Integrand size = 27, antiderivative size = 458 \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {16 b d^2 x \sqrt {d-c^2 d x^2}}{3003 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^2 x^3 \sqrt {d-c^2 d x^2}}{9009 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^5 \sqrt {d-c^2 d x^2}}{5005 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^7 \sqrt {d-c^2 d x^2}}{21021 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {53 b c d^2 x^9 \sqrt {d-c^2 d x^2}}{3861 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {27 b c^3 d^2 x^{11} \sqrt {d-c^2 d x^2}}{1573 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{13} \sqrt {d-c^2 d x^2}}{169 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^2}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{13/2} (a+b \text {arccosh}(c x))}{13 c^8 d^4} \]
-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^8/d+1/3*(-c^2*d*x^2+d)^(9/2 )*(a+b*arccosh(c*x))/c^8/d^2-3/11*(-c^2*d*x^2+d)^(11/2)*(a+b*arccosh(c*x)) /c^8/d^3+1/13*(-c^2*d*x^2+d)^(13/2)*(a+b*arccosh(c*x))/c^8/d^4+16/3003*b*d ^2*x*(-c^2*d*x^2+d)^(1/2)/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)+8/9009*b*d^2*x^3 *(-c^2*d*x^2+d)^(1/2)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/5005*b*d^2*x^5*(-c ^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/21021*b*d^2*x^7*(-c^2* d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-53/3861*b*c*d^2*x^9*(-c^2*d*x ^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+27/1573*b*c^3*d^2*x^11*(-c^2*d*x^2 +d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/169*b*c^5*d^2*x^13*(-c^2*d*x^2+d)^ (1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.39 \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (b c \left (720720 x+120120 c^2 x^3+54054 c^4 x^5+32175 c^6 x^7-1856855 c^8 x^9+2321865 c^{10} x^{11}-800415 c^{12} x^{13}\right )+10405395 c^6 x^6 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+90090 (-1+c x)^{7/2} (1+c x)^{7/2} \left (8+28 c^2 x^2+63 c^4 x^4\right ) (a+b \text {arccosh}(c x))\right )}{135270135 c^8 \sqrt {-1+c x} \sqrt {1+c x}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(b*c*(720720*x + 120120*c^2*x^3 + 54054*c^4*x^5 + 32175*c^6*x^7 - 1856855*c^8*x^9 + 2321865*c^10*x^11 - 800415*c^12*x^13) + 10405395*c^6*x^6*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 90090*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4)*(a + b*ArcCosh[c*x])))/(135270135*c^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Time = 0.71 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.53, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6337, 27, 2341, 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.
| \(\displaystyle \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (231 c^6 x^6+126 c^4 x^4+56 c^2 x^2+16\right )}{3003 c^8}dx}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{13/2} (a+b \text {arccosh}(c x))}{13 c^8 d^4}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^3 \left (231 c^6 x^6+126 c^4 x^4+56 c^2 x^2+16\right )dx}{3003 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{13/2} (a+b \text {arccosh}(c x))}{13 c^8 d^4}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (-231 c^{12} x^{12}+567 c^{10} x^{10}-371 c^8 x^8+5 c^6 x^6+6 c^4 x^4+8 c^2 x^2+16\right )dx}{3003 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\left (d-c^2 d x^2\right )^{13/2} (a+b \text {arccosh}(c x))}{13 c^8 d^4}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (d-c^2 d x^2\right )^{13/2} (a+b \text {arccosh}(c x))}{13 c^8 d^4}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d}+\frac {b d^2 \left (-\frac {231}{13} c^{12} x^{13}+\frac {567 c^{10} x^{11}}{11}-\frac {371 c^8 x^9}{9}+\frac {5 c^6 x^7}{7}+\frac {6 c^4 x^5}{5}+\frac {8 c^2 x^3}{3}+16 x\right ) \sqrt {d-c^2 d x^2}}{3003 c^7 \sqrt {c x-1} \sqrt {c x+1}}\) |
(b*d^2*Sqrt[d - c^2*d*x^2]*(16*x + (8*c^2*x^3)/3 + (6*c^4*x^5)/5 + (5*c^6* x^7)/7 - (371*c^8*x^9)/9 + (567*c^10*x^11)/11 - (231*c^12*x^13)/13))/(3003 *c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh [c*x]))/(7*c^8*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(3*c^8*d^ 2) - (3*(d - c^2*d*x^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^8*d^3) + ((d - c^2*d*x^2)^(13/2)*(a + b*ArcCosh[c*x]))/(13*c^8*d^4)
3.1.96.3.1 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[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -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])
Leaf count of result is larger than twice the leaf count of optimal. \(2373\) vs. \(2(386)=772\).
Time = 0.97 (sec) , antiderivative size = 2374, normalized size of antiderivative = 5.18
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2374\) |
| parts | \(\text {Expression too large to display}\) | \(2374\) |
a*(-1/13*x^6*(-c^2*d*x^2+d)^(7/2)/c^2/d+6/13/c^2*(-1/11*x^4*(-c^2*d*x^2+d) ^(7/2)/c^2/d+4/11/c^2*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^ 2*d*x^2+d)^(7/2))))+b*(1/1384448*(-d*(c^2*x^2-1))^(1/2)*(-1-16896*c^8*x^8- 364*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4096*c^14*x^14-9984*(c*x+1)^(1/2)* (c*x-1)^(1/2)*x^7*c^7-1204*c^4*x^4+85*c^2*x^2+13*(c*x-1)^(1/2)*(c*x+1)^(1/ 2)*c*x+2912*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5-15360*c^12*x^12+6496*c^6*x ^6+22784*c^10*x^10+4096*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^13*c^13-13312*(c*x+1 )^(1/2)*(c*x-1)^(1/2)*x^11*c^11+16640*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9) *(-1+13*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)+1/991232*(-d*(c^2*x^2-1))^(1 /2)*(1+4096*c^8*x^8+220*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+2816*(c*x+1)^( 1/2)*(c*x-1)^(1/2)*x^7*c^7+620*c^4*x^4-61*c^2*x^2-11*(c*x-1)^(1/2)*(c*x+1) ^(1/2)*c*x-1232*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+1024*c^12*x^12-2352*c^ 6*x^6-3328*c^10*x^10+1024*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^11*c^11-2816*(c*x+ 1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9)*(-1+11*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x- 1)-1/110592*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8+256*(c*x+1)^ (1/2)*(c*x-1)^(1/2)*x^9*c^9+688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^ 7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+41*c^2*x^2-120*( c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*(- 1+9*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-3/200704*(-d*(c^2*x^2-1))^(1/2)* (64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*...
Time = 0.28 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.77 \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {45045 \, {\left (231 \, b c^{14} d^{2} x^{14} - 798 \, b c^{12} d^{2} x^{12} + 938 \, b c^{10} d^{2} x^{10} - 376 \, b c^{8} d^{2} x^{8} - b c^{6} d^{2} x^{6} - 2 \, b c^{4} d^{2} x^{4} - 8 \, b c^{2} d^{2} x^{2} + 16 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (800415 \, b c^{13} d^{2} x^{13} - 2321865 \, b c^{11} d^{2} x^{11} + 1856855 \, b c^{9} d^{2} x^{9} - 32175 \, b c^{7} d^{2} x^{7} - 54054 \, b c^{5} d^{2} x^{5} - 120120 \, b c^{3} d^{2} x^{3} - 720720 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 45045 \, {\left (231 \, a c^{14} d^{2} x^{14} - 798 \, a c^{12} d^{2} x^{12} + 938 \, a c^{10} d^{2} x^{10} - 376 \, a c^{8} d^{2} x^{8} - a c^{6} d^{2} x^{6} - 2 \, a c^{4} d^{2} x^{4} - 8 \, a c^{2} d^{2} x^{2} + 16 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{135270135 \, {\left (c^{10} x^{2} - c^{8}\right )}} \]
1/135270135*(45045*(231*b*c^14*d^2*x^14 - 798*b*c^12*d^2*x^12 + 938*b*c^10 *d^2*x^10 - 376*b*c^8*d^2*x^8 - b*c^6*d^2*x^6 - 2*b*c^4*d^2*x^4 - 8*b*c^2* d^2*x^2 + 16*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (8 00415*b*c^13*d^2*x^13 - 2321865*b*c^11*d^2*x^11 + 1856855*b*c^9*d^2*x^9 - 32175*b*c^7*d^2*x^7 - 54054*b*c^5*d^2*x^5 - 120120*b*c^3*d^2*x^3 - 720720* b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 45045*(231*a*c^14*d^2* x^14 - 798*a*c^12*d^2*x^12 + 938*a*c^10*d^2*x^10 - 376*a*c^8*d^2*x^8 - a*c ^6*d^2*x^6 - 2*a*c^4*d^2*x^4 - 8*a*c^2*d^2*x^2 + 16*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^10*x^2 - c^8)
Timed out. \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.68 \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{3003} \, {\left (\frac {231 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{6}}{c^{2} d} + \frac {126 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{4} d} + \frac {56 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{8} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{3003} \, {\left (\frac {231 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{6}}{c^{2} d} + \frac {126 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{4} d} + \frac {56 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{8} d}\right )} a - \frac {{\left (800415 \, c^{12} \sqrt {-d} d^{2} x^{13} - 2321865 \, c^{10} \sqrt {-d} d^{2} x^{11} + 1856855 \, c^{8} \sqrt {-d} d^{2} x^{9} - 32175 \, c^{6} \sqrt {-d} d^{2} x^{7} - 54054 \, c^{4} \sqrt {-d} d^{2} x^{5} - 120120 \, c^{2} \sqrt {-d} d^{2} x^{3} - 720720 \, \sqrt {-d} d^{2} x\right )} b}{135270135 \, c^{7}} \]
-1/3003*(231*(-c^2*d*x^2 + d)^(7/2)*x^6/(c^2*d) + 126*(-c^2*d*x^2 + d)^(7/ 2)*x^4/(c^4*d) + 56*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^6*d) + 16*(-c^2*d*x^2 + d)^(7/2)/(c^8*d))*b*arccosh(c*x) - 1/3003*(231*(-c^2*d*x^2 + d)^(7/2)*x^6/ (c^2*d) + 126*(-c^2*d*x^2 + d)^(7/2)*x^4/(c^4*d) + 56*(-c^2*d*x^2 + d)^(7/ 2)*x^2/(c^6*d) + 16*(-c^2*d*x^2 + d)^(7/2)/(c^8*d))*a - 1/135270135*(80041 5*c^12*sqrt(-d)*d^2*x^13 - 2321865*c^10*sqrt(-d)*d^2*x^11 + 1856855*c^8*sq rt(-d)*d^2*x^9 - 32175*c^6*sqrt(-d)*d^2*x^7 - 54054*c^4*sqrt(-d)*d^2*x^5 - 120120*c^2*sqrt(-d)*d^2*x^3 - 720720*sqrt(-d)*d^2*x)*b/c^7
Exception generated. \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^7 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^7\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]