Integrand size = 27, antiderivative size = 293 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {-1+c x} \sqrt {1+c x}} \]
5/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x-1/3*(-c^2*d*x^2+d)^(5/ 2)*(a+b*arccosh(c*x))/x^3+5/2*c^4*d^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^ (1/2)-1/6*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/4 *b*c^5*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/4*c^3*d^ 2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/(c*x-1)^(1/2)/(c*x+1)^(1/2)- 7/3*b*c^3*d^2*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 1.31 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {30 b c^3 d^3 x^3 (-1+c x) \text {arccosh}(c x)^2-60 a c^3 d^{5/2} x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+3 b c^3 d^3 x^3 (-1+c x) \cosh (2 \text {arccosh}(c x))-4 d^3 \left (b c x (1-c x)+a \sqrt {\frac {-1+c x}{1+c x}} \left (2-16 c^2 x^2+11 c^4 x^4+3 c^6 x^6\right )-14 b c^3 x^3 (-1+c x) \log (c x)\right )-2 b d^3 (-1+c x) \text {arccosh}(c x) \left (4 \sqrt {\frac {-1+c x}{1+c x}} \left (-1-c x+7 c^2 x^2+7 c^3 x^3\right )+3 c^3 x^3 \sinh (2 \text {arccosh}(c x))\right )}{24 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \]
(30*b*c^3*d^3*x^3*(-1 + c*x)*ArcCosh[c*x]^2 - 60*a*c^3*d^(5/2)*x^3*Sqrt[(- 1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/( Sqrt[d]*(-1 + c^2*x^2))] + 3*b*c^3*d^3*x^3*(-1 + c*x)*Cosh[2*ArcCosh[c*x]] - 4*d^3*(b*c*x*(1 - c*x) + a*Sqrt[(-1 + c*x)/(1 + c*x)]*(2 - 16*c^2*x^2 + 11*c^4*x^4 + 3*c^6*x^6) - 14*b*c^3*x^3*(-1 + c*x)*Log[c*x]) - 2*b*d^3*(-1 + c*x)*ArcCosh[c*x]*(4*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 - c*x + 7*c^2*x^2 + 7*c^3*x^3) + 3*c^3*x^3*Sinh[2*ArcCosh[c*x]]))/(24*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])
Time = 1.12 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6343, 82, 243, 49, 2009, 6343, 25, 82, 244, 2009, 6310, 15, 6308}
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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx\) |
\(\Big \downarrow \) 6343 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^2 (c x+1)^2}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^3}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^4}dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4-\frac {2 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6343 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int -\frac {(1-c x) (c x+1)}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {(1-c x) (c x+1)}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x}dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x}-c^2 x\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6310 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int xdx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 6308 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/3*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^3 - (5*c^2*d*(-(((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x) - 3*c^2*d*(-1/4*(b*c*x^2*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c* Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/2*(c^2*x^2 ) + Log[x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/3 + (b*c*d^2*Sqrt[d - c^2*d* x^2]*(-x^(-2) + c^4*x^2 - 2*c^2*Log[x^2]))/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x] )
3.1.91.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(a + b*ArcC osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[x*(a + b*ArcCosh[c*x])^ (n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n , 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Cosh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m + 1) *(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && G tQ[p, 0] && LtQ[m, -1]
Time = 1.14 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.16
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-56 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(340\) |
parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+6 c^{5} x^{5}+30 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}+56 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}-56 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-56 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-3 c^{3} x^{3}+8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) d^{2}}{24 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}\) | \(340\) |
-1/3*a/d/x^3*(-c^2*d*x^2+d)^(7/2)+4/3*a*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+4/3*a *c^4*x*(-c^2*d*x^2+d)^(5/2)+5/3*a*c^4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d ^2*x*(-c^2*d*x^2+d)^(1/2)+5/2*a*c^4*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) *x/(-c^2*d*x^2+d)^(1/2))-1/24*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+ 1)^(1/2)/x^3*(-12*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4+6*c^5*x ^5+30*arccosh(c*x)^2*x^3*c^3+56*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)* x^3*c^3-56*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-56*c^3*x^3*arc cosh(c*x)-3*c^3*x^3+8*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*d^2
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c ^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/x^4, x)
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
1/6*(10*(-c^2*d*x^2 + d)^(3/2)*c^4*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^4*d^2*x + 15*c^3*d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^2/x - 2*(-c^2*d *x^2 + d)^(7/2)/(d*x^3))*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^4, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \]