\(\int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx\) [273]

Optimal result

Integrand size = 23, antiderivative size = 101 \[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {c-\sqrt {a} d}}\right )}{\sqrt {c-\sqrt {a} d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {c+\sqrt {a} d}}\right )}{\sqrt {c+\sqrt {a} d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=\frac {\arctan \left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {-c-\sqrt {a} d}}\right )}{\sqrt {-c-\sqrt {a} d}}+\frac {\arctan \left (\frac {\sqrt {c+d \sqrt {a+b x^2}}}{\sqrt {-c+\sqrt {a} d}}\right )}{\sqrt {-c+\sqrt {a} d}} \] Input:

Integrate[1/(x*Sqrt[c + d*Sqrt[a + b*x^2]]),x]
 

Output:

ArcTan[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[-c - Sqrt[a]*d]]/Sqrt[-c - Sqrt[a] 
*d] + ArcTan[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[-c + Sqrt[a]*d]]/Sqrt[-c + S 
qrt[a]*d]
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {7282, 896, 25, 1732, 561, 25, 27, 1480, 221}

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.

\(\displaystyle \int \frac {1}{x \sqrt {d \sqrt {a+b x^2}+c}} \, dx\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \sqrt {c+d \sqrt {b x^2+a}}}dx^2\)

\(\Big \downarrow \) 896

\(\displaystyle \frac {1}{2} \int \frac {1}{b x^2 \sqrt {c+d \sqrt {b x^2+a}}}d\left (b x^2+a\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{2} \int -\frac {1}{b x^2 \sqrt {c+d \sqrt {b x^2+a}}}d\left (b x^2+a\right )\)

\(\Big \downarrow \) 1732

\(\displaystyle -\int \frac {\sqrt {b x^2+a}}{\left (a-x^4\right ) \sqrt {c+d \sqrt {b x^2+a}}}d\sqrt {b x^2+a}\)

\(\Big \downarrow \) 561

\(\displaystyle -\frac {2 \int -\frac {c-x^4}{d \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {c-x^4}{d \left (-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}\right )}d\sqrt {c+d \sqrt {b x^2+a}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {c-x^4}{-\frac {x^8}{d^2}+\frac {2 c x^4}{d^2}+a-\frac {c^2}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}}{d^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (-\frac {1}{2} \int \frac {1}{\frac {c-\sqrt {a} d}{d^2}-\frac {x^4}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}-\frac {1}{2} \int \frac {1}{\frac {c+\sqrt {a} d}{d^2}-\frac {x^4}{d^2}}d\sqrt {c+d \sqrt {b x^2+a}}\right )}{d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (-\frac {d^2 \text {arctanh}\left (\frac {\sqrt {d \sqrt {a+b x^2}+c}}{\sqrt {c-\sqrt {a} d}}\right )}{2 \sqrt {c-\sqrt {a} d}}-\frac {d^2 \text {arctanh}\left (\frac {\sqrt {d \sqrt {a+b x^2}+c}}{\sqrt {\sqrt {a} d+c}}\right )}{2 \sqrt {\sqrt {a} d+c}}\right )}{d^2}\)

Input:

Int[1/(x*Sqrt[c + d*Sqrt[a + b*x^2]]),x]
 

Output:

(2*(-1/2*(d^2*ArcTanh[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[c - Sqrt[a]*d]])/Sq 
rt[c - Sqrt[a]*d] - (d^2*ArcTanh[Sqrt[c + d*Sqrt[a + b*x^2]]/Sqrt[c + Sqrt 
[a]*d]])/(2*Sqrt[c + Sqrt[a]*d])))/d^2
 

Defintions of rubi rules used

rule 25

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

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1732

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb 
ol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(d + e*x^(g* 
n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} 
, x] && EqQ[n2, 2*n] && FractionQ[n]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [F]

\[\int \frac {1}{x \sqrt {c +d \sqrt {b \,x^{2}+a}}}d x\]

Input:

int(1/x/(c+d*(b*x^2+a)^(1/2))^(1/2),x)
 

Output:

int(1/x/(c+d*(b*x^2+a)^(1/2))^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=\int \frac {1}{x \sqrt {c + d \sqrt {a + b x^{2}}}}\, dx \] Input:

integrate(1/x/(c+d*(b*x**2+a)**(1/2))**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=\int { \frac {1}{\sqrt {\sqrt {b x^{2} + a} d + c} x} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(sqrt(b*x^2 + a)*d + c)*x), x)
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (77) = 154\).

Time = 0.13 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.25 \[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=\frac {\frac {{\left (\sqrt {a} d^{3} {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right ) + c d^{3} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right )\right )} \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{\sqrt {-c + \sqrt {a d^{2}}}}\right )}{{\left (\sqrt {a} d + c\right )} \sqrt {\sqrt {a} d - c}} + \frac {{\left (\sqrt {a} d^{3} {\left | d \right |} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right ) - c d^{3} \mathrm {sgn}\left ({\left (\sqrt {b x^{2} + a} d + c\right )} d - c d\right )\right )} \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} d + c}}{\sqrt {-c - \sqrt {a d^{2}}}}\right )}{{\left (\sqrt {a} d - c\right )} \sqrt {-\sqrt {a} d - c}}}{d^{3}} \] Input:

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

Output:

((sqrt(a)*d^3*abs(d)*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) + c*d^3*sgn((sqr 
t(b*x^2 + a)*d + c)*d - c*d))*arctan(sqrt(sqrt(b*x^2 + a)*d + c)/sqrt(-c + 
 sqrt(a*d^2)))/((sqrt(a)*d + c)*sqrt(sqrt(a)*d - c)) + (sqrt(a)*d^3*abs(d) 
*sgn((sqrt(b*x^2 + a)*d + c)*d - c*d) - c*d^3*sgn((sqrt(b*x^2 + a)*d + c)* 
d - c*d))*arctan(sqrt(sqrt(b*x^2 + a)*d + c)/sqrt(-c - sqrt(a*d^2)))/((sqr 
t(a)*d - c)*sqrt(-sqrt(a)*d - c)))/d^3
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx=\int \frac {1}{x\,\sqrt {c+d\,\sqrt {b\,x^2+a}}} \,d x \] Input:

int(1/(x*(c + d*(a + b*x^2)^(1/2))^(1/2)),x)
 

Output:

int(1/(x*(c + d*(a + b*x^2)^(1/2))^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 778, normalized size of antiderivative = 7.70 \[ \int \frac {1}{x \sqrt {c+d \sqrt {a+b x^2}}} \, dx =\text {Too large to display} \] Input:

int(1/x/(c+d*(b*x^2+a)^(1/2))^(1/2),x)
 

Output:

( - sqrt(a)*sqrt(sqrt(a)*d - c)*atan((sqrt(a + b*x**2)*sqrt(sqrt(a)*d - c) 
*sqrt(sqrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a + b*x**2)*sqrt(sqrt(a)*d - c 
)*sqrt(sqrt(a + b*x**2)*d + c)*c**2*d - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(s 
qrt(a + b*x**2)*d + c)*a*d**3 - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + 
b*x**2)*d + c)*b*d**3*x**2 + sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x 
**2)*d + c)*c**2*d - 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*a* 
c*d**2 - sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*c*d**2*x**2 + 
2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**3)/(2*a**2*d**4 + 2* 
a*b*d**4*x**2 - 4*a*c**2*d**2 - 2*b*c**2*d**2*x**2 + 2*c**4))*d - sqrt(sqr 
t(a)*d - c)*atan((sqrt(a + b*x**2)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x** 
2)*d + c)*a*d**3 - sqrt(a + b*x**2)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x* 
*2)*d + c)*c**2*d - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + 
c)*a*d**3 - sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*d** 
3*x**2 + sqrt(a)*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*c**2*d - 
 2*sqrt(sqrt(a)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*a*c*d**2 - sqrt(sqrt(a 
)*d - c)*sqrt(sqrt(a + b*x**2)*d + c)*b*c*d**2*x**2 + 2*sqrt(sqrt(a)*d - c 
)*sqrt(sqrt(a + b*x**2)*d + c)*c**3)/(2*a**2*d**4 + 2*a*b*d**4*x**2 - 4*a* 
c**2*d**2 - 2*b*c**2*d**2*x**2 + 2*c**4))*c + sqrt(a)*sqrt(sqrt(a)*d + c)* 
log(sqrt(sqrt(a + b*x**2)*d + c) - sqrt(sqrt(a)*d + c))*d - sqrt(a)*sqrt(s 
qrt(a)*d + c)*log(sqrt(sqrt(a + b*x**2)*d + c) + sqrt(sqrt(a)*d + c))*d...