\(\int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [160]

Optimal result

Integrand size = 23, antiderivative size = 139 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}+\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 b^{3/2} \sqrt {a+b} d}-\frac {(a+b) \tanh (c+d x)}{4 a b d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {(a-4 b) \tanh (c+d x)}{8 a^2 b d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

x/a^3+1/8*(a^2-4*a*b-8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^3/b 
^(3/2)/(a+b)^(1/2)/d-1/4*(a+b)*tanh(d*x+c)/a/b/d/(a+b-b*tanh(d*x+c)^2)^2+1 
/8*(a-4*b)*tanh(d*x+c)/a^2/b/d/(a+b-b*tanh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1317\) vs. \(2(139)=278\).

Time = 9.08 (sec) , antiderivative size = 1317, normalized size of antiderivative = 9.47 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((6*a*(a + 2*b)*ArcTanh 
[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (4*(3*a^2 + 8*a*b + 
 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (4*a 
*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[ 
2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) - (2*Sqrt[b]*( 
3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cosh[2*(c 
 + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2) 
 + (Sqrt[b]*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 
+ 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] 
 - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cos 
h[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2 
*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*( 
a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 
 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] 
 + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] 
+ 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 
128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 256*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 12 
8*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - 9*a^6*Sinh[2*c] + 12*a^5*b*Sinh[2*c] + 6 
84*a^4*b^2*Sinh[2*c] + 2880*a^3*b^3*Sinh[2*c] + 5280*a^2*b^4*Sinh[2*c] + 4 
608*a*b^5*Sinh[2*c] + 1536*b^6*Sinh[2*c] + 9*a^6*Sinh[2*d*x] - 14*a^5*b...
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4629, 2075, 372, 402, 25, 27, 397, 219, 221}

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.

\(\displaystyle \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (i c+i d x)^4}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\)

\(\Big \downarrow \) 4629

\(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 2075

\(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {\frac {\int \frac {-\left ((a-3 b) \tanh ^2(c+d x)\right )+a+b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {(a-4 b) \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\int -\frac {(a+b) \left (-\left ((a-4 b) \tanh ^2(c+d x)\right )+a+4 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {(a+b) \left (-\left ((a-4 b) \tanh ^2(c+d x)\right )+a+4 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}+\frac {(a-4 b) \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {-\left ((a-4 b) \tanh ^2(c+d x)\right )+a+4 b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a}+\frac {(a-4 b) \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2-4 a b-8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}+\frac {8 b \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}}{2 a}+\frac {(a-4 b) \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2-4 a b-8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}+\frac {8 b \text {arctanh}(\tanh (c+d x))}{a}}{2 a}+\frac {(a-4 b) \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b} \sqrt {a+b}}+\frac {8 b \text {arctanh}(\tanh (c+d x))}{a}}{2 a}+\frac {(a-4 b) \tanh (c+d x)}{2 a \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a b}-\frac {(a+b) \tanh (c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

Input:

Int[Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

(-1/4*((a + b)*Tanh[c + d*x])/(a*b*(a + b - b*Tanh[c + d*x]^2)^2) + (((8*b 
*ArcTanh[Tanh[c + d*x]])/a + ((a^2 - 4*a*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[ 
c + d*x])/Sqrt[a + b]])/(a*Sqrt[b]*Sqrt[a + b]))/(2*a) + ((a - 4*b)*Tanh[c 
 + d*x])/(2*a*(a + b - b*Tanh[c + d*x]^2)))/(4*a*b))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 221

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

rule 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 2075

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa 
ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi 
alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] &&  ! 
BinomialMatchQ[{u, v}, x]
 

rule 3042

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

rule 4629

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f 
_.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim 
p[ff/f   Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 
)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte 
gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(125)=250\).

Time = 89.64 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.47

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {a \left (a^{2}+5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b}-\frac {\left (3 a^{2}+19 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 b}-\frac {\left (3 a^{2}+19 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 b}-\frac {a \left (a^{2}+5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (a^{2}-4 a b -8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b}}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}}{d}\)	\(343\)
default	\(\frac {\frac {\frac {2 \left (-\frac {a \left (a^{2}+5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b}-\frac {\left (3 a^{2}+19 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 b}-\frac {\left (3 a^{2}+19 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 b}-\frac {a \left (a^{2}+5 a b +4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (a^{2}-4 a b -8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b}}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}}{d}\)	\(343\)
risch	\(\frac {x}{a^{3}}+\frac {a^{3} {\mathrm e}^{6 d x +6 c}+12 a^{2} b \,{\mathrm e}^{6 d x +6 c}+16 a \,b^{2} {\mathrm e}^{6 d x +6 c}+3 a^{3} {\mathrm e}^{4 d x +4 c}+26 a^{2} b \,{\mathrm e}^{4 d x +4 c}+56 a \,b^{2} {\mathrm e}^{4 d x +4 c}+48 b^{3} {\mathrm e}^{4 d x +4 c}+3 a^{3} {\mathrm e}^{2 d x +2 c}+20 a^{2} b \,{\mathrm e}^{2 d x +2 c}+32 a \,b^{2} {\mathrm e}^{2 d x +2 c}+a^{3}+6 a^{2} b}{4 a^{3} b d \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}\, d b a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, d \,a^{2}}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}\, d b a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}\, d \,a^{2}}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, d \,a^{3}}\)	\(664\)

Input:

int(tanh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(2/a^3*((-1/8*a*(a^2+5*a*b+4*b^2)/b*tanh(1/2*d*x+1/2*c)^7-1/8*(3*a^2+1 
9*a*b-4*b^2)*a/b*tanh(1/2*d*x+1/2*c)^5-1/8*(3*a^2+19*a*b-4*b^2)*a/b*tanh(1 
/2*d*x+1/2*c)^3-1/8*a*(a^2+5*a*b+4*b^2)/b*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d 
*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2 
*d*x+1/2*c)^2*b+a+b)^2+1/8*(a^2-4*a*b-8*b^2)/b*(-1/4/b^(1/2)/(a+b)^(1/2)*l 
n((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1 
/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1 
/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))))+1/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-1/a^3 
*ln(tanh(1/2*d*x+1/2*c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3111 vs. \(2 (131) = 262\).

Time = 0.52 (sec) , antiderivative size = 6464, normalized size of antiderivative = 46.50 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(tanh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\tanh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(tanh(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Integral(tanh(c + d*x)**4/(a + b*sech(c + d*x)**2)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2201 vs. \(2 (131) = 262\).

Time = 0.58 (sec) , antiderivative size = 2201, normalized size of antiderivative = 15.83 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(tanh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

1/256*(a^4 - 20*a^3*b - 120*a^2*b^2 - 160*a*b^3 - 64*b^4)*log((a*e^(2*d*x 
+ 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqr 
t((a + b)*b)))/((a^5*b + 2*a^4*b^2 + a^3*b^3)*sqrt((a + b)*b)*d) + 1/64*(a 
 - 2*b)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x 
+ 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b + 2*a*b^2 + b^3)*sqrt((a + 
b)*b)*d) - 1/256*(a^4 - 20*a^3*b - 120*a^2*b^2 - 160*a*b^3 - 64*b^4)*log(( 
a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a 
+ 2*b + 2*sqrt((a + b)*b)))/((a^5*b + 2*a^4*b^2 + a^3*b^3)*sqrt((a + b)*b) 
*d) - 3/128*(a + 4*b)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b 
))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b + 2*a*b^2 + 
 b^3)*sqrt((a + b)*b)*d) - 1/64*(a - 2*b)*log((a*e^(-2*d*x - 2*c) + a + 2* 
b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b))) 
/((a^2*b + 2*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/64*(a^5 + 38*a^4*b + 88*a 
^3*b^2 + 48*a^2*b^3 + (a^5 + 76*a^4*b + 392*a^3*b^2 + 576*a^2*b^3 + 256*a* 
b^4)*e^(6*d*x + 6*c) + (3*a^5 + 186*a^4*b + 1024*a^3*b^2 + 2240*a^2*b^3 + 
2176*a*b^4 + 768*b^5)*e^(4*d*x + 4*c) + (3*a^5 + 148*a^4*b + 648*a^3*b^2 + 
 896*a^2*b^3 + 384*a*b^4)*e^(2*d*x + 2*c))/((a^7*b + 2*a^6*b^2 + a^5*b^3 + 
 (a^7*b + 2*a^6*b^2 + a^5*b^3)*e^(8*d*x + 8*c) + 4*(a^7*b + 4*a^6*b^2 + 5* 
a^5*b^3 + 2*a^4*b^4)*e^(6*d*x + 6*c) + 2*(3*a^7*b + 14*a^6*b^2 + 27*a^5*b^ 
3 + 24*a^4*b^4 + 8*a^3*b^5)*e^(4*d*x + 4*c) + 4*(a^7*b + 4*a^6*b^2 + 5*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (131) = 262\).

Time = 1.19 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.02 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {8 \, {\left (d x + c\right )}}{a^{3}} + \frac {{\left (a^{2} - 4 \, a b - 8 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{3} b} + \frac {2 \, {\left (a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 12 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 26 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 20 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 6 \, a^{2} b\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2} a^{3} b}}{8 \, d} \] Input:

integrate(tanh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

1/8*(8*(d*x + c)/a^3 + (a^2 - 4*a*b - 8*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) 
 + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^3*b) + 2*(a^3*e^(6*d*x + 
 6*c) + 12*a^2*b*e^(6*d*x + 6*c) + 16*a*b^2*e^(6*d*x + 6*c) + 3*a^3*e^(4*d 
*x + 4*c) + 26*a^2*b*e^(4*d*x + 4*c) + 56*a*b^2*e^(4*d*x + 4*c) + 48*b^3*e 
^(4*d*x + 4*c) + 3*a^3*e^(2*d*x + 2*c) + 20*a^2*b*e^(2*d*x + 2*c) + 32*a*b 
^2*e^(2*d*x + 2*c) + a^3 + 6*a^2*b)/((a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2 
*c) + 4*b*e^(2*d*x + 2*c) + a)^2*a^3*b))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {tanh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:

int(tanh(c + d*x)^4/(a + b/cosh(c + d*x)^2)^3,x)
 

Output:

int((cosh(c + d*x)^6*tanh(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 4136, normalized size of antiderivative = 29.76 \[ \int \frac {\tanh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(tanh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

(e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - 
a - 2*b) + e**(c + d*x)*sqrt(a))*a**5 - 2*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a 
+ b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))* 
a**4*b - 16*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqr 
t(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**2 - 16*e**(8*c + 8*d*x 
)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c 
 + d*x)*sqrt(a))*a**2*b**3 + e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt 
(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5 - 2*e**(8*c 
 + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + 
e**(c + d*x)*sqrt(a))*a**4*b - 16*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log 
(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**2 - 
 16*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - 
a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b**3 - e**(8*c + 8*d*x)*sqrt(b)*sqrt 
(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**5 + 2 
*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c 
+ 2*d*x)*a + a + 2*b)*a**4*b + 16*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log 
(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**3*b**2 + 16*e**( 
8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d 
*x)*a + a + 2*b)*a**2*b**3 + 4*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - 
 sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5 - 8...