Integrand size = 15, antiderivative size = 40 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 \sqrt {a+b}} \] Output:
1/2*arctanh((a+b*tanh(x)^2)/(a+b)^(1/2)/(a+b*tanh(x)^4)^(1/2))/(a+b)^(1/2)
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a+b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 \sqrt {a+b}} \] Input:
Integrate[Tanh[x]/Sqrt[a + b*Tanh[x]^4],x]
Output:
ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*Sqrt[a + b])
Time = 0.47 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 26, 4153, 26, 1577, 488, 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.
\(\displaystyle \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i x)}{\sqrt {a+b \tan (i x)^4}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i x)}{\sqrt {b \tan (i x)^4+a}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -i \int \frac {i \tanh (x)}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh (x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\tanh (x)}{\left (1-\tanh ^2(x)\right ) \sqrt {a+b \tanh ^4(x)}}d\tanh (x)\) |
\(\Big \downarrow \) 1577 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^4(x)+a}}d\tanh ^2(x)\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{-\tanh ^4(x)+a+b}d\frac {-b \tanh ^2(x)-a}{\sqrt {b \tanh ^4(x)+a}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {-a-b \tanh ^2(x)}{\sqrt {a+b} \sqrt {a+b \tanh ^4(x)}}\right )}{2 \sqrt {a+b}}\) |
Input:
Int[Tanh[x]/Sqrt[a + b*Tanh[x]^4],x]
Output:
-1/2*ArcTanh[(-a - b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/Sqrt[ a + b]
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/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}\) | \(37\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {2 b \tanh \left (x \right )^{2}+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \tanh \left (x \right )^{4}}}\right )}{2 \sqrt {a +b}}\) | \(37\) |
Input:
int(tanh(x)/(a+b*tanh(x)^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/(a+b)^(1/2)*arctanh(1/2*(2*b*tanh(x)^2+2*a)/(a+b)^(1/2)/(a+b*tanh(x)^4 )^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (32) = 64\).
Time = 0.40 (sec) , antiderivative size = 1286, normalized size of antiderivative = 32.15 \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)/(a+b*tanh(x)^4)^(1/2),x, algorithm="fricas")
Output:
[1/4*log(((a^2 + 2*a*b + b^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(x)*si nh(x)^7 + (a^2 + 2*a*b + b^2)*sinh(x)^8 + 4*(a^2 - b^2)*cosh(x)^6 + 4*(7*( a^2 + 2*a*b + b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^3 + 3*(a^2 - b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b + 3*b ^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 30*(a^2 - b^2)*cosh( x)^2 + 3*a^2 + 2*a*b + 3*b^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(x) ^5 + 10*(a^2 - b^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 3*b^2)*cosh(x))*sinh(x)^3 + 4*(a^2 - b^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(x)^6 + 15*(a^2 - b^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b + 3*b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x )^2 + sqrt(2)*((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*s inh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a - b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)*sqrt(a + b)*sq rt(((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 + 4*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + 2*a - 2*b)*sinh(x)^2 + 3*a + 3*b)/(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 + 8*((a^2 + 2*a*b + b^2)*cosh(x)^7 + 3*(a^2 - b^2)*cosh(x) ^5 + (3*a^2 + 2*a*b + 3*b^2)*cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x))/(co sh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^ 3 + sinh(x)^4))/sqrt(a + b), -1/2*sqrt(-a - b)*arctan(sqrt(2)*((a + b)*cos h(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*(a - b)*co...
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \tanh ^{4}{\left (x \right )}}}\, dx \] Input:
integrate(tanh(x)/(a+b*tanh(x)**4)**(1/2),x)
Output:
Integral(tanh(x)/sqrt(a + b*tanh(x)**4), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \tanh \left (x\right )^{4} + a}} \,d x } \] Input:
integrate(tanh(x)/(a+b*tanh(x)^4)^(1/2),x, algorithm="maxima")
Output:
integrate(tanh(x)/sqrt(b*tanh(x)^4 + a), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \tanh \left (x\right )^{4} + a}} \,d x } \] Input:
integrate(tanh(x)/(a+b*tanh(x)^4)^(1/2),x, algorithm="giac")
Output:
integrate(tanh(x)/sqrt(b*tanh(x)^4 + a), x)
Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {b\,{\mathrm {tanh}\left (x\right )}^4+a}} \,d x \] Input:
int(tanh(x)/(a + b*tanh(x)^4)^(1/2),x)
Output:
int(tanh(x)/(a + b*tanh(x)^4)^(1/2), x)
\[ \int \frac {\tanh (x)}{\sqrt {a+b \tanh ^4(x)}} \, dx=\int \frac {\sqrt {\tanh \left (x \right )^{4} b +a}\, \tanh \left (x \right )}{\tanh \left (x \right )^{4} b +a}d x \] Input:
int(tanh(x)/(a+b*tanh(x)^4)^(1/2),x)
Output:
int((sqrt(tanh(x)**4*b + a)*tanh(x))/(tanh(x)**4*b + a),x)