Integrand size = 27, antiderivative size = 378 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {8 b d^2 x \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 x^3 \sqrt {d-c^2 d x^2}}{2079 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^5 \sqrt {d-c^2 d x^2}}{1155 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {113 b c d^2 x^7 \sqrt {d-c^2 d x^2}}{4851 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^3 d^2 x^9 \sqrt {d-c^2 d x^2}}{891 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^6 d^3} \] Output:
8/693*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4/2079* b*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/1155*b*d^ 2*x^5*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-113/4851*b*c*d^2* x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+23/891*b*c^3*d^2*x^9* (-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/121*b*c^5*d^2*x^11*(-c^ 2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b *arccosh(c*x))/c^6/d+2/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arccosh(c*x))/c^6/d^2-1 /11*(-c^2*d*x^2+d)^(11/2)*(a+b*arccosh(c*x))/c^6/d^3
Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.43 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (b c \left (27720 x+4620 c^2 x^3+2079 c^4 x^5-55935 c^6 x^7+61985 c^8 x^9-19845 c^{10} x^{11}\right )+218295 c^4 x^4 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+13860 (-1+c x)^{7/2} (1+c x)^{7/2} \left (2+7 c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )}{2401245 c^6 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:
Integrate[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*(b*c*(27720*x + 4620*c^2*x^3 + 2079*c^4*x^5 - 559 35*c^6*x^7 + 61985*c^8*x^9 - 19845*c^10*x^11) + 218295*c^4*x^4*(-1 + c*x)^ (7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 13860*(-1 + c*x)^(7/2)*(1 + c *x)^(7/2)*(2 + 7*c^2*x^2)*(a + b*ArcCosh[c*x])))/(2401245*c^6*Sqrt[-1 + c* x]*Sqrt[1 + c*x])
Time = 0.87 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6337, 27, 1467, 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.
| \(\displaystyle \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )}{693 c^6}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )dx}{693 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (-63 c^{10} x^{10}+161 c^8 x^8-113 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{693 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d}+\frac {b d^2 \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right ) \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]
Output:
(b*d^2*Sqrt[d - c^2*d*x^2]*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (113*c^6 *x^7)/7 + (161*c^8*x^9)/9 - (63*c^10*x^11)/11))/(693*c^5*Sqrt[-1 + c*x]*Sq rt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^6*d) + (2 *(d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(9*c^6*d^2) - ((d - c^2*d*x^2 )^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^6*d^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo sh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c *x])] Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b , c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Time = 0.54 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.68
| method | result | size |
| orering | \(\frac {\left (83349 x^{12} c^{12}-299047 c^{10} x^{10}+363737 c^{8} x^{8}-140481 c^{6} x^{6}-7854 c^{4} x^{4}-53592 c^{2} x^{2}+33264\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{480249 c^{6} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (19845 c^{10} x^{10}-61985 c^{8} x^{8}+55935 c^{6} x^{6}-2079 c^{4} x^{4}-4620 c^{2} x^{2}-27720\right ) \left (5 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-5 x^{6} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d +\frac {x^{5} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{2401245 x^{4} c^{6} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\) | \(257\) |
| default | \(\text {Expression too large to display}\) | \(1840\) |
| parts | \(\text {Expression too large to display}\) | \(1840\) |
Input:
int(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
1/480249*(83349*c^12*x^12-299047*c^10*x^10+363737*c^8*x^8-140481*c^6*x^6-7 854*c^4*x^4-53592*c^2*x^2+33264)/c^6/(c*x-1)^2/(c*x+1)^2/(c^2*x^2-1)*(-c^2 *d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-1/2401245/x^4*(19845*c^10*x^10-61985*c^ 8*x^8+55935*c^6*x^6-2079*c^4*x^4-4620*c^2*x^2-27720)/c^6/(c*x-1)^2/(c*x+1) ^2*(5*x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-5*x^6*(-c^2*d*x^2+d)^(3/ 2)*(a+b*arccosh(c*x))*c^2*d+x^5*(-c^2*d*x^2+d)^(5/2)*b*c/(c*x-1)^(1/2)/(c* x+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.84 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3465 \, {\left (63 \, b c^{12} d^{2} x^{12} - 224 \, b c^{10} d^{2} x^{10} + 274 \, b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} - b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + 8 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (19845 \, b c^{11} d^{2} x^{11} - 61985 \, b c^{9} d^{2} x^{9} + 55935 \, b c^{7} d^{2} x^{7} - 2079 \, b c^{5} d^{2} x^{5} - 4620 \, b c^{3} d^{2} x^{3} - 27720 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 3465 \, {\left (63 \, a c^{12} d^{2} x^{12} - 224 \, a c^{10} d^{2} x^{10} + 274 \, a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} - a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + 8 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{2401245 \, {\left (c^{8} x^{2} - c^{6}\right )}} \] Input:
integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas ")
Output:
1/2401245*(3465*(63*b*c^12*d^2*x^12 - 224*b*c^10*d^2*x^10 + 274*b*c^8*d^2* x^8 - 116*b*c^6*d^2*x^6 - b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + 8*b*d^2)*sqrt( -c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (19845*b*c^11*d^2*x^11 - 61 985*b*c^9*d^2*x^9 + 55935*b*c^7*d^2*x^7 - 2079*b*c^5*d^2*x^5 - 4620*b*c^3* d^2*x^3 - 27720*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 3465*( 63*a*c^12*d^2*x^12 - 224*a*c^10*d^2*x^10 + 274*a*c^8*d^2*x^8 - 116*a*c^6*d ^2*x^6 - a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + 8*a*d^2)*sqrt(-c^2*d*x^2 + d))/ (c^8*x^2 - c^6)
Timed out. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:
integrate(x**5*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.66 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} a - \frac {{\left (19845 \, c^{10} \sqrt {-d} d^{2} x^{11} - 61985 \, c^{8} \sqrt {-d} d^{2} x^{9} + 55935 \, c^{6} \sqrt {-d} d^{2} x^{7} - 2079 \, c^{4} \sqrt {-d} d^{2} x^{5} - 4620 \, c^{2} \sqrt {-d} d^{2} x^{3} - 27720 \, \sqrt {-d} d^{2} x\right )} b}{2401245 \, c^{5}} \] Input:
integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima ")
Output:
-1/693*(63*(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)* x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*b*arccosh(c*x) - 1/693*(63 *(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^4*d ) + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*a - 1/2401245*(19845*c^10*sqrt(-d)*d ^2*x^11 - 61985*c^8*sqrt(-d)*d^2*x^9 + 55935*c^6*sqrt(-d)*d^2*x^7 - 2079*c ^4*sqrt(-d)*d^2*x^5 - 4620*c^2*sqrt(-d)*d^2*x^3 - 27720*sqrt(-d)*d^2*x)*b/ c^5
Exception generated. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),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. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int(x^5*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2),x)
Output:
int(x^5*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2), x)
\[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (63 \sqrt {-c^{2} x^{2}+1}\, a \,c^{10} x^{10}-161 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}+113 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a +693 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{9}d x \right ) b \,c^{10}-1386 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+693 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}\right )}{693 c^{6}} \] Input:
int(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x)),x)
Output:
(sqrt(d)*d**2*(63*sqrt( - c**2*x**2 + 1)*a*c**10*x**10 - 161*sqrt( - c**2* x**2 + 1)*a*c**8*x**8 + 113*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 3*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 - 4*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 8*sqr t( - c**2*x**2 + 1)*a + 693*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**9,x)* b*c**10 - 1386*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**7,x)*b*c**8 + 693* int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**5,x)*b*c**6))/(693*c**6)