Integrand size = 25, antiderivative size = 184 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \text {arccosh}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}+d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
-1/16*b*c*d^2*x*(c*x-1)^(3/2)*(c*x+1)^(3/2)-11/32*b*d^2*arccosh(c*x)+1/2*d ^2*(-c^2*x^2+1)*(a+b*arccosh(c*x))+1/4*d^2*(-c^2*x^2+1)^2*(a+b*arccosh(c*x ))+1/2*d^2*(a+b*arccosh(c*x))^2/b+d^2*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2))^2)-1/2*b*d^2*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))^2)+11/32*b*c*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)
Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{4} d^2 \left (-4 a c^2 x^2+a c^4 x^4-4 b c^2 x^2 \text {arccosh}(c x)+b c^4 x^4 \text {arccosh}(c x)+2 b \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )-\frac {1}{8} b \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+2 b \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+4 a \log (x)-2 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
(d^2*(-4*a*c^2*x^2 + a*c^4*x^4 - 4*b*c^2*x^2*ArcCosh[c*x] + b*c^4*x^4*ArcC osh[c*x] + 2*b*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c* x)/(1 + c*x)]]) - (b*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(3 + 3*c*x + 2*c^2*x^ 2 + 2*c^3*x^3) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/8 + 2*b*ArcCosh[c *x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) + 4*a*Log[x] - 2*b*Pol yLog[2, -E^(-2*ArcCosh[c*x])]))/4
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {6334, 27, 40, 40, 43, 6334, 40, 43, 6297, 25, 3042, 26, 4201, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle d \int \frac {d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c d^2 \int (c x-1)^{3/2} (c x+1)^{3/2}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c d^2 \int (c x-1)^{3/2} (c x+1)^{3/2}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \int \sqrt {c x-1} \sqrt {c x+1}dx\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle d^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \int \sqrt {c x-1} \sqrt {c x+1}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle d^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle d^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle d^2 \left (\frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^2 \left (-\frac {\int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^2 \left (-\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle d^2 \left (\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle d^2 \left (\frac {i \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle d^2 \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle d^2 \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle d^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {1}{4} b c d^2 \left (\frac {1}{4} x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{4} \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )\) |
(d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/4 - (b*c*d^2*((x*(-1 + c*x)^(3/ 2)*(1 + c*x)^(3/2))/4 - (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[c *x]/(2*c)))/4))/4 + d^2*(((1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/2 + (b*c*((x *Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[c*x]/(2*c)))/2 + (I*((-1/2*I)*( a + b*ArcCosh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c*x])*Log[1 + E^(-2* ArcCosh[c*x])]) + (b^2*PolyLog[2, -a - b*ArcCosh[c*x]])/4)))/b)
3.1.15.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* (a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1)) Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ b*c + a*d, 0] && IGtQ[m + 1/2, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcCosh[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcCosh[c*x])/x), x], x] - Simp[b*c*((-d )^p/(2*p)) Int[(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], x]) /; FreeQ [{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.95 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(d^{2} a \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (x \right )\right )+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}+\frac {d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {13 b \,d^{2} \operatorname {arccosh}\left (c x \right )}{32}-\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\) | \(192\) |
| derivativedivides | \(d^{2} a \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 b \,d^{2} \operatorname {arccosh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}\) | \(194\) |
| default | \(d^{2} a \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 b \,d^{2} \operatorname {arccosh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}\) | \(194\) |
d^2*a*(1/4*c^4*x^4-c^2*x^2+ln(x))+1/4*d^2*b*arccosh(c*x)*c^4*x^4-d^2*b*arc cosh(c*x)*c^2*x^2+1/2*d^2*b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2 )-1/2*d^2*b*arccosh(c*x)^2+13/32*b*d^2*arccosh(c*x)-1/16*d^2*b*(c*x+1)^(1/ 2)*(c*x-1)^(1/2)*c^3*x^3+13/32*b*c*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)+d^2*b *arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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))/x, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=d^{2} \left (\int \frac {a}{x}\, dx + \int \left (- 2 a c^{2} x\right )\, dx + \int a c^{4} x^{3}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x}\, dx + \int \left (- 2 b c^{2} x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a/x, x) + Integral(-2*a*c**2*x, x) + Integral(a*c**4*x**3, x) + Integral(b*acosh(c*x)/x, x) + Integral(-2*b*c**2*x*acosh(c*x), x) + I ntegral(b*c**4*x**3*acosh(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
1/4*a*c^4*d^2*x^4 - a*c^2*d^2*x^2 + a*d^2*log(x) + integrate(b*c^4*d^2*x^3 *log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) - 2*b*c^2*d^2*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + b*d^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x} \,d x \]