01 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {24 a c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b c \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+24 b c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {arccosh}(c x)-9 b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{96 c^4} \]
(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*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*b*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*b*e*(8*c^2*d^2 + e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(96*c^4)
Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b c \int \frac {(d+e x)^4}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 e}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b c \left (\frac {\int \frac {(d+e x)^2 \left (4 d^2 c^2+7 d e x c^2+3 e^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {\int \frac {c^2 (d+e x) \left (d \left (12 c^2 d^2+23 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {1}{3} \int \frac {(d+e x) \left (d \left (12 c^2 d^2+23 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {1}{3} \left (\frac {3 \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{2 c^2}\right )+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arccosh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {1}{3} \left (\frac {3 \text {arccosh}(c x) \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right )}{2 c^3}+\frac {e \sqrt {c x-1} \sqrt {c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{2 c^2}\right )+\frac {7}{3} d e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
((d + e*x)^4*(a + b*ArcCosh[c*x]))/(4*e) - (b*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)
3.1.14.3.1 Defintions of rubi rules used
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 = 0.67 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.64
| method | result | size |
| parts | \(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{2}}{2}+c \,e^{2} \operatorname {arccosh}\left (c x \right ) d \,x^{3}+\frac {c \,e^{3} \operatorname {arccosh}\left (c x \right ) x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 c^{3} e \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(314\) |
| derivativedivides | \(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\) | \(331\) |
| default | \(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccosh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccosh}\left (c x \right ) c^{4} d \,x^{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{4} x^{4}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (24 c^{4} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+96 c^{3} d^{3} e \sqrt {c^{2} x^{2}-1}+72 c^{3} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+32 c^{3} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{2}+6 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+72 c^{2} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+64 c d \,e^{3} \sqrt {c^{2} x^{2}-1}+9 e^{4} c x \sqrt {c^{2} x^{2}-1}+9 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{96 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{3}}}{c}\) | \(331\) |
1/4*a*(e*x+d)^4/e+b/c*(1/4*c/e*arccosh(c*x)*d^4+arccosh(c*x)*c*x*d^3+3/2*c *arccosh(c*x)*d^2*e*x^2+c*e^2*arccosh(c*x)*d*x^3+1/4*c*e^3*arccosh(c*x)*x^ 4-1/96/c^3/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(24*c^4*d^4*ln(c*x+(c^2*x^2-1)^(1 /2))+96*c^3*d^3*e*(c^2*x^2-1)^(1/2)+72*c^3*d^2*e^2*x*(c^2*x^2-1)^(1/2)+32* c^3*d*e^3*(c^2*x^2-1)^(1/2)*x^2+6*e^4*c^3*x^3*(c^2*x^2-1)^(1/2)+72*c^2*d^2 *e^2*ln(c*x+(c^2*x^2-1)^(1/2))+64*c*d*e^3*(c^2*x^2-1)^(1/2)+9*e^4*c*x*(c^2 *x^2-1)^(1/2)+9*e^4*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {24 \, a c^{4} e^{3} x^{4} + 96 \, a c^{4} d e^{2} x^{3} + 144 \, a c^{4} d^{2} e x^{2} + 96 \, a c^{4} d^{3} x + 3 \, {\left (8 \, b c^{4} e^{3} x^{4} + 32 \, b c^{4} d e^{2} x^{3} + 48 \, b c^{4} d^{2} e x^{2} + 32 \, b c^{4} d^{3} x - 24 \, b c^{2} d^{2} e - 3 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{3} e^{3} x^{3} + 32 \, b c^{3} d e^{2} x^{2} + 96 \, b c^{3} d^{3} + 64 \, b c d e^{2} + 9 \, {\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{96 \, c^{4}} \]
1/96*(24*a*c^4*e^3*x^4 + 96*a*c^4*d*e^2*x^3 + 144*a*c^4*d^2*e*x^2 + 96*a*c ^4*d^3*x + 3*(8*b*c^4*e^3*x^4 + 32*b*c^4*d*e^2*x^3 + 48*b*c^4*d^2*e*x^2 + 32*b*c^4*d^3*x - 24*b*c^2*d^2*e - 3*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (6*b*c^3*e^3*x^3 + 32*b*c^3*d*e^2*x^2 + 96*b*c^3*d^3 + 64*b*c*d*e^2 + 9*(8 *b*c^3*d^2*e + b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^4
\[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x\right )^{3}\, dx \]
Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.39 \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \]
1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/4*(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))*b* d^2*e + 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^ 2*x^2 - 1)/c^4))*b*d*e^2 + 1/32*(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)*b*e^3 + a*d^3*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3 /c
Exception generated. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (d+e x)^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]