Integrand size = 23, antiderivative size = 50 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}-\frac {\tanh (c+d x)}{b d} \] Output:
(a+b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(1/2)/b^(3/2)/d-tanh(d*x+c)/b/ d
Time = 0.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}-\frac {\tanh (c+d x)}{b d} \] Input:
Integrate[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]
Output:
((a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d) - Ta nh[c + d*x]/(b*d)
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4158, 299, 218}
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.
\(\displaystyle \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (i c+i d x)^4}{a-b \tan (i c+i d x)^2}dx\) |
\(\Big \downarrow \) 4158 |
\(\displaystyle \frac {\int \frac {1-\tanh ^2(c+d x)}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\frac {(a+b) \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{b}-\frac {\tanh (c+d x)}{b}}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {(a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\tanh (c+d x)}{b}}{d}\) |
Input:
Int[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]
Output:
(((a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)) - Tan h[c + d*x]/b)/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/(c^(m - 1)*f) Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^ p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && I ntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(42)=84\).
Time = 38.36 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.94
method | result | size |
derivativedivides | \(\frac {-\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )}+\frac {2 a \left (a +b \right ) \left (\frac {\left (-a -\sqrt {\left (a +b \right ) b}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (a -\sqrt {\left (a +b \right ) b}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{b}}{d}\) | \(197\) |
default | \(\frac {-\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )}+\frac {2 a \left (a +b \right ) \left (\frac {\left (-a -\sqrt {\left (a +b \right ) b}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (a -\sqrt {\left (a +b \right ) b}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{b}}{d}\) | \(197\) |
risch | \(\frac {2}{b d \left ({\mathrm e}^{2 d x +2 c}+1\right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{2 \sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{2 \sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}\) | \(255\) |
Input:
int(sech(d*x+c)^4/(a+tanh(d*x+c)^2*b),x,method=_RETURNVERBOSE)
Output:
1/d*(-2/b*tanh(1/2*d*x+1/2*c)/(tanh(1/2*d*x+1/2*c)^2+1)+2/b*a*(a+b)*(1/2*( -a-((a+b)*b)^(1/2)-b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2 )*arctan(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))-1/2*(a -((a+b)*b)^(1/2)+b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)* arctanh(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (42) = 84\).
Time = 0.11 (sec) , antiderivative size = 649, normalized size of antiderivative = 12.98 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")
Output:
[-1/2*(((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a + b)*sqrt(-a*b)*log(((a^2 + 2*a*b + b^2)*cosh( d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c)) *sinh(d*x + c) - 4*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh (d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(-a*b))/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^ 4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh (d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) - 4*a*b)/(a*b^2*d*cosh(d*x + c)^2 + 2*a*b^2*d*cosh(d*x + c )*sinh(d*x + c) + a*b^2*d*sinh(d*x + c)^2 + a*b^2*d), (((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a + b)*sqrt(a*b)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(a*b)/(a*b)) + 2 *a*b)/(a*b^2*d*cosh(d*x + c)^2 + 2*a*b^2*d*cosh(d*x + c)*sinh(d*x + c) + a *b^2*d*sinh(d*x + c)^2 + a*b^2*d)]
\[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(sech(d*x+c)**4/(a+b*tanh(d*x+c)**2),x)
Output:
Integral(sech(c + d*x)**4/(a + b*tanh(c + d*x)**2), x)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {{\left (a + b\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b d} - \frac {2}{{\left (b e^{\left (-2 \, d x - 2 \, c\right )} + b\right )} d} \] Input:
integrate(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")
Output:
-(a + b)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a* b)*b*d) - 2/((b*e^(-2*d*x - 2*c) + b)*d)
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.40 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\frac {{\left (a + b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2}{b {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \] Input:
integrate(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2),x, algorithm="giac")
Output:
((a + b)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a *b))/(sqrt(a*b)*b) + 2/(b*(e^(2*d*x + 2*c) + 1)))/d
Time = 3.14 (sec) , antiderivative size = 176, normalized size of antiderivative = 3.52 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}}{b}-\frac {2\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{3/2}\,d}\right )\,\left (a+b\right )}{2\,\sqrt {-a}\,b^{3/2}\,d}-\frac {\ln \left (\frac {2\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{3/2}\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}}{b}\right )\,\left (a+b\right )}{2\,\sqrt {-a}\,b^{3/2}\,d} \] Input:
int(1/(cosh(c + d*x)^4*(a + b*tanh(c + d*x)^2)),x)
Output:
2/(b*d*(exp(2*c + 2*d*x) + 1)) + (log(- (4*exp(2*c + 2*d*x))/b - (2*(a*d + b*d + a*d*exp(2*c + 2*d*x) - b*d*exp(2*c + 2*d*x)))/((-a)^(1/2)*b^(3/2)*d ))*(a + b))/(2*(-a)^(1/2)*b^(3/2)*d) - (log((2*(a*d + b*d + a*d*exp(2*c + 2*d*x) - b*d*exp(2*c + 2*d*x)))/((-a)^(1/2)*b^(3/2)*d) - (4*exp(2*c + 2*d* x))/b)*(a + b))/(2*(-a)^(1/2)*b^(3/2)*d)
Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 6.14 \[ \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a +e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b -e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a -e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b -\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a -\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b -2 e^{2 d x +2 c} a b}{a \,b^{2} d \left (e^{2 d x +2 c}+1\right )} \] Input:
int(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2),x)
Output:
(e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b) )/sqrt(a))*a + e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b + sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b ) - sqrt(b))/sqrt(a))*a + sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b - e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x) *sqrt(a + b) + sqrt(b))/sqrt(a))*a - e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan ((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b - sqrt(b)*sqrt(a)*atan((e **(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a - sqrt(b)*sqrt(a)*atan((e**( c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b - 2*e**(2*c + 2*d*x)*a*b)/(a*b* *2*d*(e**(2*c + 2*d*x) + 1))