\(\int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx\) [32]

Optimal result

Integrand size = 22, antiderivative size = 59 \[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d} \]

[Out]

2*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d+b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2)
)/c/d-b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5903, 4267, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\frac {2 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d} \]

[In]

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

[Out]

(2*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c*d) + (b*PolyLog[2, -E^ArcCosh[c*x]])/(c*d) - (b*PolyLog[2,
E^ArcCosh[c*x]])/(c*d)

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 4267

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

Rule 5903

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx=\frac {-\left ((a+b \text {arccosh}(c x)) \left (\log \left (1-e^{\text {arccosh}(c x)}\right )-\log \left (1+e^{\text {arccosh}(c x)}\right )\right )\right )+b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c d} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.05

method	result	size
derivativedivides	\(\frac {\frac {a \,\operatorname {arctanh}\left (c x \right )}{d}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\)	\(180\)
default	\(\frac {\frac {a \,\operatorname {arctanh}\left (c x \right )}{d}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d}}{c}\)	\(180\)
parts	\(\frac {a \ln \left (c x +1\right )}{2 d c}-\frac {a \ln \left (c x -1\right )}{2 d c}-\frac {b \left (-\operatorname {arctanh}\left (c x \right ) \operatorname {arccosh}\left (c x \right )-\frac {2 i \left (\operatorname {arctanh}\left (c x \right ) \ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (c x \right ) \ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right ) \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c x}{2}+\frac {1}{2}}\, \sqrt {\frac {c x}{2}-\frac {1}{2}}}{c^{2} x^{2}-1}\right )}{d c}\)	\(200\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/8*b*((4*(log(c*x + 1) - log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) - log(c*x + 1)^2 - 2*log(c*x +
1)*log(c*x - 1))/(c*d) + 8*integrate(1/4*(3*c*x - 1)*log(c*x - 1)/(c^2*d*x^2 - d), x) + 8*integrate(1/2*(log(c
*x + 1) - log(c*x - 1))/(c^3*d*x^3 - c*d*x + (c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(c*x - 1)), x)) + 1/2*a*(log(c*
x + 1)/(c*d) - log(c*x - 1)/(c*d))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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