Integrand size = 21, antiderivative size = 58 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \] Output:
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*t anh(x)^4)^(1/2))/(a+b+c)^(1/2)
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \] Input:
Integrate[Tanh[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]
Output:
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])
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.50, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4183, 1576, 1154, 219}
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.
\(\displaystyle \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i x)}{\sqrt {a-b \tan (i x)^2+c \tan (i x)^4}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i x)}{\sqrt {c \tan (i x)^4-b \tan (i x)^2+a}}dx\) |
\(\Big \downarrow \) 4183 |
\(\displaystyle -\int \frac {i \tanh (x)}{\left (1-\tanh ^2(x)\right ) \sqrt {c \tanh ^4(x)+b \tanh ^2(x)+a}}d(i \tanh (x))\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {-c \tanh ^2(x)-i b \tanh (x)+a}}d\left (-\tanh ^2(x)\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \int \frac {1}{4 (a+b+c)+\tanh ^2(x)}d\frac {2 a-i (b+2 c) \tanh (x)+b}{\sqrt {a-i b \tanh (x)-c \tanh ^2(x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {i \arctan \left (\frac {\tanh (x)}{2 \sqrt {a+b+c}}\right )}{2 \sqrt {a+b+c}}\) |
Input:
Int[Tanh[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]
Output:
((I/2)*ArcTan[Tanh[x]/(2*Sqrt[a + b + c])])/Sqrt[a + b + c]
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[((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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[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]
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Simp[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[n 2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Time = 2.15 (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 | \(\frac {\operatorname {arctanh}\left (\frac {b \tanh \left (x \right )^{2}+2 c \tanh \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \tanh \left (x \right )^{2}+c \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\) | \(52\) |
Input:
int(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (48) = 96\).
Time = 0.56 (sec) , antiderivative size = 1748, normalized size of antiderivative = 30.14 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="fricas")
Output:
[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)*si nh(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)*cosh(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 + 1 0*(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)*cos h(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)*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 +...
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \tanh ^{2}{\left (x \right )} + c \tanh ^{4}{\left (x \right )}}}\, dx \] Input:
integrate(tanh(x)/(a+b*tanh(x)**2+c*tanh(x)**4)**(1/2),x)
Output:
Integral(tanh(x)/sqrt(a + b*tanh(x)**2 + c*tanh(x)**4), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {c \tanh \left (x\right )^{4} + b \tanh \left (x\right )^{2} + a}} \,d x } \] Input:
integrate(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="maxima")
Output:
integrate(tanh(x)/sqrt(c*tanh(x)^4 + b*tanh(x)^2 + a), x)
Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Timed out} \] Input:
integrate(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {c\,{\mathrm {tanh}\left (x\right )}^4+b\,{\mathrm {tanh}\left (x\right )}^2+a}} \,d x \] Input:
int(tanh(x)/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2),x)
Output:
int(tanh(x)/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\sqrt {\tanh \left (x \right )^{4} c +\tanh \left (x \right )^{2} b +a}\, \tanh \left (x \right )}{\tanh \left (x \right )^{4} c +\tanh \left (x \right )^{2} b +a}d x \] Input:
int(tanh(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)
Output:
int((sqrt(tanh(x)**4*c + tanh(x)**2*b + a)*tanh(x))/(tanh(x)**4*c + tanh(x )**2*b + a),x)