Integrand size = 27, antiderivative size = 385 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^3 d^2 \sqrt {d-c^2 d x^2}}{792 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {113 b c^5 d^2 \sqrt {d-c^2 d x^2}}{4158 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{924 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2}}{693 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}-\frac {8 b c^{11} d^2 \sqrt {d-c^2 d x^2} \log (x)}{693 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/11*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/d/x^11-4/99*c^2*(-c^2*d*x^2+ d)^(7/2)*(a+b*arccosh(c*x))/d/x^9-8/693*c^4*(-c^2*d*x^2+d)^(7/2)*(a+b*arcc osh(c*x))/d/x^7-1/110*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^10/(c*x-1)^(1/2)/(c*x +1)^(1/2)+23/792*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^ (1/2)-113/4158*b*c^5*d^2*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1 /2)+1/924*b*c^7*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2 /693*b*c^9*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-8/693* b*c^11*d^2*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.43 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (7560 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+480 c^2 x^2 (-1+c x)^{7/2} (1+c x)^{7/2} \left (7+2 c^2 x^2\right ) (a+b \text {arccosh}(c x))-b c x \left (756-2415 c^2 x^2+2260 c^4 x^4-90 c^6 x^6-240 c^8 x^8+960 c^{10} x^{10} \log (x)\right )\right )}{83160 x^{11} \sqrt {-1+c x} \sqrt {1+c x}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(7560*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*Arc Cosh[c*x]) + 480*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(7 + 2*c^2*x^2)* (a + b*ArcCosh[c*x]) - b*c*x*(756 - 2415*c^2*x^2 + 2260*c^4*x^4 - 90*c^6*x ^6 - 240*c^8*x^8 + 960*c^10*x^10*Log[x])))/(83160*x^11*Sqrt[-1 + c*x]*Sqrt [1 + c*x])
Time = 0.52 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 1578, 1195, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 6337 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (8 c^4 x^4+28 c^2 x^2+63\right )}{693 x^{11}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3 \left (8 c^4 x^4+28 c^2 x^2+63\right )}{x^{11}}dx}{693 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3 \left (8 c^4 x^4+28 c^2 x^2+63\right )}{x^{12}}dx^2}{1386 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (-\frac {8 c^{10}}{x^2}-\frac {4 c^8}{x^4}-\frac {3 c^6}{x^6}+\frac {113 c^4}{x^8}-\frac {161 c^2}{x^{10}}+\frac {63}{x^{12}}\right )dx^2}{1386 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{99 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{693 d x^7}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-8 c^{10} \log \left (x^2\right )+\frac {4 c^8}{x^2}+\frac {3 c^6}{2 x^4}-\frac {113 c^4}{3 x^6}+\frac {161 c^2}{4 x^8}-\frac {63}{5 x^{10}}\right )}{1386 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/11*((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(d*x^11) - (4*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(99*d*x^9) - (8*c^4*(d - c^2*d*x^2) ^(7/2)*(a + b*ArcCosh[c*x]))/(693*d*x^7) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(- 63/(5*x^10) + (161*c^2)/(4*x^8) - (113*c^4)/(3*x^6) + (3*c^6)/(2*x^4) + (4 *c^8)/x^2 - 8*c^10*Log[x^2]))/(1386*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.1.95.3.1 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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/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. \(6381\) vs. \(2(325)=650\).
Time = 1.47 (sec) , antiderivative size = 6382, normalized size of antiderivative = 16.58
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(6382\) |
| parts | \(\text {Expression too large to display}\) | \(6382\) |
Time = 0.33 (sec) , antiderivative size = 879, normalized size of antiderivative = 2.28 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\left [\frac {120 \, {\left (8 \, b c^{12} d^{2} x^{12} - 4 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} + 274 \, b c^{4} d^{2} x^{4} - 224 \, b c^{2} d^{2} x^{2} + 63 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 480 \, {\left (b c^{13} d^{2} x^{13} - b c^{11} d^{2} x^{11}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (240 \, b c^{9} d^{2} x^{9} + 90 \, b c^{7} d^{2} x^{7} - {\left (240 \, b c^{9} + 90 \, b c^{7} - 2260 \, b c^{5} + 2415 \, b c^{3} - 756 \, b c\right )} d^{2} x^{11} - 2260 \, b c^{5} d^{2} x^{5} + 2415 \, b c^{3} d^{2} x^{3} - 756 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 120 \, {\left (8 \, a c^{12} d^{2} x^{12} - 4 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} + 274 \, a c^{4} d^{2} x^{4} - 224 \, a c^{2} d^{2} x^{2} + 63 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{83160 \, {\left (c^{2} x^{13} - x^{11}\right )}}, -\frac {960 \, {\left (b c^{13} d^{2} x^{13} - b c^{11} d^{2} x^{11}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 120 \, {\left (8 \, b c^{12} d^{2} x^{12} - 4 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} + 274 \, b c^{4} d^{2} x^{4} - 224 \, b c^{2} d^{2} x^{2} + 63 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (240 \, b c^{9} d^{2} x^{9} + 90 \, b c^{7} d^{2} x^{7} - {\left (240 \, b c^{9} + 90 \, b c^{7} - 2260 \, b c^{5} + 2415 \, b c^{3} - 756 \, b c\right )} d^{2} x^{11} - 2260 \, b c^{5} d^{2} x^{5} + 2415 \, b c^{3} d^{2} x^{3} - 756 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 120 \, {\left (8 \, a c^{12} d^{2} x^{12} - 4 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} + 274 \, a c^{4} d^{2} x^{4} - 224 \, a c^{2} d^{2} x^{2} + 63 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{83160 \, {\left (c^{2} x^{13} - x^{11}\right )}}\right ] \]
[1/83160*(120*(8*b*c^12*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 116 *b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 - 224*b*c^2*d^2*x^2 + 63*b*d^2)*sqrt(-c ^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 480*(b*c^13*d^2*x^13 - b*c^11 *d^2*x^11)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) + (240*b*c^ 9*d^2*x^9 + 90*b*c^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b *c^3 - 756*b*c)*d^2*x^11 - 2260*b*c^5*d^2*x^5 + 2415*b*c^3*d^2*x^3 - 756*b *c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 120*(8*a*c^12*d^2*x^12 - 4*a*c^10*d^2*x^10 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^ 4 - 224*a*c^2*d^2*x^2 + 63*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11), -1/83160*(960*(b*c^13*d^2*x^13 - b*c^11*d^2*x^11)*sqrt(d)*arctan(sqrt(-c^ 2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*sqrt(d)/(c^2*d*x^4 - (c^2 + 1)*d* x^2 + d)) - 120*(8*b*c^12*d^2*x^12 - 4*b*c^10*d^2*x^10 - b*c^8*d^2*x^8 - 1 16*b*c^6*d^2*x^6 + 274*b*c^4*d^2*x^4 - 224*b*c^2*d^2*x^2 + 63*b*d^2)*sqrt( -c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (240*b*c^9*d^2*x^9 + 90*b*c ^7*d^2*x^7 - (240*b*c^9 + 90*b*c^7 - 2260*b*c^5 + 2415*b*c^3 - 756*b*c)*d^ 2*x^11 - 2260*b*c^5*d^2*x^5 + 2415*b*c^3*d^2*x^3 - 756*b*c*d^2*x)*sqrt(-c^ 2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 120*(8*a*c^12*d^2*x^12 - 4*a*c^10*d^2*x^1 0 - a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 + 274*a*c^4*d^2*x^4 - 224*a*c^2*d^2* x^2 + 63*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11)]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.65 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {1}{83160} \, {\left (960 \, c^{10} \sqrt {-d} d^{2} \log \left (x\right ) - \frac {240 \, c^{8} \sqrt {-d} d^{2} x^{8} + 90 \, c^{6} \sqrt {-d} d^{2} x^{6} - 2260 \, c^{4} \sqrt {-d} d^{2} x^{4} + 2415 \, c^{2} \sqrt {-d} d^{2} x^{2} - 756 \, \sqrt {-d} d^{2}}{x^{10}}\right )} b c - \frac {1}{693} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{4}}{d x^{7}} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{9}} + \frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{11}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{693} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{4}}{d x^{7}} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{9}} + \frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{11}}\right )} \]
-1/83160*(960*c^10*sqrt(-d)*d^2*log(x) - (240*c^8*sqrt(-d)*d^2*x^8 + 90*c^ 6*sqrt(-d)*d^2*x^6 - 2260*c^4*sqrt(-d)*d^2*x^4 + 2415*c^2*sqrt(-d)*d^2*x^2 - 756*sqrt(-d)*d^2)/x^10)*b*c - 1/693*b*(8*(-c^2*d*x^2 + d)^(7/2)*c^4/(d* x^7) + 28*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^9) + 63*(-c^2*d*x^2 + d)^(7/2)/( d*x^11))*arccosh(c*x) - 1/693*a*(8*(-c^2*d*x^2 + d)^(7/2)*c^4/(d*x^7) + 28 *(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^9) + 63*(-c^2*d*x^2 + d)^(7/2)/(d*x^11))
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^{12}} \,d x \]