\(\int \frac {\text {sech}^3(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [120]

Optimal result

Integrand size = 23, antiderivative size = 72 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {a+b} d}+\frac {\sinh (c+d x)}{2 a d \left (a+(a+b) \sinh ^2(c+d x)\right )} \] Output:

1/2*arctan((a+b)^(1/2)*sinh(d*x+c)/a^(1/2))/a^(3/2)/(a+b)^(1/2)/d+1/2*sinh 
(d*x+c)/a/d/(a+(a+b)*sinh(d*x+c)^2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {\arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a+b}}+\frac {\sqrt {a} \sinh (c+d x)}{a+(a+b) \sinh ^2(c+d x)}}{2 a^{3/2} d} \] Input:

Integrate[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]
 

Output:

(ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]]/Sqrt[a + b] + (Sqrt[a]*Sinh[c 
 + d*x])/(a + (a + b)*Sinh[c + d*x]^2))/(2*a^(3/2)*d)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4159, 215, 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.

\(\displaystyle \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sec (i c+i d x)^3}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\)

\(\Big \downarrow \) 4159

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {a+b}}+\frac {\sinh (c+d x)}{2 a \left ((a+b) \sinh ^2(c+d x)+a\right )}}{d}\)

Input:

Int[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]
 

Output:

(ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a + b]) + Sin 
h[c + d*x]/(2*a*(a + (a + b)*Sinh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 218

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4159

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
  Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2 
*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f} 
, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(60)=120\).

Time = 60.37 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.54

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, d a}\)	\(183\)
derivativedivides	\(\frac {\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}}{d}\)	\(228\)
default	\(\frac {\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}}{d}\)	\(228\)

Input:

int(sech(d*x+c)^3/(a+tanh(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)
 

Output:

exp(d*x+c)*(exp(2*d*x+2*c)-1)/a/d/(exp(4*d*x+4*c)*a+b*exp(4*d*x+4*c)+2*exp 
(2*d*x+2*c)*a-2*exp(2*d*x+2*c)*b+a+b)-1/4/(-a^2-a*b)^(1/2)/d/a*ln(exp(2*d* 
x+2*c)-2*a/(-a^2-a*b)^(1/2)*exp(d*x+c)-1)+1/4/(-a^2-a*b)^(1/2)/d/a*ln(exp( 
2*d*x+2*c)+2*a/(-a^2-a*b)^(1/2)*exp(d*x+c)-1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (60) = 120\).

Time = 0.12 (sec) , antiderivative size = 1563, normalized size of antiderivative = 21.71 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
 

Output:

[1/4*(4*(a^2 + a*b)*cosh(d*x + c)^3 + 12*(a^2 + a*b)*cosh(d*x + c)*sinh(d* 
x + c)^2 + 4*(a^2 + a*b)*sinh(d*x + c)^3 - ((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) 
*sqrt(-a^2 - a*b)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*s 
inh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(3*a + b)*cosh(d*x + c)^2 + 2 
*(3*(a + b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d 
*x + c)^3 - (3*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 
3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1 
)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + 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)*si 
nh(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^2 + a*b)*cosh(d*x + c) + 4*(3*(a^2 + a*b)*cosh(d* 
x + c)^2 - a^2 - a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x 
 + c)^4 + 4*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a 
^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*(a^4 - a^2*b^2)*d*cosh(d*x + 
 c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 - a^2*b^2) 
*d)*sinh(d*x + c)^2 + (a^4 + 2*a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b +...
 

Sympy [F]

\[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:

integrate(sech(d*x+c)**3/(a+b*tanh(d*x+c)**2)**2,x)
 

Output:

Integral(sech(c + d*x)**3/(a + b*tanh(c + d*x)**2)**2, x)
 

Maxima [F]

\[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
 

Output:

((a*e^(3*c) + b*e^(3*c))*e^(3*d*x) - (a*e^c + b*e^c)*e^(d*x))/(a^3*d + 2*a 
^2*b*d + a*b^2*d + (a^3*d*e^(4*c) + 2*a^2*b*d*e^(4*c) + a*b^2*d*e^(4*c))*e 
^(4*d*x) + 2*(a^3*d*e^(2*c) - a*b^2*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/ 
8*(e^(3*d*x + 3*c) + e^(d*x + c))/(a^2 + a*b + (a^2*e^(4*c) + a*b*e^(4*c)) 
*e^(4*d*x) + 2*(a^2*e^(2*c) - a*b*e^(2*c))*e^(2*d*x)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:

int(1/(cosh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^2),x)
 

Output:

int(1/(cosh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 590, normalized size of antiderivative = 8.19 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {e^{4 d x +4 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a +e^{4 d x +4 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b +2 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a -2 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b +\sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a +\sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b +e^{4 d x +4 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a +e^{4 d x +4 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b +2 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a -2 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b +\sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a +\sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b +2 e^{3 d x +3 c} a^{2}+2 e^{3 d x +3 c} a b -2 e^{d x +c} a^{2}-2 e^{d x +c} a b}{2 a^{2} d \left (e^{4 d x +4 c} a^{2}+2 e^{4 d x +4 c} a b +e^{4 d x +4 c} b^{2}+2 e^{2 d x +2 c} a^{2}-2 e^{2 d x +2 c} b^{2}+a^{2}+2 a b +b^{2}\right )} \] Input:

int(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x)
 

Output:

(e**(4*c + 4*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqr 
t(b))/sqrt(a))*a + e**(4*c + 4*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x) 
*sqrt(a + b) - sqrt(b))/sqrt(a))*b + 2*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a + b 
)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a - 2*e**(2*c + 2*d*x 
)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b 
 + sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))* 
a + sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a)) 
*b + e**(4*c + 4*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) + 
 sqrt(b))/sqrt(a))*a + e**(4*c + 4*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + 
d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b + 2*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a 
 + b)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a - 2*e**(2*c + 2 
*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a 
))*b + sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt( 
a))*a + sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt 
(a))*b + 2*e**(3*c + 3*d*x)*a**2 + 2*e**(3*c + 3*d*x)*a*b - 2*e**(c + d*x) 
*a**2 - 2*e**(c + d*x)*a*b)/(2*a**2*d*(e**(4*c + 4*d*x)*a**2 + 2*e**(4*c + 
 4*d*x)*a*b + e**(4*c + 4*d*x)*b**2 + 2*e**(2*c + 2*d*x)*a**2 - 2*e**(2*c 
+ 2*d*x)*b**2 + a**2 + 2*a*b + b**2))