Integrand size = 12, antiderivative size = 178 \[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=-\frac {\text {arccosh}(c x)^2}{2 e}+\frac {\text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \] Output:
-1/2*arccosh(c*x)^2/e+arccosh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) )/(c*d-(c^2*d^2-e^2)^(1/2)))/e+arccosh(c*x)*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x +1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e+polylog(2,-e*(c*x+(c*x-1)^(1/2)*(c *x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e+polylog(2,-e*(c*x+(c*x-1)^(1/2)* (c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e
Time = 0.01 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=-\frac {\text {arccosh}(c x)^2}{2 e}+\frac {\text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,\frac {e e^{\text {arccosh}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \] Input:
Integrate[ArcCosh[c*x]/(d + e*x),x]
Output:
-1/2*ArcCosh[c*x]^2/e + (ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sq rt[c^2*d^2 - e^2])])/e + (ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d + S qrt[c^2*d^2 - e^2])])/e + PolyLog[2, (e*E^ArcCosh[c*x])/(-(c*d) + Sqrt[c^2 *d^2 - e^2])]/e + PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^ 2]))]/e
Time = 0.66 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6377, 6096, 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.
\(\displaystyle \int \frac {\text {arccosh}(c x)}{d+e x} \, dx\) |
\(\Big \downarrow \) 6377 |
\(\displaystyle \int \frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {arccosh}(c x)}{c d+c e x}d\text {arccosh}(c x)\) |
\(\Big \downarrow \) 6096 |
\(\displaystyle \int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)+\int \frac {e^{\text {arccosh}(c x)} \text {arccosh}(c x)}{c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)-\frac {\text {arccosh}(c x)^2}{2 e}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {\int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}-\frac {\int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}-\frac {\int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\text {arccosh}(c x) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\text {arccosh}(c x)^2}{2 e}\) |
Input:
Int[ArcCosh[c*x]/(d + e*x),x]
Output:
-1/2*ArcCosh[c*x]^2/e + (ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sq rt[c^2*d^2 - e^2])])/e + (ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d + S qrt[c^2*d^2 - e^2])])/e + PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2* d^2 - e^2]))]/e + PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^ 2]))]/e
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[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_ .)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo l] :> Subst[Int[(a + b*x)^n*(Sinh[x]/(c*d + e*Cosh[x])), x], x, ArcCosh[c*x ]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
Time = 0.44 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.71
method | result | size |
derivativedivides | \(\frac {-\frac {c \operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {c \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {c \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}}{c}\) | \(304\) |
default | \(\frac {-\frac {c \operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {c \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {c \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}}{c}\) | \(304\) |
Input:
int(arccosh(c*x)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/c*(-1/2*c*arccosh(c*x)^2/e+c/e*arccosh(c*x)*ln((-c*d-e*(c*x+(c*x-1)^(1/2 )*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2)))+c/e*arcc osh(c*x)*ln((c*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/ (c*d+(c^2*d^2-e^2)^(1/2)))+c/e*dilog((-c*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1 /2))+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2)))+c/e*dilog((c*d+e*(c* x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/ 2))))
\[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )}{e x + d} \,d x } \] Input:
integrate(arccosh(c*x)/(e*x+d),x, algorithm="fricas")
Output:
integral(arccosh(c*x)/(e*x + d), x)
\[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=\int \frac {\operatorname {acosh}{\left (c x \right )}}{d + e x}\, dx \] Input:
integrate(acosh(c*x)/(e*x+d),x)
Output:
Integral(acosh(c*x)/(d + e*x), x)
\[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )}{e x + d} \,d x } \] Input:
integrate(arccosh(c*x)/(e*x+d),x, algorithm="maxima")
Output:
integrate(arccosh(c*x)/(e*x + d), x)
\[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=\int { \frac {\operatorname {arcosh}\left (c x\right )}{e x + d} \,d x } \] Input:
integrate(arccosh(c*x)/(e*x+d),x, algorithm="giac")
Output:
integrate(arccosh(c*x)/(e*x + d), x)
Timed out. \[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=\int \frac {\mathrm {acosh}\left (c\,x\right )}{d+e\,x} \,d x \] Input:
int(acosh(c*x)/(d + e*x),x)
Output:
int(acosh(c*x)/(d + e*x), x)
\[ \int \frac {\text {arccosh}(c x)}{d+e x} \, dx=\int \frac {\mathit {acosh} \left (c x \right )}{e x +d}d x \] Input:
int(acosh(c*x)/(e*x+d),x)
Output:
int(acosh(c*x)/(d + e*x),x)