\(\int \frac {\log (h (f+g x)^m)}{\sqrt {1+c^2 x^2}} \, dx\) [56]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {m \text {arcsinh}(c x)^2}{2 c}-\frac {m \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {m \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \] Output:

1/2*m*arcsinh(c*x)^2/c-m*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f- 
(c^2*f^2+g^2)^(1/2)))/c-m*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f 
+(c^2*f^2+g^2)^(1/2)))/c+arcsinh(c*x)*ln(h*(g*x+f)^m)/c-m*polylog(2,-(c*x+ 
(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))/c-m*polylog(2,-(c*x+(c^2*x 
^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/c
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {m \text {arcsinh}(c x)^2}{2 c}-\frac {m \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)} g}{c^2 f-c \sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)} g}{c^2 f+c \sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {m \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \] Input:

Integrate[Log[h*(f + g*x)^m]/Sqrt[1 + c^2*x^2],x]
 

Output:

(m*ArcSinh[c*x]^2)/(2*c) - (m*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x]*g)/(c 
^2*f - c*Sqrt[c^2*f^2 + g^2])])/c - (m*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c 
*x]*g)/(c^2*f + c*Sqrt[c^2*f^2 + g^2])])/c + (ArcSinh[c*x]*Log[h*(f + g*x) 
^m])/c - (m*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))]) 
/c - (m*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2851, 27, 6242, 6095, 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.

\(\displaystyle \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {c^2 x^2+1}} \, dx\)

\(\Big \downarrow \) 2851

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-g m \int \frac {\text {arcsinh}(c x)}{c (f+g x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {g m \int \frac {\text {arcsinh}(c x)}{f+g x}dx}{c}\)

\(\Big \downarrow \) 6242

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {g m \int \frac {\sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{c f+c g x}d\text {arcsinh}(c x)}{c}\)

\(\Big \downarrow \) 6095

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {g m \left (\int \frac {e^{\text {arcsinh}(c x)} \text {arcsinh}(c x)}{c f+e^{\text {arcsinh}(c x)} g-\sqrt {c^2 f^2+g^2}}d\text {arcsinh}(c x)+\int \frac {e^{\text {arcsinh}(c x)} \text {arcsinh}(c x)}{c f+e^{\text {arcsinh}(c x)} g+\sqrt {c^2 f^2+g^2}}d\text {arcsinh}(c x)-\frac {\text {arcsinh}(c x)^2}{2 g}\right )}{c}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {g m \left (-\frac {\int \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )d\text {arcsinh}(c x)}{g}-\frac {\int \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )d\text {arcsinh}(c x)}{g}+\frac {\text {arcsinh}(c x) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {\text {arcsinh}(c x) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {\text {arcsinh}(c x)^2}{2 g}\right )}{c}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {g m \left (-\frac {\int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )de^{\text {arcsinh}(c x)}}{g}-\frac {\int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )de^{\text {arcsinh}(c x)}}{g}+\frac {\text {arcsinh}(c x) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {\text {arcsinh}(c x) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {\text {arcsinh}(c x)^2}{2 g}\right )}{c}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\text {arcsinh}(c x) \log \left (h (f+g x)^m\right )}{c}-\frac {g m \left (\frac {\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g}+\frac {\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g}+\frac {\text {arcsinh}(c x) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {\text {arcsinh}(c x) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {\text {arcsinh}(c x)^2}{2 g}\right )}{c}\)

Input:

Int[Log[h*(f + g*x)^m]/Sqrt[1 + c^2*x^2],x]
 

Output:

(ArcSinh[c*x]*Log[h*(f + g*x)^m])/c - (g*m*(-1/2*ArcSinh[c*x]^2/g + (ArcSi 
nh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/g + (ArcS 
inh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g + Poly 
Log[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))]/g + PolyLog[2, - 
((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]/g))/c
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2851

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)* 
(x_)^2], x_Symbol] :> With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + 
 b*Log[c*(d + e*x)^n]), x] - Simp[b*e*n   Int[SimplifyIntegrand[u/(d + e*x) 
, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), 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]
 

rule 6242

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo 
l] :> Subst[Int[(a + b*x)^n*(Cosh[x]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x 
]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
 

Maple [F]

\[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {c^{2} x^{2}+1}}d x\]

Input:

int(ln(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x)
 

Output:

int(ln(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \] Input:

integrate(log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(log((g*x + f)^m*h)/sqrt(c^2*x^2 + 1), x)
 

Sympy [F]

\[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \] Input:

integrate(ln(h*(g*x+f)**m)/(c**2*x**2+1)**(1/2),x)
 

Output:

Integral(log(h*(f + g*x)**m)/sqrt(c**2*x**2 + 1), x)
 

Maxima [F]

\[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \] Input:

integrate(log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(log((g*x + f)^m*h)/sqrt(c^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \] Input:

integrate(log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(log((g*x + f)^m*h)/sqrt(c^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\sqrt {c^2\,x^2+1}} \,d x \] Input:

int(log(h*(f + g*x)^m)/(c^2*x^2 + 1)^(1/2),x)
 

Output:

int(log(h*(f + g*x)^m)/(c^2*x^2 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\mathrm {log}\left (\left (g x +f \right )^{m} h \right )}{\sqrt {c^{2} x^{2}+1}}d x \] Input:

int(log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x)
 

Output:

int(log((f + g*x)**m*h)/sqrt(c**2*x**2 + 1),x)