Integrand size = 22, antiderivative size = 250 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}} \]
1/2*b*n*arcsinh(x*e^(1/2)/d^(1/2))^2*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e* x^2+d)^(1/2)-b*n*arcsinh(x*e^(1/2)/d^(1/2))*ln(1-(x*e^(1/2)/d^(1/2)+(1+e*x ^2/d)^(1/2))^2)*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)+arcsinh( x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^ 2+d)^(1/2)-1/2*b*n*polylog(2,(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1 /2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)
Time = 0.36 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{\sqrt {e}}+\frac {b n \sqrt {1+\frac {e x^2}{d}} \left (-\text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )^2-2 \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )}\right )+2 \log (x) \log \left (\sqrt {\frac {e}{d}} x+\sqrt {1+\frac {e x^2}{d}}\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )}\right )\right )}{2 \sqrt {\frac {e}{d}} \sqrt {d+e x^2}} \]
((a - b*n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/Sqrt[ e] + (b*n*Sqrt[1 + (e*x^2)/d]*(-ArcSinh[Sqrt[e/d]*x]^2 - 2*ArcSinh[Sqrt[e/ d]*x]*Log[1 - E^(-2*ArcSinh[Sqrt[e/d]*x])] + 2*Log[x]*Log[Sqrt[e/d]*x + Sq rt[1 + (e*x^2)/d]] + PolyLog[2, E^(-2*ArcSinh[Sqrt[e/d]*x])]))/(2*Sqrt[e/d ]*Sqrt[d + e*x^2])
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.66, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2764, 2762, 6190, 3042, 26, 4199, 25, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 2764 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e x^2}{d}+1}}dx}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2762 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int \frac {\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} x}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int -i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \int \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )de^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\) |
(Sqrt[1 + (e*x^2)/d]*((Sqrt[d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x ^n]))/Sqrt[e] + (I*b*Sqrt[d]*n*((-1/2*I)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2 - (2*I)*(-1/2*(ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x) /Sqrt[d]])]) - PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]/4)))/Sqrt[e] ))/Sqrt[d + e*x^2]
3.3.82.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symb ol] :> Simp[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]*((a + b*Log[c*x^n])/Rt[e, 2]), x] - Simp[b*(n/Rt[e, 2]) Int[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]/x, x], x] /; Fre eQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symb ol] :> Simp[Sqrt[1 + (e/d)*x^2]/Sqrt[d + e*x^2] Int[(a + b*Log[c*x^n])/Sq rt[1 + (e/d)*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && !GtQ[d, 0]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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 ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{\sqrt {e \,x^{2}+d}}d x\]
\[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x^{2} + d}} \,d x } \]
\[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{\sqrt {e\,x^2+d}} \,d x \]