\(\int \frac {a+b \log (c (d+e x)^n)}{\sqrt {2+g x^2}} \, dx\) [275]

Optimal result

Integrand size = 26, antiderivative size = 326 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )^2}{2 \sqrt {g}}-\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}}-\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}}+\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}} \] Output:

1/2*b*n*arcsinh(1/2*g^(1/2)*x*2^(1/2))^2/g^(1/2)-b*n*arcsinh(1/2*g^(1/2)*x 
*2^(1/2))*ln(1+2^(1/2)*e*(1/2*g^(1/2)*x*2^(1/2)+1/2*(2*g*x^2+4)^(1/2))/(d* 
g^(1/2)-(d^2*g+2*e^2)^(1/2)))/g^(1/2)-b*n*arcsinh(1/2*g^(1/2)*x*2^(1/2))*l 
n(1+2^(1/2)*e*(1/2*g^(1/2)*x*2^(1/2)+1/2*(2*g*x^2+4)^(1/2))/(d*g^(1/2)+(d^ 
2*g+2*e^2)^(1/2)))/g^(1/2)+arcsinh(1/2*g^(1/2)*x*2^(1/2))*(a+b*ln(c*(e*x+d 
)^n))/g^(1/2)-b*n*polylog(2,-2^(1/2)*e*(1/2*g^(1/2)*x*2^(1/2)+1/2*(2*g*x^2 
+4)^(1/2))/(d*g^(1/2)-(d^2*g+2*e^2)^(1/2)))/g^(1/2)-b*n*polylog(2,-2^(1/2) 
*e*(1/2*g^(1/2)*x*2^(1/2)+1/2*(2*g*x^2+4)^(1/2))/(d*g^(1/2)+(d^2*g+2*e^2)^ 
(1/2)))/g^(1/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (2 a+b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )-2 b n \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {2 e^2+d^2 g}}\right )-2 b n \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )+2 b \log \left (c (d+e x)^n\right )\right )-2 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{-d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )-2 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )}{2 \sqrt {g}} \] Input:

Integrate[(a + b*Log[c*(d + e*x)^n])/Sqrt[2 + g*x^2],x]
 

Output:

(ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*(2*a + b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]] - 2* 
b*n*Log[1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2 
*e^2 + d^2*g])] - 2*b*n*Log[1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]]) 
/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g])] + 2*b*Log[c*(d + e*x)^n]) - 2*b*n*Poly 
Log[2, (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(-(d*Sqrt[g]) + Sqrt[2*e 
^2 + d^2*g])] - 2*b*n*PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2 
]])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g]))])/(2*Sqrt[g])
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, 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.

\(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {g x^2+2}} \, dx\)

\(\Big \downarrow \) 2851

\(\displaystyle \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-b e n \int \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}{\sqrt {g} (d+e x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b e n \int \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}{d+e x}dx}{\sqrt {g}}\)

\(\Big \downarrow \) 6242

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

\(\Big \downarrow \) 6095

\(\displaystyle \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b e n \left (\int \frac {e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )} \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}{\frac {\sqrt {g} d}{\sqrt {2}}+e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}-\frac {\sqrt {g d^2+2 e^2}}{\sqrt {2}}}d\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )+\int \frac {e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )} \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}{\frac {\sqrt {g} d}{\sqrt {2}}+e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}+\frac {\sqrt {g d^2+2 e^2}}{\sqrt {2}}}d\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )-\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )^2}{2 e}\right )}{\sqrt {g}}\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b e n \left (-\frac {\int e^{-\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )} \log \left (\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )} e}{d \sqrt {g}-\sqrt {g d^2+2 e^2}}+1\right )de^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{e}-\frac {\int e^{-\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )} \log \left (\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )} e}{\sqrt {g} d+\sqrt {g d^2+2 e^2}}+1\right )de^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{e}+\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {d^2 g+2 e^2}}+1\right )}{e}+\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{\sqrt {d^2 g+2 e^2}+d \sqrt {g}}+1\right )}{e}-\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )^2}{2 e}\right )}{\sqrt {g}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b e n \left (\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {g d^2+2 e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{\sqrt {g} d+\sqrt {g d^2+2 e^2}}\right )}{e}+\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {d^2 g+2 e^2}}+1\right )}{e}+\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{\sqrt {d^2 g+2 e^2}+d \sqrt {g}}+1\right )}{e}-\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )^2}{2 e}\right )}{\sqrt {g}}\)

Input:

Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[2 + g*x^2],x]
 

Output:

(ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*(a + b*Log[c*(d + e*x)^n]))/Sqrt[g] - (b*e*n 
*(-1/2*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]^2/e + (ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*Lo 
g[1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2*e^2 + 
 d^2*g])])/e + (ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*Log[1 + (Sqrt[2]*e*E^ArcSinh[ 
(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g])])/e + PolyLog[2, - 
((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2*e^2 + d^2* 
g]))]/e + PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[ 
g] + Sqrt[2*e^2 + d^2*g]))]/e))/Sqrt[g]
 

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]

\[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\sqrt {g \,x^{2}+2}}d x\]

Input:

int((a+b*ln(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x)
 

Output:

int((a+b*ln(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x^{2} + 2}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x, algorithm="fricas")
 

Output:

integral((sqrt(g*x^2 + 2)*b*log((e*x + d)^n*c) + sqrt(g*x^2 + 2)*a)/(g*x^2 
 + 2), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\sqrt {g x^{2} + 2}}\, dx \] Input:

integrate((a+b*ln(c*(e*x+d)**n))/(g*x**2+2)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))/sqrt(g*x**2 + 2), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x^{2} + 2}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x, algorithm="maxima")
 

Output:

b*integrate((log((e*x + d)^n) + log(c))/sqrt(g*x^2 + 2), x) + a*arcsinh(1/ 
2*sqrt(2)*sqrt(g)*x)/sqrt(g)
 

Giac [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x^{2} + 2}} \,d x } \] Input:

integrate((a+b*log(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*log((e*x + d)^n*c) + a)/sqrt(g*x^2 + 2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {g\,x^2+2}} \,d x \] Input:

int((a + b*log(c*(d + e*x)^n))/(g*x^2 + 2)^(1/2),x)
 

Output:

int((a + b*log(c*(d + e*x)^n))/(g*x^2 + 2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b +a}{\sqrt {g \,x^{2}+2}}d x \] Input:

int((a+b*log(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x)
 

Output:

int((a+b*log(c*(e*x+d)^n))/(g*x^2+2)^(1/2),x)