\(\int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx\) [202]

Optimal result

Integrand size = 21, antiderivative size = 58 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]

[Out]

1/2*arctanh(1/2*(2*a+b+(b+2*c)*tanh(x)^2)/(a+b+c)^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/(a+b+c)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3781, 1261, 738, 212} \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+(b+2 c) \tanh ^2(x)+b}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]

[In]

Int[Tanh[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]

[Out]

ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b
+ c])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)
), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a-b x^2+c x^4}} \, dx,x,i \tanh (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right )\right ) \\ & = \text {Subst}\left (\int \frac {1}{4 a+4 b+4 c-x^2} \, dx,x,\frac {2 a+b+(b+2 c) \tanh ^2(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]

[In]

Integrate[Tanh[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]

[Out]

ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b
+ c])

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\operatorname {arctanh}\left (\frac {b \tanh \left (x \right )^{2}+2 c \tanh \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \tanh \left (x \right )^{2}+c \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\)	\(52\)
default	\(\frac {\operatorname {arctanh}\left (\frac {b \tanh \left (x \right )^{2}+2 c \tanh \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \tanh \left (x \right )^{2}+c \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\)	\(52\)

[In]

int(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/(a+b+c)^(1/2)*arctanh(1/2*(b*tanh(x)^2+2*c*tanh(x)^2+2*a+b)/(a+b+c)^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2
))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (48) = 96\).

Time = 0.68 (sec) , antiderivative size = 1748, normalized size of antiderivative = 30.14 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Too large to display} \]

[In]

integrate(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/4*log(((a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x
)*sinh(x)^7 + (a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*sinh(x)^8 + 4*(a^2 + a*b - b*c - c^2)*cosh(x)^6 + 4*(7*(
a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^2 + a^2 + a*b - b*c - c^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2
+ 2*(a + b)*c + c^2)*cosh(x)^3 + 3*(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b + 2*(a + b)*c
+ 3*c^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^4 + 30*(a^2 + a*b - b*c - c^2)*cos
h(x)^2 + 3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)
^5 + 10*(a^2 + a*b - b*c - c^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x))*sinh(x)^3 + 4*(a^2
+ a*b - b*c - c^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^6 + 15*(a^2 + a*b - b*c -
c^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^2 + a^2 + a*b - b*c - c^2)*sinh(x)^2 + sqrt(2
)*((a + b + c)*cosh(x)^4 + 4*(a + b + c)*cosh(x)*sinh(x)^3 + (a + b + c)*sinh(x)^4 + 2*(a - c)*cosh(x)^2 + 2*(
3*(a + b + c)*cosh(x)^2 + a - c)*sinh(x)^2 + 4*((a + b + c)*cosh(x)^3 + (a - c)*cosh(x))*sinh(x) + a + b + c)*
sqrt(a + b + c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 + 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*c
osh(x)^2 + 2*a - 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*
cosh(x)*sinh(x)^3 + sinh(x)^4)) + a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2 + 8*((a^2 + 2*a*b + b^2 + 2*(a + b)*c
+ c^2)*cosh(x)^7 + 3*(a^2 + a*b - b*c - c^2)*cosh(x)^5 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^3 + (a^
2 + a*b - b*c - c^2)*cosh(x))*sinh(x))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*si
nh(x)^3 + sinh(x)^4))/sqrt(a + b + c), -1/2*sqrt(-a - b - c)*arctan(sqrt(2)*((a + b + c)*cosh(x)^4 + 4*(a + b
+ c)*cosh(x)*sinh(x)^3 + (a + b + c)*sinh(x)^4 + 2*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 + a - c)*sin
h(x)^2 + 4*((a + b + c)*cosh(x)^3 + (a - c)*cosh(x))*sinh(x) + a + b + c)*sqrt(-a - b - c)*sqrt(((a + b + c)*c
osh(x)^4 + (a + b + c)*sinh(x)^4 + 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 + 2*a - 2*c)*sinh(x)^2 + 3
*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4))/((a
^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)*sinh(x)^7
+ (a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*sinh(x)^8 + 4*(a^2 + a*b - b*c - c^2)*cosh(x)^6 + 4*(7*(a^2 + 2*a*b
+ b^2 + 2*(a + b)*c + c^2)*cosh(x)^2 + a^2 + a*b - b*c - c^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*
c + c^2)*cosh(x)^3 + 3*(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3
*c^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^4 + 30*(a^2 + a*b - b*c - c^2)*cosh(x)
^2 + 3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cos
h(x)^5 + 10*(a^2 + a*b - b*c - c^2)*cosh(x)^3 + (3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*cosh(x))*sinh(x)
^3 + 4*(a^2 + a*b - b*c - c^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^6 + 15*(a^2 +
a*b - b*c - c^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*cosh(x)^2 + a^2 + a*b - b*c - c^2
)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2 + 8*((a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^7 + 3
*(a^2 + a*b - b*c - c^2)*cosh(x)^5 + (3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*cosh(x)^3 + (a^2 + a*b - b*
c - c^2)*cosh(x))*sinh(x)))/(a + b + c)]

Sympy [F]

\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \tanh ^{2}{\left (x \right )} + c \tanh ^{4}{\left (x \right )}}}\, dx \]

[In]

integrate(tanh(x)/(a+b*tanh(x)**2+c*tanh(x)**4)**(1/2),x)

[Out]

Integral(tanh(x)/sqrt(a + b*tanh(x)**2 + c*tanh(x)**4), x)

Maxima [F]

\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {c \tanh \left (x\right )^{4} + b \tanh \left (x\right )^{2} + a}} \,d x } \]

[In]

integrate(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(tanh(x)/sqrt(c*tanh(x)^4 + b*tanh(x)^2 + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Timed out} \]

[In]

integrate(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {c\,{\mathrm {tanh}\left (x\right )}^4+b\,{\mathrm {tanh}\left (x\right )}^2+a}} \,d x \]

[In]

int(tanh(x)/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2),x)

[Out]

int(tanh(x)/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2), x)