\(\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx\) [110]

Optimal result

Integrand size = 27, antiderivative size = 151 \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]

[Out]

2*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-
I*b*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+I*b*polyl
og(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5946, 4265, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x*Sqrt[d - c^2*d*x^2]),x]

[Out]

(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - (I*b*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] + (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]*PolyLog[2, I*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5946

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c
*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \\ & = \frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \\ & = \frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {a \log (x)}{\sqrt {d}}-\frac {a \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}-\frac {i b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x) \left (\log \left (1-i e^{-\text {arccosh}(c x)}\right )-\log \left (1+i e^{-\text {arccosh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}} \]

[In]

Integrate[(a + b*ArcCosh[c*x])/(x*Sqrt[d - c^2*d*x^2]),x]

[Out]

(a*Log[x])/Sqrt[d] - (a*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] - (I*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c
*x)*(ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]) + PolyLog[2, (-I)/E^ArcCosh[c*x]] -
PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2]

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.16

method	result	size
default	\(-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+b \left (\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(326\)
parts	\(-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+b \left (\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\right )\)	\(326\)

[In]

int((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-a/d^(1/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+b*(I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/
d/(c^2*x^2-1)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c
*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+I*(-d*(c^2*x^2-1))^(1/2)*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-I*(-d*(c^2*x^2-1))^(1/2)*(c
*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))))

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^2*d*x^3 - d*x), x)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

[In]

integrate((a+b*acosh(c*x))/x/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral((a + b*acosh(c*x))/(x*sqrt(-d*(c*x - 1)*(c*x + 1))), x)

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(sqrt(-c^2*d*x^2 + d)*x), x) - a*log(2*sqrt(-c^2*d*x^2 + d)
*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d)

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,\sqrt {d-c^2\,d\,x^2}} \,d x \]

[In]

int((a + b*acosh(c*x))/(x*(d - c^2*d*x^2)^(1/2)),x)

[Out]

int((a + b*acosh(c*x))/(x*(d - c^2*d*x^2)^(1/2)), x)