\(\int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [139]

Optimal result

Integrand size = 23, antiderivative size = 59 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {x}{a}-\frac {(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a b^{3/2} d}+\frac {\tanh (c+d x)}{b d} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(196\) vs. \(2(59)=118\).

Time = 1.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 3.32 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left ((a+b)^2 \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (-\cosh (2 c)+\sinh (2 c))+\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} (b d x+a \text {sech}(c) \text {sech}(c+d x) \sinh (d x))\right )}{2 a b \sqrt {a+b} d \left (a+b \text {sech}^2(c+d x)\right ) \sqrt {b (\cosh (c)-\sinh (c))^4}} \] Input:

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

Output:

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*((a + b)^2*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])]*(-Cosh[2*c] + Sinh[2*c]) + Sq 
rt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*(b*d*x + a*Sech[c]*Sech[c + d*x]*S 
inh[d*x])))/(2*a*b*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - 
 Sinh[c])^4])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4629, 2075, 381, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4629

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

\(\Big \downarrow \) 2075

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

\(\Big \downarrow \) 381

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\tanh (c+d x)}{b}-\frac {\frac {(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b}}-\frac {b \text {arctanh}(\tanh (c+d x))}{a}}{b}}{d}\)

Input:

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

Output:

(-((-((b*ArcTanh[Tanh[c + d*x]])/a) + ((a + b)^(3/2)*ArcTanh[(Sqrt[b]*Tanh 
[c + d*x])/Sqrt[a + b]])/(a*Sqrt[b]))/b) + Tanh[c + d*x]/b)/d
 

Defintions of rubi rules used

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 381

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
+ 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) 
Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 
2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q 
}, x] && NeQ[b*c - a*d, 0] && 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 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. \(183\) vs. \(2(51)=102\).

Time = 2.66 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.12

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a b}}{d}\)	\(184\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a b}}{d}\)	\(184\)
risch	\(\frac {x}{a}-\frac {2}{b d \left ({\mathrm e}^{2 d x +2 c}+1\right )}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 b^{2} d}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 b d a}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 b^{2} d}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 b d a}\)	\(216\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (51) = 102\).

Time = 0.32 (sec) , antiderivative size = 683, normalized size of antiderivative = 11.58 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*(2*b*d*x*cosh(d*x + c)^2 + 4*b*d*x*cosh(d*x + c)*sinh(d*x + c) + 2*b* 
d*x*sinh(d*x + c)^2 + 2*b*d*x + ((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)/b)* 
log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh( 
d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + 
a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^ 
3 + (a^2 + 2*a*b)*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 + 2*b^2)*sqr 
t((a + b)/b))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*s 
inh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a 
+ 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*s 
inh(d*x + c) + a)) - 4*a)/(a*b*d*cosh(d*x + c)^2 + 2*a*b*d*cosh(d*x + c)*s 
inh(d*x + c) + a*b*d*sinh(d*x + c)^2 + a*b*d), (b*d*x*cosh(d*x + c)^2 + 2* 
b*d*x*cosh(d*x + c)*sinh(d*x + c) + b*d*x*sinh(d*x + c)^2 + b*d*x - ((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)/b)*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*c 
osh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-(a + b)/b) 
/(a + b)) - 2*a)/(a*b*d*cosh(d*x + c)^2 + 2*a*b*d*cosh(d*x + c)*sinh(d*x + 
 c) + a*b*d*sinh(d*x + c)^2 + a*b*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (51) = 102\).

Time = 0.26 (sec) , antiderivative size = 637, normalized size of antiderivative = 10.80 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*(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)))/(sqrt((a + b)*b)*b*d) + 1/8* 
(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)))/(sqrt((a + b)*b)*b*d) + 3/16*a* 
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)))/(sqrt((a + b)*b)*b*d) + 1/8*(a + 2*b)*log 
(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a*b*d) + 1/4*log(a* 
e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(b*d) - 1/8*(a + 2*b)*l 
og(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a*b*d) - 1/4*lo 
g(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(b*d) - 3/4*log(e 
^(2*d*x + 2*c) + 1)/(b*d) + 3/4*log(e^(-2*d*x - 2*c) + 1)/(b*d) - 1/32*(a^ 
2 + 8*a*b + 8*b^2)*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)))/(sqrt((a + b)*b)*a*b*d) 
+ 1/32*(a^2 + 8*a*b + 8*b^2)*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)))/(sqrt((a + b 
)*b)*a*b*d) - 5/8/((b*e^(2*d*x + 2*c) + b)*d) + 11/8/((b*e^(-2*d*x - 2*c) 
+ b)*d)
 

Giac [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.59 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\frac {d x + c}{a} - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a b} - \frac {2}{b {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \] Input:

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

Output:

((d*x + c)/a - (a^2 + 2*a*b + 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*b) - 2/(b*(e^(2*d*x + 2*c) + 1)))/ 
d
 

Mupad [B] (verification not implemented)

Time = 2.72 (sec) , antiderivative size = 183, normalized size of antiderivative = 3.10 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {x}{a}-\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{a^2\,b}-\frac {2\,{\left (a+b\right )}^{3/2}\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,b^{3/2}}\right )\,{\left (a+b\right )}^{3/2}}{2\,a\,b^{3/2}\,d}-\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{a^2\,b}+\frac {2\,{\left (a+b\right )}^{3/2}\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,b^{3/2}}\right )\,{\left (a+b\right )}^{3/2}}{2\,a\,b^{3/2}\,d} \] Input:

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

Output:

x/a - 2/(b*d*(exp(2*c + 2*d*x) + 1)) + (log((4*exp(2*c + 2*d*x)*(a + b)^2) 
/(a^2*b) - (2*(a + b)^(3/2)*(a + a*exp(2*c + 2*d*x) + 2*b*exp(2*c + 2*d*x) 
))/(a^2*b^(3/2)))*(a + b)^(3/2))/(2*a*b^(3/2)*d) - (log((4*exp(2*c + 2*d*x 
)*(a + b)^2)/(a^2*b) + (2*(a + b)^(3/2)*(a + a*exp(2*c + 2*d*x) + 2*b*exp( 
2*c + 2*d*x)))/(a^2*b^(3/2)))*(a + b)^(3/2))/(2*a*b^(3/2)*d)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 564, normalized size of antiderivative = 9.56 \[ \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {-e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) a -e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b -e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) a -e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b +e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {b}\, \sqrt {a +b}+e^{2 d x +2 c} a +a +2 b \right ) a +e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {b}\, \sqrt {a +b}+e^{2 d x +2 c} a +a +2 b \right ) b -\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) a -\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b -\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) a -\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b +\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {b}\, \sqrt {a +b}+e^{2 d x +2 c} a +a +2 b \right ) a +\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {b}\, \sqrt {a +b}+e^{2 d x +2 c} a +a +2 b \right ) b +4 e^{2 d x +2 c} a b +2 e^{2 d x +2 c} b^{2} d x +2 b^{2} d x}{2 a \,b^{2} d \left (e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

( - e**(2*c + 2*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 - e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + 
b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b 
- e**(2*c + 2*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 - e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*lo 
g(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b + e**(2* 
c + 2*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 + e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt 
(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*b - sqrt(b)*sqrt(a + b)*log( - sqr 
t(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a - sqrt(b)*sqr 
t(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt( 
a))*b - sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e* 
*(c + d*x)*sqrt(a))*a - sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) 
 - a - 2*b) + e**(c + d*x)*sqrt(a))*b + sqrt(b)*sqrt(a + b)*log(2*sqrt(b)* 
sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a + sqrt(b)*sqrt(a + b)*log(2* 
sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*b + 4*e**(2*c + 2*d*x) 
*a*b + 2*e**(2*c + 2*d*x)*b**2*d*x + 2*b**2*d*x)/(2*a*b**2*d*(e**(2*c + 2* 
d*x) + 1))