\(\int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) [113]

Optimal result

Integrand size = 23, antiderivative size = 86 \[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {(2 a+3 b) \arctan (\sinh (c+d x))}{2 b^2 d}+\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}-\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3757, 425, 536, 209, 211} \[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}-\frac {(2 a+3 b) \arctan (\sinh (c+d x))}{2 b^2 d}-\frac {\tanh (c+d x) \text {sech}(c+d x)}{2 b d} \]

[In]

Int[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2),x]

[Out]

-1/2*((2*a + 3*b)*ArcTan[Sinh[c + d*x]])/(b^2*d) + ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])
/(Sqrt[a]*b^2*d) - (Sech[c + d*x]*Tanh[c + d*x])/(2*b*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 211

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

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

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

Rule 3757

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = -\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac {\text {Subst}\left (\int \frac {a+2 b+(-a-b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\sinh (c+d x)\right )}{2 b d} \\ & = -\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{b^2 d}-\frac {(2 a+3 b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 b^2 d} \\ & = -\frac {(2 a+3 b) \arctan (\sinh (c+d x))}{2 b^2 d}+\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}-\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92 \[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\frac {2 (a+b)^{3/2} \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a}}+2 (2 a+3 b) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+b \text {sech}(c+d x) \tanh (c+d x)}{2 b^2 d} \]

[In]

Integrate[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2),x]

[Out]

-1/2*((2*(a + b)^(3/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/Sqrt[a] + 2*(2*a + 3*b)*ArcTan[Tanh[(c + d
*x)/2]] + b*Sech[c + d*x]*Tanh[c + d*x])/(b^2*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(74)=148\).

Time = 50.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.74

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{2}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{2}}+\frac {\left (2 a +3 b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{2}}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) a \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{b^{2}}}{d}\)	\(236\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{2}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{2}}+\frac {\left (2 a +3 b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{2}}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) a \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{b^{2}}}{d}\)	\(236\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d b \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d \,b^{2}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d b}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d \,b^{2}}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d b}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{2 d \,b^{2}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{2 a d b}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{2 d \,b^{2}}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}-1\right )}{2 a d b}\)	\(320\)

[In]

int(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (74) = 148\).

Time = 0.29 (sec) , antiderivative size = 1584, normalized size of antiderivative = 18.42 \[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/2*(2*b*cosh(d*x + c)^3 + 6*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*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 + b)/a)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*s
inh(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*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh
(d*x + c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c))*sqrt(-(a + b)/a)
+ 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)) + 2*((2*a + 3*b)*cosh(d*x + c)^4 + 4*(2*a + 3*b)*cosh(d*x + c)*sin
h(d*x + c)^3 + (2*a + 3*b)*sinh(d*x + c)^4 + 2*(2*a + 3*b)*cosh(d*x + c)^2 + 2*(3*(2*a + 3*b)*cosh(d*x + c)^2
+ 2*a + 3*b)*sinh(d*x + c)^2 + 4*((2*a + 3*b)*cosh(d*x + c)^3 + (2*a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 2*a
+ 3*b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*b*cosh(d*x + c) + 2*(3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c))
/(b^2*d*cosh(d*x + c)^4 + 4*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*d*sinh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c
)^2 + b^2*d + 2*(3*b^2*d*cosh(d*x + c)^2 + b^2*d)*sinh(d*x + c)^2 + 4*(b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x
+ c))*sinh(d*x + c)), -(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 - ((a + b)*c
osh(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 + b)/a)*arctan(1/2*sqrt((a + b)/a)*(cosh(d*x + c) + sinh(d*x + c))) - ((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 + b)/a)*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh(d
*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + 3*a - b)*sinh(d*x
+ c))*sqrt((a + b)/a)/(a + b)) + ((2*a + 3*b)*cosh(d*x + c)^4 + 4*(2*a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^3 +
(2*a + 3*b)*sinh(d*x + c)^4 + 2*(2*a + 3*b)*cosh(d*x + c)^2 + 2*(3*(2*a + 3*b)*cosh(d*x + c)^2 + 2*a + 3*b)*s
inh(d*x + c)^2 + 4*((2*a + 3*b)*cosh(d*x + c)^3 + (2*a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 2*a + 3*b)*arctan
(cosh(d*x + c) + sinh(d*x + c)) - b*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c))/(b^2*d*cosh(d*x +
c)^4 + 4*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*d*sinh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + b^2*d + 2*(3
*b^2*d*cosh(d*x + c)^2 + b^2*d)*sinh(d*x + c)^2 + 4*(b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c))*sinh(d*x + c
))]

Sympy [F]

\[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\operatorname {sech}^{5}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]

[In]

integrate(sech(d*x+c)**5/(a+b*tanh(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**5/(a + b*tanh(c + d*x)**2), x)

Maxima [F]

\[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{5}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]

[In]

integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")

[Out]

-(e^(3*d*x + 3*c) - e^(d*x + c))/(b*d*e^(4*d*x + 4*c) + 2*b*d*e^(2*d*x + 2*c) + b*d) - (2*a*e^c + 3*b*e^c)*arc
tan(e^(d*x + c))*e^(-c)/(b^2*d) + 32*integrate(1/16*((a^2*e^(3*c) + 2*a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (
a^2*e^c + 2*a*b*e^c + b^2*e^c)*e^(d*x))/(a*b^2 + b^3 + (a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a*b^2*e^(2
*c) - b^3*e^(2*c))*e^(2*d*x)), x)

Giac [F]

\[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{5}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]

[In]

integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 2.60 (sec) , antiderivative size = 1012, normalized size of antiderivative = 11.77 \[ \int \frac {\text {sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\left (2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (12\,a^2\,b^4\,d\,\sqrt {a^3+3\,a^2\,b+3\,a\,b^2+b^3}-2\,a\,b^5\,d\,\sqrt {a^3+3\,a^2\,b+3\,a\,b^2+b^3}+18\,a^3\,b^3\,d\,\sqrt {a^3+3\,a^2\,b+3\,a\,b^2+b^3}+6\,a^4\,b^2\,d\,\sqrt {a^3+3\,a^2\,b+3\,a\,b^2+b^3}\right )}{a^3\,b^9\,d^2\,{\left (a+b\right )}^2\,\left (a^2+2\,a\,b+b^2\right )}-\frac {32\,\left (3\,a^5\,\sqrt {a\,b^4\,d^2}-b^5\,\sqrt {a\,b^4\,d^2}+4\,a\,b^4\,\sqrt {a\,b^4\,d^2}+15\,a^4\,b\,\sqrt {a\,b^4\,d^2}+20\,a^2\,b^3\,\sqrt {a\,b^4\,d^2}+27\,a^3\,b^2\,\sqrt {a\,b^4\,d^2}\right )}{a^3\,b^7\,d\,\sqrt {{\left (a+b\right )}^3}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {a\,b^4\,d^2}}\right )+\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,a^5\,\sqrt {a\,b^4\,d^2}-b^5\,\sqrt {a\,b^4\,d^2}+4\,a\,b^4\,\sqrt {a\,b^4\,d^2}+15\,a^4\,b\,\sqrt {a\,b^4\,d^2}+20\,a^2\,b^3\,\sqrt {a\,b^4\,d^2}+27\,a^3\,b^2\,\sqrt {a\,b^4\,d^2}\right )}{a^3\,b^7\,d\,\sqrt {{\left (a+b\right )}^3}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {a\,b^4\,d^2}}\right )\,\left (a^2\,b^7\,\sqrt {a\,b^4\,d^2}+2\,a^3\,b^6\,\sqrt {a\,b^4\,d^2}+a^4\,b^5\,\sqrt {a\,b^4\,d^2}\right )}{192\,a^3+576\,a^2\,b+384\,a\,b^2-64\,b^3}\right )+2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,{\left (a+b\right )}^2\,\sqrt {a\,b^4\,d^2}}{2\,a\,b^2\,d\,\sqrt {{\left (a+b\right )}^3}}\right )\right )\,\sqrt {a^3+3\,a^2\,b+3\,a\,b^2+b^3}}{2\,\sqrt {a\,b^4\,d^2}}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (18\,a^7\,\sqrt {b^4\,d^2}+3\,b^7\,\sqrt {b^4\,d^2}+30\,a^2\,b^5\,\sqrt {b^4\,d^2}+342\,a^3\,b^4\,\sqrt {b^4\,d^2}+555\,a^4\,b^3\,\sqrt {b^4\,d^2}+396\,a^5\,b^2\,\sqrt {b^4\,d^2}-34\,a\,b^6\,\sqrt {b^4\,d^2}+135\,a^6\,b\,\sqrt {b^4\,d^2}\right )}{b^8\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}-12\,a\,b^7\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}+18\,a^2\,b^6\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}+102\,a^3\,b^5\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}+117\,a^4\,b^4\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}+54\,a^5\,b^3\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}+9\,a^6\,b^2\,d\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}}\right )\,\sqrt {4\,a^2+12\,a\,b+9\,b^2}}{\sqrt {b^4\,d^2}}-\frac {{\mathrm {e}}^{c+d\,x}}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]

[In]

int(1/(cosh(c + d*x)^5*(a + b*tanh(c + d*x)^2)),x)

[Out]

((2*atan(((exp(d*x)*exp(c)*((64*(12*a^2*b^4*d*(3*a*b^2 + 3*a^2*b + a^3 + b^3)^(1/2) - 2*a*b^5*d*(3*a*b^2 + 3*a
^2*b + a^3 + b^3)^(1/2) + 18*a^3*b^3*d*(3*a*b^2 + 3*a^2*b + a^3 + b^3)^(1/2) + 6*a^4*b^2*d*(3*a*b^2 + 3*a^2*b
+ a^3 + b^3)^(1/2)))/(a^3*b^9*d^2*(a + b)^2*(2*a*b + a^2 + b^2)) - (32*(3*a^5*(a*b^4*d^2)^(1/2) - b^5*(a*b^4*d
^2)^(1/2) + 4*a*b^4*(a*b^4*d^2)^(1/2) + 15*a^4*b*(a*b^4*d^2)^(1/2) + 20*a^2*b^3*(a*b^4*d^2)^(1/2) + 27*a^3*b^2
*(a*b^4*d^2)^(1/2)))/(a^3*b^7*d*((a + b)^3)^(1/2)*(2*a*b + a^2 + b^2)*(a*b^4*d^2)^(1/2))) + (32*exp(3*c)*exp(3
*d*x)*(3*a^5*(a*b^4*d^2)^(1/2) - b^5*(a*b^4*d^2)^(1/2) + 4*a*b^4*(a*b^4*d^2)^(1/2) + 15*a^4*b*(a*b^4*d^2)^(1/2
) + 20*a^2*b^3*(a*b^4*d^2)^(1/2) + 27*a^3*b^2*(a*b^4*d^2)^(1/2)))/(a^3*b^7*d*((a + b)^3)^(1/2)*(2*a*b + a^2 +
b^2)*(a*b^4*d^2)^(1/2)))*(a^2*b^7*(a*b^4*d^2)^(1/2) + 2*a^3*b^6*(a*b^4*d^2)^(1/2) + a^4*b^5*(a*b^4*d^2)^(1/2))
)/(384*a*b^2 + 576*a^2*b + 192*a^3 - 64*b^3)) + 2*atan((exp(d*x)*exp(c)*(a + b)^2*(a*b^4*d^2)^(1/2))/(2*a*b^2*
d*((a + b)^3)^(1/2))))*(3*a*b^2 + 3*a^2*b + a^3 + b^3)^(1/2))/(2*(a*b^4*d^2)^(1/2)) - (atan((exp(d*x)*exp(c)*(
18*a^7*(b^4*d^2)^(1/2) + 3*b^7*(b^4*d^2)^(1/2) + 30*a^2*b^5*(b^4*d^2)^(1/2) + 342*a^3*b^4*(b^4*d^2)^(1/2) + 55
5*a^4*b^3*(b^4*d^2)^(1/2) + 396*a^5*b^2*(b^4*d^2)^(1/2) - 34*a*b^6*(b^4*d^2)^(1/2) + 135*a^6*b*(b^4*d^2)^(1/2)
))/(b^8*d*(12*a*b + 4*a^2 + 9*b^2)^(1/2) - 12*a*b^7*d*(12*a*b + 4*a^2 + 9*b^2)^(1/2) + 18*a^2*b^6*d*(12*a*b +
4*a^2 + 9*b^2)^(1/2) + 102*a^3*b^5*d*(12*a*b + 4*a^2 + 9*b^2)^(1/2) + 117*a^4*b^4*d*(12*a*b + 4*a^2 + 9*b^2)^(
1/2) + 54*a^5*b^3*d*(12*a*b + 4*a^2 + 9*b^2)^(1/2) + 9*a^6*b^2*d*(12*a*b + 4*a^2 + 9*b^2)^(1/2)))*(12*a*b + 4*
a^2 + 9*b^2)^(1/2))/(b^4*d^2)^(1/2) - exp(c + d*x)/(b*d*(exp(2*c + 2*d*x) + 1)) + (2*exp(c + d*x))/(b*d*(2*exp
(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))