### 3.122 $$\int \frac{\text{sech}^5(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=102 $-\frac{(2 a-b) \sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}+\frac{(a+b) \sinh (c+d x)}{2 a b d \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d}$

[Out]

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

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Rubi [A]  time = 0.127104, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.217, Rules used = {3676, 414, 522, 203, 205} $-\frac{(2 a-b) \sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}+\frac{(a+b) \sinh (c+d x)}{2 a b d \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3676

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]

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

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 203

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\text{sech}^5(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a+b) \sinh (c+d x)}{2 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{a-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 a b d}\\ &=\frac{(a+b) \sinh (c+d x)}{2 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b^2 d}-\frac{((2 a-b) (a+b)) \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{2 a b^2 d}\\ &=\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d}-\frac{(2 a-b) \sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^2 d}+\frac{(a+b) \sinh (c+d x)}{2 a b d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.483468, size = 203, normalized size = 1.99 $\frac{(a-b) \left (\left (2 a^2+a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right )+4 a^{3/2} \sqrt{a+b} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )+(a+b) \cosh (2 (c+d x)) \left (\left (2 a^2+a b-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right )+4 a^{3/2} \sqrt{a+b} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )+2 \sqrt{a} b (a+b)^{3/2} \sinh (c+d x)}{2 a^{3/2} b^2 d \sqrt{a+b} ((a+b) \cosh (2 (c+d x))+a-b)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a - b)*((2*a^2 + a*b - b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]] + 4*a^(3/2)*Sqrt[a + b]*ArcTan[Tanh[
(c + d*x)/2]]) + (a + b)*((2*a^2 + a*b - b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]] + 4*a^(3/2)*Sqrt[a +
b]*ArcTan[Tanh[(c + d*x)/2]])*Cosh[2*(c + d*x)] + 2*Sqrt[a]*b*(a + b)^(3/2)*Sinh[c + d*x])/(2*a^(3/2)*b^2*Sqr
t[a + b]*d*(a - b + (a + b)*Cosh[2*(c + d*x)]))

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Maple [B]  time = 0.099, size = 1007, normalized size = 9.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a e^{\left (3 \, c\right )} + b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (a e^{c} + b e^{c}\right )} e^{\left (d x\right )}}{a^{2} b d + a b^{2} d +{\left (a^{2} b d e^{\left (4 \, c\right )} + a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} b d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \frac{2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b^{2} d} - 32 \, \int \frac{{\left (2 \, a^{2} e^{\left (3 \, c\right )} + a b e^{\left (3 \, c\right )} - b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (2 \, a^{2} e^{c} + a b e^{c} - b^{2} e^{c}\right )} e^{\left (d x\right )}}{32 \,{\left (a^{2} b^{2} + a b^{3} +{\left (a^{2} b^{2} e^{\left (4 \, c\right )} + a b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} b^{2} e^{\left (2 \, c\right )} - a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*e^(3*c) + b*e^(3*c))*e^(3*d*x) - (a*e^c + b*e^c)*e^(d*x))/(a^2*b*d + a*b^2*d + (a^2*b*d*e^(4*c) + a*b^2*d*
e^(4*c))*e^(4*d*x) + 2*(a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e^(d*x + c))/(b^2*d) - 32*int
egrate(1/32*((2*a^2*e^(3*c) + a*b*e^(3*c) - b^2*e^(3*c))*e^(3*d*x) + (2*a^2*e^c + a*b*e^c - b^2*e^c)*e^(d*x))/
(a^2*b^2 + a*b^3 + (a^2*b^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(a^2*b^2*e^(2*c) - a*b^3*e^(2*c))*e^(2*d*x)
), x)

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Fricas [B]  time = 2.58701, size = 5292, normalized size = 51.88 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a*b + b^2)*cosh(d*x + c)^3 + 12*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + 4*(a*b + b^2)*sinh(d*x +
c)^3 - ((2*a^2 + a*b - b^2)*cosh(d*x + c)^4 + 4*(2*a^2 + a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + a
*b - b^2)*sinh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(2*a^2 + a*b - b^2)*cosh(d*x + c)^2
+ 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + 2*a^2 + a*b - b^2 + 4*((2*a^2 + a*b - b^2)*cosh(d*x + c)^3 + (2*a^2
- 3*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*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)*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*(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)) + 8*((a^2 + a*b)*cosh(d*x + c)^4
+ 4*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + a*b)*sinh(d*x + c)^4 + 2*(a^2 - a*b)*cosh(d*x + c)^2 +
2*(3*(a^2 + a*b)*cosh(d*x + c)^2 + a^2 - a*b)*sinh(d*x + c)^2 + a^2 + a*b + 4*((a^2 + a*b)*cosh(d*x + c)^3 +
(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 4*(a*b + b^2)*cosh(d*x + c)
+ 4*(3*(a*b + b^2)*cosh(d*x + c)^2 - a*b - b^2)*sinh(d*x + c))/((a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 4*(a^2*b
^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b^2 + a*b^3)*d*sinh(d*x + c)^4 + 2*(a^2*b^2 - a*b^3)*d*cosh
(d*x + c)^2 + 2*(3*(a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 + (a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^2 + (a^2*b^2 + a*b
^3)*d + 4*((a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 + (a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*(a*
b + b^2)*cosh(d*x + c)^3 + 6*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a*b + b^2)*sinh(d*x + c)^3 - ((2*a
^2 + a*b - b^2)*cosh(d*x + c)^4 + 4*(2*a^2 + a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + a*b - b^2)*si
nh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(2*a^2 + a*b - b^2)*cosh(d*x + c)^2 + 2*a^2 - 3
*a*b + b^2)*sinh(d*x + c)^2 + 2*a^2 + a*b - b^2 + 4*((2*a^2 + a*b - b^2)*cosh(d*x + c)^3 + (2*a^2 - 3*a*b + b^
2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a + b)/a)*arctan(1/2*sqrt((a + b)/a)*(cosh(d*x + c) + sinh(d*x + c))) -
((2*a^2 + a*b - b^2)*cosh(d*x + c)^4 + 4*(2*a^2 + a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + a*b - b
^2)*sinh(d*x + c)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(2*a^2 + a*b - b^2)*cosh(d*x + c)^2 + 2*a
^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + 2*a^2 + a*b - b^2 + 4*((2*a^2 + a*b - b^2)*cosh(d*x + c)^3 + (2*a^2 - 3*a*
b + b^2)*cosh(d*x + c))*sinh(d*x + c))*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)) + 4*((a^2 + a*b)*cosh(d*x + c)^4 + 4*(a^2 + a*b)*cosh(d*x + c)*si
nh(d*x + c)^3 + (a^2 + a*b)*sinh(d*x + c)^4 + 2*(a^2 - a*b)*cosh(d*x + c)^2 + 2*(3*(a^2 + a*b)*cosh(d*x + c)^2
+ a^2 - a*b)*sinh(d*x + c)^2 + a^2 + a*b + 4*((a^2 + a*b)*cosh(d*x + c)^3 + (a^2 - a*b)*cosh(d*x + c))*sinh(d
*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(a*b + b^2)*cosh(d*x + c) + 2*(3*(a*b + b^2)*cosh(d*x + c)^
2 - a*b - b^2)*sinh(d*x + c))/((a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 4*(a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(
d*x + c)^3 + (a^2*b^2 + a*b^3)*d*sinh(d*x + c)^4 + 2*(a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^2*b^2 + a*b
^3)*d*cosh(d*x + c)^2 + (a^2*b^2 - a*b^3)*d)*sinh(d*x + c)^2 + (a^2*b^2 + a*b^3)*d + 4*((a^2*b^2 + a*b^3)*d*co
sh(d*x + c)^3 + (a^2*b^2 - a*b^3)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{5}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 2.1604, size = 5392, normalized size = 52.86 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*(3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(a
rccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a^4 + 3*a^3*b - a*b^
3)*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3
- 9*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-
a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a
+ b) + b/(a + b)))) + 3*(2*a^4 + 3*a^3*b - a*b^3)*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/
2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a^4
+ 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) +
b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a +
b))))^2 - 3*(2*a^4 + 3*a^3*b - a*b^3)*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(a
rccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a^4 + 3*a^3*b -
a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))
)*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^4 + 3*a^3*b - a*b^3)*sin(1/2*real_part(arccos(-
a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^4 + 3*a^3*b - a*b^3)*c
osh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a^4
+ 3*a^3*b - a*b^3)*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) +
b/(a + b)))))*arctan((((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a +
b))) + e^(d*x))/(((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b))
)))/(2*a^3*b^3 + (a^2*b^2 - a*b^3)*sqrt(-a*b)*abs(a)) + 2*(3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-
a/(a + b) + b/(a + b)))) - (2*a^4 + 3*a^3*b - a*b^3)*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin
(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3 - 9*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a +
b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a^4 + 3*a^3*b - a*b^3)*cosh(1/2*i
mag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag
_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(
a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b
))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a^4 + 3*a^3*b - a*b^3)*cosh(1/2*imag_part(ar
ccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(
-a/(a + b) + b/(a + b))))^2 - 3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2
*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2
*a^4 + 3*a^3*b - a*b^3)*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a +
b) + b/(a + b))))^3 + (2*a^4 + 3*a^3*b - a*b^3)*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*r
eal_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a^4 + 3*a^3*b - a*b^3)*sin(1/2*real_part(arccos(-a/(a + b) + b/
(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arctan(-(((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) +
a*b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b
^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*a^3*b^3 + (a^2*b^2 - a*b^3)*sqrt(-a*b)*abs(a)) + ((2
*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a +
b) + b/(a + b))))^3 - 3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2
*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a^4
+ 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) +
b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real
_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part
(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a^4 + 3*a^3*b -
a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))
))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a^4 + 3*a^3*b - a*b^3)*cos(1
/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + 3*(2*a
^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) +
b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real
_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a^4 + 3*a^3*b
- a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))
))*log(2*((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x)
+ sqrt((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b^3*e^(4*c))) + e^(2*d*x))/(2*a^3*b^3 + (a^2*b^2 - a*b^3)*sqrt(
-a*b)*abs(a)) - ((2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_
part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/
(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a +
b))))^2 - 3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(
arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a^4 + 3*a^3*b -
a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^
2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3
*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(
a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(2*a^4 + 3*a^3*b - a*b^3)*cos(
1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*rea
l_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a^4 + 3*a
^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a
+ b))))^3 + 3*(2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(
arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^4 + 3*a^3*b -
a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))
- (2*a^4 + 3*a^3*b - a*b^3)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a
+ b) + b/(a + b)))))*log(-2*((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a -
b)/(a + b)))*e^(d*x) + sqrt((a^2*b^2 + a*b^3)/(a^2*b^2*e^(4*c) + a*b^3*e^(4*c))) + e^(2*d*x))/(2*a^3*b^3 + (a^
2*b^2 - a*b^3)*sqrt(-a*b)*abs(a)) - 16*arctan(e^(d*x + c))/b^2 - 8*(a*e^(3*d*x + 3*c) + b*e^(3*d*x + 3*c) - a*
e^(d*x + c) - b*e^(d*x + c))/((a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*
c) + a + b)*a*b))/d