Integrand size = 21, antiderivative size = 494 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}+\frac {b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{5120 c^{10} e^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/8*d*(e*x^2+d)^4*(a+b*arccosh(c*x))/e^2+1/10*(e*x^2+d)^5*(a+b*arccosh(c* x))/e^2-1/76800*b*(1232*c^8*d^4-2536*c^6*d^3*e-7758*c^4*d^2*e^2-6615*c^2*d *e^3-1890*e^4)*x*(-c^2*x^2+1)/c^9/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/38400*b* (136*c^6*d^3-1096*c^4*d^2*e-1617*c^2*d*e^2-630*e^3)*x*(-c^2*x^2+1)*(e*x^2+ d)/c^7/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/9600*b*(26*c^4*d^2+201*c^2*d*e+126* e^2)*x*(-c^2*x^2+1)*(e*x^2+d)^2/c^5/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/1600*b *(11*c^2*d+18*e)*x*(-c^2*x^2+1)*(e*x^2+d)^3/c^3/e/(c*x-1)^(1/2)/(c*x+1)^(1 /2)+1/100*b*x*(-c^2*x^2+1)*(e*x^2+d)^4/c/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/5 120*b*(128*c^10*d^5-480*c^6*d^3*e^2-800*c^4*d^2*e^3-525*c^2*d*e^4-126*e^5) *arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^10/e^2/(c*x-1)^(1/2)/( c*x+1)^(1/2)
Time = 0.36 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.60 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1920 a x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )-\frac {b x \sqrt {-1+c x} \sqrt {1+c x} \left (1890 e^3+315 c^2 e^2 \left (25 d+4 e x^2\right )+6 c^4 e \left (2000 d^2+875 d e x^2+168 e^2 x^4\right )+8 c^6 \left (900 d^3+1000 d^2 e x^2+525 d e^2 x^4+108 e^3 x^6\right )+16 c^8 \left (300 d^3 x^2+400 d^2 e x^4+225 d e^2 x^6+48 e^3 x^8\right )\right )}{c^9}+1920 b x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right ) \text {arccosh}(c x)-\frac {30 b \left (480 c^6 d^3+800 c^4 d^2 e+525 c^2 d e^2+126 e^3\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^{10}}}{76800} \]
(1920*a*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) - (b*x*Sqrt [-1 + c*x]*Sqrt[1 + c*x]*(1890*e^3 + 315*c^2*e^2*(25*d + 4*e*x^2) + 6*c^4* e*(2000*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x^2 + 525*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 22 5*d*e^2*x^6 + 48*e^3*x^8)))/c^9 + 1920*b*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d *e^2*x^4 + 4*e^3*x^6)*ArcCosh[c*x] - (30*b*(480*c^6*d^3 + 800*c^4*d^2*e + 525*c^2*d*e^2 + 126*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/c^10)/76800
Time = 0.86 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.89, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6373, 27, 2041, 403, 27, 403, 403, 403, 299, 224, 219}
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.
| \(\displaystyle \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int -\frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{\sqrt {c x-1} \sqrt {c x+1}}dx}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 2041 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \int \frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{\sqrt {c^2 x^2-1}}dx}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {2 \left (e x^2+d\right )^3 \left (d \left (5 c^2 d-2 e\right )-e \left (11 d c^2+18 e\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{10 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {\left (e x^2+d\right )^3 \left (d \left (5 c^2 d-2 e\right )-e \left (11 d c^2+18 e\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\int \frac {\left (e x^2+d\right )^2 \left (d \left (40 d^2 c^4-27 d e c^2-18 e^2\right )-e \left (26 d^2 c^4+201 d e c^2+126 e^2\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{8 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\int \frac {\left (e x^2+d\right ) \left (e \left (136 d^3 c^6-1096 d^2 e c^4-1617 d e^2 c^2-630 e^3\right ) x^2+d \left (240 d^3 c^6-188 d^2 e c^4-309 d e^2 c^2-126 e^3\right )\right )}{\sqrt {c^2 x^2-1}}dx}{6 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {\int \frac {e \left (1232 d^4 c^8-2536 d^3 e c^6-7758 d^2 e^2 c^4-6615 d e^3 c^2-1890 e^4\right ) x^2+d \left (960 d^4 c^8-616 d^3 e c^6-2332 d^2 e^2 c^4-2121 d e^3 c^2-630 e^4\right )}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {\frac {15 \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{2 c^2}}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {\frac {15 \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{2 c^2}}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {\frac {15 \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right )}{2 c^3}+\frac {e x \sqrt {c^2 x^2-1} \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{2 c^2}}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}-\frac {e x \sqrt {c^2 x^2-1} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{8 c^2}}{5 c^2}-\frac {2 e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^4}{5 c^2}\right )}{40 e^2 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/8*(d*(d + e*x^2)^4*(a + b*ArcCosh[c*x]))/e^2 + ((d + e*x^2)^5*(a + b*Ar cCosh[c*x]))/(10*e^2) + (b*c*Sqrt[-1 + c^2*x^2]*((-2*e*x*Sqrt[-1 + c^2*x^2 ]*(d + e*x^2)^4)/(5*c^2) + (-1/8*(e*(11*c^2*d + 18*e)*x*Sqrt[-1 + c^2*x^2] *(d + e*x^2)^3)/c^2 + (-1/6*(e*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*Sqrt [-1 + c^2*x^2]*(d + e*x^2)^2)/c^2 + ((e*(136*c^6*d^3 - 1096*c^4*d^2*e - 16 17*c^2*d*e^2 - 630*e^3)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2))/(4*c^2) + ((e*(1 232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^ 4)*x*Sqrt[-1 + c^2*x^2])/(2*c^2) + (15*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 8 00*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2] ])/(2*c^3))/(4*c^2))/(6*c^2))/(8*c^2))/(5*c^2)))/(40*e^2*Sqrt[-1 + c*x]*Sq rt[1 + c*x])
3.5.80.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((e1_) + (f1_.)*(x_)^(n2_.))^(r_.)*((e2_) + (f2_.)*(x_)^(n2_.))^(r_.)*( (a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(e1 + f1*x^(n/2))^FracPart[r]*((e2 + f2*x^(n/2))^FracPart[r]/(e1*e2 + f1*f2*x^n)^FracPart[r]) Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^ n)^r, x], x] /; FreeQ[{a, b, c, d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n 2, n/2] && EqQ[e2*f1 + e1*f2, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x _)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim p[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le Q[m + p, 0]))
Time = 0.77 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.09
| method | result | size |
| parts | \(a \left (\frac {1}{10} e^{3} x^{10}+\frac {3}{8} d \,e^{2} x^{8}+\frac {1}{2} d^{2} e \,x^{6}+\frac {1}{4} d^{3} x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccosh}\left (c x \right ) e^{3} x^{10}}{10}+\frac {3 c^{4} \operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{6}}{2}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4} d^{3}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4800 c^{9} d^{3} \sqrt {c^{2} x^{2}-1}\, x^{3}+6400 c^{9} d^{2} e \sqrt {c^{2} x^{2}-1}\, x^{5}+3600 c^{9} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{7}+768 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{9} x^{9}+7200 c^{7} d^{3} x \sqrt {c^{2} x^{2}-1}+8000 \sqrt {c^{2} x^{2}-1}\, c^{7} d^{2} e \,x^{3}+4200 \sqrt {c^{2} x^{2}-1}\, c^{7} d \,e^{2} x^{5}+864 e^{3} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+7200 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+12000 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+5250 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+1008 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+12000 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+7875 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+1260 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+7875 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+1890 e^{3} c x \sqrt {c^{2} x^{2}-1}+1890 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{76800 c^{6} \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(538\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} c^{10} d^{3} x^{4}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{10} c^{10} e^{3} x^{10}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{3} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{2} e \,x^{6}}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{10} d \,e^{2} x^{8}}{8}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4800 c^{9} d^{3} \sqrt {c^{2} x^{2}-1}\, x^{3}+6400 c^{9} d^{2} e \sqrt {c^{2} x^{2}-1}\, x^{5}+3600 c^{9} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{7}+768 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{9} x^{9}+7200 c^{7} d^{3} x \sqrt {c^{2} x^{2}-1}+8000 \sqrt {c^{2} x^{2}-1}\, c^{7} d^{2} e \,x^{3}+4200 \sqrt {c^{2} x^{2}-1}\, c^{7} d \,e^{2} x^{5}+864 e^{3} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+7200 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+12000 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+5250 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+1008 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+12000 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+7875 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+1260 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+7875 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+1890 e^{3} c x \sqrt {c^{2} x^{2}-1}+1890 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{76800 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{4}}\) | \(554\) |
| default | \(\frac {\frac {a \left (\frac {1}{4} c^{10} d^{3} x^{4}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{10} c^{10} e^{3} x^{10}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{3} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{2} e \,x^{6}}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{10} d \,e^{2} x^{8}}{8}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4800 c^{9} d^{3} \sqrt {c^{2} x^{2}-1}\, x^{3}+6400 c^{9} d^{2} e \sqrt {c^{2} x^{2}-1}\, x^{5}+3600 c^{9} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{7}+768 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{9} x^{9}+7200 c^{7} d^{3} x \sqrt {c^{2} x^{2}-1}+8000 \sqrt {c^{2} x^{2}-1}\, c^{7} d^{2} e \,x^{3}+4200 \sqrt {c^{2} x^{2}-1}\, c^{7} d \,e^{2} x^{5}+864 e^{3} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+7200 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+12000 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+5250 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+1008 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+12000 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+7875 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+1260 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+7875 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+1890 e^{3} c x \sqrt {c^{2} x^{2}-1}+1890 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{76800 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{4}}\) | \(554\) |
a*(1/10*e^3*x^10+3/8*d*e^2*x^8+1/2*d^2*e*x^6+1/4*d^3*x^4)+b/c^4*(1/10*c^4* arccosh(c*x)*e^3*x^10+3/8*c^4*arccosh(c*x)*d*e^2*x^8+1/2*c^4*arccosh(c*x)* d^2*e*x^6+1/4*arccosh(c*x)*c^4*x^4*d^3-1/76800/c^6*(c*x-1)^(1/2)*(c*x+1)^( 1/2)*(4800*c^9*d^3*(c^2*x^2-1)^(1/2)*x^3+6400*c^9*d^2*e*(c^2*x^2-1)^(1/2)* x^5+3600*c^9*d*e^2*(c^2*x^2-1)^(1/2)*x^7+768*e^3*(c^2*x^2-1)^(1/2)*c^9*x^9 +7200*c^7*d^3*x*(c^2*x^2-1)^(1/2)+8000*(c^2*x^2-1)^(1/2)*c^7*d^2*e*x^3+420 0*(c^2*x^2-1)^(1/2)*c^7*d*e^2*x^5+864*e^3*c^7*x^7*(c^2*x^2-1)^(1/2)+7200*c ^6*d^3*ln(c*x+(c^2*x^2-1)^(1/2))+12000*c^5*d^2*e*x*(c^2*x^2-1)^(1/2)+5250* c^5*d*e^2*(c^2*x^2-1)^(1/2)*x^3+1008*e^3*(c^2*x^2-1)^(1/2)*c^5*x^5+12000*c ^4*d^2*e*ln(c*x+(c^2*x^2-1)^(1/2))+7875*c^3*d*e^2*x*(c^2*x^2-1)^(1/2)+1260 *e^3*c^3*x^3*(c^2*x^2-1)^(1/2)+7875*c^2*d*e^2*ln(c*x+(c^2*x^2-1)^(1/2))+18 90*e^3*c*x*(c^2*x^2-1)^(1/2)+1890*e^3*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2- 1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.67 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {7680 \, a c^{10} e^{3} x^{10} + 28800 \, a c^{10} d e^{2} x^{8} + 38400 \, a c^{10} d^{2} e x^{6} + 19200 \, a c^{10} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} e^{3} x^{10} + 1920 \, b c^{10} d e^{2} x^{8} + 2560 \, b c^{10} d^{2} e x^{6} + 1280 \, b c^{10} d^{3} x^{4} - 480 \, b c^{6} d^{3} - 800 \, b c^{4} d^{2} e - 525 \, b c^{2} d e^{2} - 126 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (768 \, b c^{9} e^{3} x^{9} + 144 \, {\left (25 \, b c^{9} d e^{2} + 6 \, b c^{7} e^{3}\right )} x^{7} + 8 \, {\left (800 \, b c^{9} d^{2} e + 525 \, b c^{7} d e^{2} + 126 \, b c^{5} e^{3}\right )} x^{5} + 10 \, {\left (480 \, b c^{9} d^{3} + 800 \, b c^{7} d^{2} e + 525 \, b c^{5} d e^{2} + 126 \, b c^{3} e^{3}\right )} x^{3} + 15 \, {\left (480 \, b c^{7} d^{3} + 800 \, b c^{5} d^{2} e + 525 \, b c^{3} d e^{2} + 126 \, b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{76800 \, c^{10}} \]
1/76800*(7680*a*c^10*e^3*x^10 + 28800*a*c^10*d*e^2*x^8 + 38400*a*c^10*d^2* e*x^6 + 19200*a*c^10*d^3*x^4 + 15*(512*b*c^10*e^3*x^10 + 1920*b*c^10*d*e^2 *x^8 + 2560*b*c^10*d^2*e*x^6 + 1280*b*c^10*d^3*x^4 - 480*b*c^6*d^3 - 800*b *c^4*d^2*e - 525*b*c^2*d*e^2 - 126*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - ( 768*b*c^9*e^3*x^9 + 144*(25*b*c^9*d*e^2 + 6*b*c^7*e^3)*x^7 + 8*(800*b*c^9* d^2*e + 525*b*c^7*d*e^2 + 126*b*c^5*e^3)*x^5 + 10*(480*b*c^9*d^3 + 800*b*c ^7*d^2*e + 525*b*c^5*d*e^2 + 126*b*c^3*e^3)*x^3 + 15*(480*b*c^7*d^3 + 800* b*c^5*d^2*e + 525*b*c^3*d*e^2 + 126*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^10
\[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]
Time = 0.23 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.99 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{3} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d^{2} e + \frac {1}{1024} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b d e^{2} + \frac {1}{12800} \, {\left (1280 \, x^{10} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} - 1} x}{c^{10}} + \frac {315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{11}}\right )} c\right )} b e^{3} \]
1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 + 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*d^3 + 1/96*(48*x^ 6*arccosh(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*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/ c^7)*c)*b*d^2*e + 1/1024*(384*x^8*arccosh(c*x) - (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*s qrt(c^2*x^2 - 1)*x/c^8 + 105*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^9)*c)* b*d*e^2 + 1/12800*(1280*x^10*arccosh(c*x) - (128*sqrt(c^2*x^2 - 1)*x^9/c^2 + 144*sqrt(c^2*x^2 - 1)*x^7/c^4 + 168*sqrt(c^2*x^2 - 1)*x^5/c^6 + 210*sqr t(c^2*x^2 - 1)*x^3/c^8 + 315*sqrt(c^2*x^2 - 1)*x/c^10 + 315*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^11)*c)*b*e^3
Exception generated. \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]