3.1.6 \(\int \frac {\arctan (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}})}{x} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 25, antiderivative size = 288 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=-\frac {\sqrt {d} \sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {e} \sqrt {d+e x^2}}+\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \log (x)+\frac {\sqrt {d} \sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}} \]

output

arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))*ln(x)-1/2*arcsinh(x*e^(1/2)/d^(1/2))^ 
2*d^(1/2)*(-e)^(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))*ln(1-(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*(-e) 
^(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) 
)*ln(x)*d^(1/2)*(-e)^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)+1/2*p 
olylog(2,(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*(-e)^(1/2)*(1+e* 
x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)
 

3.1.6.2 Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.59 \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \log (x)+\frac {\sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \left (\text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )^2+2 \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )}\right )-2 \log (x) \log \left (\sqrt {\frac {e}{d}} x+\sqrt {1+\frac {e x^2}{d}}\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )}\right )\right )}{2 \sqrt {\frac {e}{d}} \sqrt {d+e x^2}} \]

input

Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x,x]
 

output

ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]*Log[x] + (Sqrt[-e]*Sqrt[1 + (e*x^2)/d 
]*(ArcSinh[Sqrt[e/d]*x]^2 + 2*ArcSinh[Sqrt[e/d]*x]*Log[1 - E^(-2*ArcSinh[S 
qrt[e/d]*x])] - 2*Log[x]*Log[Sqrt[e/d]*x + Sqrt[1 + (e*x^2)/d]] - PolyLog[ 
2, E^(-2*ArcSinh[Sqrt[e/d]*x])]))/(2*Sqrt[e/d]*Sqrt[d + e*x^2])
 

3.1.6.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.66, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5672, 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.

\(\displaystyle \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx\)

\(\Big \downarrow \) 5672

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\sqrt {-e} \int \frac {\log (x)}{\sqrt {e x^2+d}}dx\)

\(\Big \downarrow \) 2764

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \int \frac {\log (x)}{\sqrt {\frac {e x^2}{d}+1}}dx}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 2762

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {\sqrt {d} \int \frac {\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 6190

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {\sqrt {d} \int \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} x}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {\sqrt {d} \int -i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {i \sqrt {d} \int \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 4199

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {i \sqrt {d} \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {i \sqrt {d} \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {i \sqrt {d} \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {i \sqrt {d} \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )de^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \log (x) \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {-e} \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \log (x) \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {i \sqrt {d} \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{\sqrt {d+e x^2}}\)

input

Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x,x]
 

output

ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]*Log[x] - (Sqrt[-e]*Sqrt[1 + (e*x^2)/d 
]*((Sqrt[d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[x])/Sqrt[e] + (I*Sqrt[d]*((-1 
/2*I)*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2 - (2*I)*(-1/2*(ArcSinh[(Sqrt[e]*x)/Sq 
rt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]) - PolyLog[2, E^(2*ArcS 
inh[(Sqrt[e]*x)/Sqrt[d]])]/4)))/Sqrt[e]))/Sqrt[d + e*x^2]
 

3.1.6.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2762

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]
 

rule 2764

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]
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

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]
 

rule 5672

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]/(x_), x_Symbol] :> Simp 
[ArcTan[c*(x/Sqrt[a + b*x^2])]*Log[x], x] - Simp[c   Int[Log[x]/Sqrt[a + b* 
x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]
 

rule 6190

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]
 

3.1.6.4 Maple [F]

\[\int \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{x}d x\]

input

int(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x,x)
 

output

int(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x,x)
 

3.1.6.5 Fricas [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x} \,d x } \]

input

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x,x, algorithm="fricas")
 

output

integral(arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x, x)
 

3.1.6.6 Sympy [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=\int \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{x}\, dx \]

input

integrate(atan(x*(-e)**(1/2)/(e*x**2+d)**(1/2))/x,x)
 

output

Integral(atan(x*sqrt(-e)/sqrt(d + e*x**2))/x, x)
 

3.1.6.7 Maxima [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x} \,d x } \]

input

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x,x, algorithm="maxima")
 

output

integrate(arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x, x)
 

3.1.6.8 Giac [F]

\[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=\int { \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x} \,d x } \]

input

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x,x, algorithm="giac")
 

output

integrate(arctan(sqrt(-e)*x/sqrt(e*x^2 + d))/x, x)
 

3.1.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx=\int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x} \,d x \]

input

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x,x)
 

output

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x, x)