Integrand size = 25, antiderivative size = 200 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \text {arccosh}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d^2 (a+b \text {arccosh}(c x))^2}{b}-2 c^2 d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+b c^2 d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
-1/2*b*c*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/x-1/4*b*c^2*d^2*arccosh(c*x)-c^2* d^2*(-c^2*x^2+1)*(a+b*arccosh(c*x))-1/2*d^2*(-c^2*x^2+1)^2*(a+b*arccosh(c* x))/x^2-c^2*d^2*(a+b*arccosh(c*x))^2/b-2*c^2*d^2*(a+b*arccosh(c*x))*ln(1+1 /(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+b*c^2*d^2*polylog(2,-1/(c*x+(c*x-1)^ (1/2)*(c*x+1)^(1/2))^2)+1/4*b*c^3*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)
Time = 0.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {d^2 \left (-2 a+2 a c^4 x^4+2 b c x \sqrt {-1+c x} \sqrt {1+c x}-b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}-4 b c^2 x^2 \text {arccosh}(c x)^2-2 b c^2 x^2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )+2 b \text {arccosh}(c x) \left (-1+c^4 x^4-4 c^2 x^2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-8 a c^2 x^2 \log (x)+4 b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{4 x^2} \]
(d^2*(-2*a + 2*a*c^4*x^4 + 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b*c^3*x^ 3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 4*b*c^2*x^2*ArcCosh[c*x]^2 - 2*b*c^2*x^2* ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 2*b*ArcCosh[c*x]*(-1 + c^4*x^4 - 4*c ^2*x^2*Log[1 + E^(-2*ArcCosh[c*x])]) - 8*a*c^2*x^2*Log[x] + 4*b*c^2*x^2*Po lyLog[2, -E^(-2*ArcCosh[c*x])]))/(4*x^2)
Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.24, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {6335, 27, 108, 27, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 6335 |
| \(\displaystyle -2 c^2 d \int \frac {d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c d^2 \int \frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x^2}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c d^2 \int \frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x^2}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c d^2 \left (\int 3 c^2 \sqrt {c x-1} \sqrt {c x+1}dx-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c d^2 \left (3 c^2 \int \sqrt {c x-1} \sqrt {c x+1}dx-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 6334 |
| \(\displaystyle -2 c^2 d^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \int \sqrt {c x-1} \sqrt {c x+1}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 40 |
| \(\displaystyle -2 c^2 d^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{2} \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -2 c^2 d^2 \left (\int \frac {a+b \text {arccosh}(c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {\int -\left ((a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 c^2 d^2 \left (-\frac {\int (a+b \text {arccosh}(c x)) \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 c^2 d^2 \left (-\frac {\int -i (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {i \int (a+b \text {arccosh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )d(a+b \text {arccosh}(c x))}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {i \left (2 i \int \frac {e^{-2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1+e^{-2 \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{-2 \text {arccosh}(c x)}\right )d(a+b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{2 \text {arccosh}(c x)} \log \left (1+e^{-2 \text {arccosh}(c x)}\right )de^{-2 \text {arccosh}(c x)}-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arccosh}(c x))-\frac {1}{2} b \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))\right )-\frac {1}{2} i (a+b \text {arccosh}(c x))^2\right )}{b}+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{2} b c \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c x-1} \sqrt {c x+1}-\frac {\text {arccosh}(c x)}{2 c}\right )-\frac {(c x-1)^{3/2} (c x+1)^{3/2}}{x}\right )\) |
-1/2*(d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/x^2 + (b*c*d^2*(-(((-1 + c *x)^(3/2)*(1 + c*x)^(3/2))/x) + 3*c^2*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/2 - ArcCosh[c*x]/(2*c))))/2 - 2*c^2*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)
3.1.17.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcCosh[c *x])/(f*(m + 1))), x] + (-Simp[b*c*((-d)^p/(f*(m + 1))) Int[(f*x)^(m + 1) *(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x], 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]), x], x]) / ; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[( m + 1)/2, 0]
Time = 0.90 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}-2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \operatorname {arccosh}\left (c x \right )^{2}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )}{4}-\frac {d^{2} b}{2}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-2 d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\right )\) | \(206\) |
| default | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}-2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \operatorname {arccosh}\left (c x \right )^{2}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )}{4}-\frac {d^{2} b}{2}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-2 d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\right )\) | \(206\) |
| parts | \(d^{2} a \left (\frac {c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}-2 c^{2} \ln \left (x \right )\right )+d^{2} b \,c^{2} \operatorname {arccosh}\left (c x \right )^{2}+\frac {d^{2} b \,c^{4} \operatorname {arccosh}\left (c x \right ) x^{2}}{2}-\frac {b \,c^{3} d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b \,c^{2} d^{2} \operatorname {arccosh}\left (c x \right )}{4}-\frac {d^{2} b \,c^{2}}{2}+\frac {b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{2 x}-\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right )}{2 x^{2}}-2 d^{2} b \,c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \,c^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\) | \(212\) |
c^2*(d^2*a*(1/2*c^2*x^2-2*ln(c*x)-1/2/c^2/x^2)+d^2*b*arccosh(c*x)^2-1/4*b* c*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/2*d^2*b*arccosh(c*x)*c^2*x^2-1/4*b*d ^2*arccosh(c*x)-1/2*d^2*b+1/2*d^2*b/c/x*(c*x+1)^(1/2)*(c*x-1)^(1/2)-1/2*d^ 2*b*arccosh(c*x)/c^2/x^2-2*d^2*b*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x +1)^(1/2))^2)-d^2*b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \left (- \frac {2 a c^{2}}{x}\right )\, dx + \int a c^{4} x\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\, dx + \int \left (- \frac {2 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int b c^{4} x \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a/x**3, x) + Integral(-2*a*c**2/x, x) + Integral(a*c**4*x, x) + Integral(b*acosh(c*x)/x**3, x) + Integral(-2*b*c**2*acosh(c*x)/x, x) + Integral(b*c**4*x*acosh(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/2*a*c^4*d^2*x^2 - 2*a*c^2*d^2*log(x) + 1/2*b*d^2*(sqrt(c^2*x^2 - 1)*c/x - arccosh(c*x)/x^2) - 1/2*a*d^2/x^2 + integrate(b*c^4*d^2*x*log(c*x + sqrt (c*x + 1)*sqrt(c*x - 1)) - 2*b*c^2*d^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x, x)
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^3} \,d x \]