\(\int \frac {\text {arccosh}(a x^5)}{x} \, dx\) [140]

Optimal result

Integrand size = 10, antiderivative size = 54 \[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=-\frac {1}{10} \text {arccosh}\left (a x^5\right )^2+\frac {1}{5} \text {arccosh}\left (a x^5\right ) \log \left (1+e^{2 \text {arccosh}\left (a x^5\right )}\right )+\frac {1}{10} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (a x^5\right )}\right ) \] Output:

-1/10*arccosh(a*x^5)^2+1/5*arccosh(a*x^5)*ln(1+(a*x^5+(a*x^5-1)^(1/2)*(a*x 
^5+1)^(1/2))^2)+1/10*polylog(2,-(a*x^5+(a*x^5-1)^(1/2)*(a*x^5+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \left (\text {arccosh}\left (a x^5\right ) \left (\text {arccosh}\left (a x^5\right )+2 \log \left (1+e^{-2 \text {arccosh}\left (a x^5\right )}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (a x^5\right )}\right )\right ) \] Input:

Integrate[ArcCosh[a*x^5]/x,x]
 

Output:

(ArcCosh[a*x^5]*(ArcCosh[a*x^5] + 2*Log[1 + E^(-2*ArcCosh[a*x^5])]) - Poly 
Log[2, -E^(-2*ArcCosh[a*x^5])])/10
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6426, 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.

\(\displaystyle \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx\)

\(\Big \downarrow \) 6426

\(\displaystyle \frac {1}{5} \int \frac {\sqrt {\frac {a x^5-1}{a x^5+1}} \left (a x^5+1\right ) \text {arccosh}\left (a x^5\right )}{a x^5}d\text {arccosh}\left (a x^5\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \int -i \text {arccosh}\left (a x^5\right ) \tan \left (i \text {arccosh}\left (a x^5\right )\right )d\text {arccosh}\left (a x^5\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {1}{5} i \int \text {arccosh}\left (a x^5\right ) \tan \left (i \text {arccosh}\left (a x^5\right )\right )d\text {arccosh}\left (a x^5\right )\)

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {1}{5} i \left (2 i \int \frac {e^{2 \text {arccosh}\left (a x^5\right )} \text {arccosh}\left (a x^5\right )}{1+e^{2 \text {arccosh}\left (a x^5\right )}}d\text {arccosh}\left (a x^5\right )-\frac {1}{2} i \text {arccosh}\left (a x^5\right )^2\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {1}{5} i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (a x^5\right ) \log \left (e^{2 \text {arccosh}\left (a x^5\right )}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}\left (a x^5\right )}\right )d\text {arccosh}\left (a x^5\right )\right )-\frac {1}{2} i \text {arccosh}\left (a x^5\right )^2\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {1}{5} i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (a x^5\right ) \log \left (e^{2 \text {arccosh}\left (a x^5\right )}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}\left (a x^5\right )} \log \left (1+e^{2 \text {arccosh}\left (a x^5\right )}\right )de^{2 \text {arccosh}\left (a x^5\right )}\right )-\frac {1}{2} i \text {arccosh}\left (a x^5\right )^2\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {1}{5} i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (a x^5\right )}\right )+\frac {1}{2} \text {arccosh}\left (a x^5\right ) \log \left (e^{2 \text {arccosh}\left (a x^5\right )}+1\right )\right )-\frac {1}{2} i \text {arccosh}\left (a x^5\right )^2\right )\)

Input:

Int[ArcCosh[a*x^5]/x,x]
 

Output:

(-1/5*I)*((-1/2*I)*ArcCosh[a*x^5]^2 + (2*I)*((ArcCosh[a*x^5]*Log[1 + E^(2* 
ArcCosh[a*x^5])])/2 + PolyLog[2, -E^(2*ArcCosh[a*x^5])]/4))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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]
 

rule 6426

Int[ArcCosh[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Simp[1/p   Subst[Int[ 
x^n*Tanh[x], x], x, ArcCosh[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
 

Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a \,x^{5}\right )}{x}d x\]

Input:

int(arccosh(a*x^5)/x,x)
 

Output:

int(arccosh(a*x^5)/x,x)
 

Fricas [F]

\[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x^{5}\right )}{x} \,d x } \] Input:

integrate(arccosh(a*x^5)/x,x, algorithm="fricas")
 

Output:

integral(arccosh(a*x^5)/x, x)
 

Sympy [F]

\[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\int \frac {\operatorname {acosh}{\left (a x^{5} \right )}}{x}\, dx \] Input:

integrate(acosh(a*x**5)/x,x)
 

Output:

Integral(acosh(a*x**5)/x, x)
 

Maxima [F]

\[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x^{5}\right )}{x} \,d x } \] Input:

integrate(arccosh(a*x^5)/x,x, algorithm="maxima")
 

Output:

integrate(arccosh(a*x^5)/x, x)
 

Giac [F]

\[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x^{5}\right )}{x} \,d x } \] Input:

integrate(arccosh(a*x^5)/x,x, algorithm="giac")
 

Output:

integrate(arccosh(a*x^5)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\int \frac {\mathrm {acosh}\left (a\,x^5\right )}{x} \,d x \] Input:

int(acosh(a*x^5)/x,x)
 

Output:

int(acosh(a*x^5)/x, x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}\left (a x^5\right )}{x} \, dx=\int \frac {\mathit {acosh} \left (a \,x^{5}\right )}{x}d x \] Input:

int(acosh(a*x^5)/x,x)
 

Output:

int(acosh(a*x**5)/x,x)