Integrand size = 27, antiderivative size = 409 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/11*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^11-2/33*c^2*(-c^2*d*x^2+ d)^(5/2)*(a+b*arccosh(c*x))/d/x^9-8/231*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcc osh(c*x))/d/x^7-16/1155*c^6*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^5- 1/110*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^10/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/66*b*c ^3*d*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/1386*b*c^5*d*( -c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/770*b*c^7*d*(-c^2*d* x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/1155*b*c^9*d*(-c^2*d*x^2+d) ^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+16/1155*b*c^11*d*ln(x)*(-c^2*d*x^2+ d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.42 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (630 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+12 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (35+20 c^2 x^2+8 c^4 x^4\right ) (a+b \text {arccosh}(c x))+b c x \left (63-105 c^2 x^2+5 c^4 x^4+9 c^6 x^6+24 c^8 x^8-96 c^{10} x^{10} \log (x)\right )\right )}{6930 x^{11} \sqrt {-1+c x} \sqrt {1+c x}} \]
-1/6930*(d*Sqrt[d - c^2*d*x^2]*(630*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(35 + 20*c^2 *x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]) + b*c*x*(63 - 105*c^2*x^2 + 5*c^4*x ^4 + 9*c^6*x^6 + 24*c^8*x^8 - 96*c^10*x^10*Log[x])))/(x^11*Sqrt[-1 + c*x]* Sqrt[1 + c*x])
Time = 0.72 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 2331, 2123, 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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx\) |
\(\Big \downarrow \) 6337 |
\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{1155 x^{11}}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{x^{11}}dx}{1155 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\) |
\(\Big \downarrow \) 2331 |
\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right )}{x^{12}}dx^2}{2310 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {16 c^{10}}{x^2}+\frac {8 c^8}{x^4}+\frac {6 c^6}{x^6}+\frac {5 c^4}{x^8}-\frac {140 c^2}{x^{10}}+\frac {105}{x^{12}}\right )dx^2}{2310 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{231 d x^7}+\frac {b c d \sqrt {d-c^2 d x^2} \left (16 c^{10} \log \left (x^2\right )-\frac {8 c^8}{x^2}-\frac {3 c^6}{x^4}-\frac {5 c^4}{3 x^6}+\frac {35 c^2}{x^8}-\frac {21}{x^{10}}\right )}{2310 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/11*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(d*x^11) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2) ^(5/2)*(a + b*ArcCosh[c*x]))/(231*d*x^7) - (16*c^6*(d - c^2*d*x^2)^(5/2)*( a + b*ArcCosh[c*x]))/(1155*d*x^5) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-21/x^10 + (35*c^2)/x^8 - (5*c^4)/(3*x^6) - (3*c^6)/x^4 - (8*c^8)/x^2 + 16*c^10*Log[ x^2]))/(2310*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.1.79.3.1 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[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(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. \(5522\) vs. \(2(345)=690\).
Time = 1.39 (sec) , antiderivative size = 5523, normalized size of antiderivative = 13.50
method | result | size |
default | \(\text {Expression too large to display}\) | \(5523\) |
parts | \(\text {Expression too large to display}\) | \(5523\) |
Time = 0.33 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.94 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\left [-\frac {6 \, {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 48 \, {\left (b c^{13} d x^{13} - b c^{11} d 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 (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}, \frac {96 \, {\left (b c^{13} d x^{13} - b c^{11} d 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 ) - 6 \, {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}\right ] \]
[-1/6930*(6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b*c^6*d* x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*lo g(c*x + sqrt(c^2*x^2 - 1)) - 48*(b*c^13*d*x^13 - b*c^11*d*x^11)*sqrt(-d)*l og((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)) + (24*b*c^9*d*x^9 + 9*b*c^7*d*x^ 7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*d *x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1 ) + 6*(16*a*c^12*d*x^12 - 8*a*c^10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^ 13 - x^11), 1/6930*(96*(b*c^13*d*x^13 - b*c^11*d*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)) - 6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b* c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b *c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*d*x^5 - 10 5*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*(16 *a*c^12*d*x^12 - 8*a*c^10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4 *d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^13 - x^11 )]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Timed out} \]
Time = 0.45 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\frac {1}{6930} \, {\left (96 \, c^{10} \sqrt {-d} d \log \left (x\right ) - \frac {24 \, c^{8} \sqrt {-d} d x^{8} + 9 \, c^{6} \sqrt {-d} d x^{6} + 5 \, c^{4} \sqrt {-d} d x^{4} - 105 \, c^{2} \sqrt {-d} d x^{2} + 63 \, \sqrt {-d} d}{x^{10}}\right )} b c - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} a \]
1/6930*(96*c^10*sqrt(-d)*d*log(x) - (24*c^8*sqrt(-d)*d*x^8 + 9*c^6*sqrt(-d )*d*x^6 + 5*c^4*sqrt(-d)*d*x^4 - 105*c^2*sqrt(-d)*d*x^2 + 63*sqrt(-d)*d)/x ^10)*b*c - 1/1155*(16*(-c^2*d*x^2 + d)^(5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^9) + 105*(-c^2 *d*x^2 + d)^(5/2)/(d*x^11))*b*arccosh(c*x) - 1/1155*(16*(-c^2*d*x^2 + d)^( 5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^9) + 105*(-c^2*d*x^2 + d)^(5/2)/(d*x^11))*a
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^{12}} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{12}} \,d x \]