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

Optimal. Leaf size=129 \[ \frac {(4 a+3 b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^{3/2}}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {b \sinh (c+d x)}{4 a d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )^2} \]

[Out]

1/8*(4*a+3*b)*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/a^(5/2)/(a+b)^(3/2)/d+1/4*b*sinh(d*x+c)/a/(a+b)/d/(a+(a+
b)*sinh(d*x+c)^2)^2+1/8*(4*a+3*b)*sinh(d*x+c)/a^2/(a+b)/d/(a+(a+b)*sinh(d*x+c)^2)

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Rubi [A]  time = 0.12, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3676, 385, 199, 205} \[ \frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {(4 a+3 b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^{3/2}}+\frac {b \sinh (c+d x)}{4 a d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a + 3*b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(3/2)*d) + (b*Sinh[c + d*x])/(4*a
*(a + b)*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + ((4*a + 3*b)*Sinh[c + d*x])/(8*a^2*(a + b)*d*(a + (a + b)*Sinh[c
+ d*x]^2))

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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]

Rule 385

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

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]

Rubi steps

\begin {align*} \int \frac {\text {sech}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{\left (a+(a+b) x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {\left (\frac {3}{a}+\frac {1}{a+b}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {(4 a+3 b) \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a+b) d}\\ &=\frac {(4 a+3 b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{3/2} d}+\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {(4 a+3 b) \sinh (c+d x)}{8 a^2 (a+b) d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.72, size = 123, normalized size = 0.95 \[ \frac {(4 a+3 b) \left (\frac {3 \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {3 (a+b) \sinh ^3(c+d x)+5 a \sinh (c+d x)}{a^2 \left ((a+b) \sinh ^2(c+d x)+a\right )^2}\right )-\frac {8 \sinh (c+d x)}{\left ((a+b) \sinh ^2(c+d x)+a\right )^2}}{24 d (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*Sinh[c + d*x])/(a + (a + b)*Sinh[c + d*x]^2)^2 + (4*a + 3*b)*((3*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[
a]])/(a^(5/2)*Sqrt[a + b]) + (5*a*Sinh[c + d*x] + 3*(a + b)*Sinh[c + d*x]^3)/(a^2*(a + (a + b)*Sinh[c + d*x]^2
)^2)))/(24*(a + b)*d)

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fricas [B]  time = 0.57, size = 6614, normalized size = 51.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^7 + 28*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b
^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*sinh(d*x + c)^7 + 4*(4*a^4 + 3
*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^5 + 4*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3 + 21*(4*a^4 + 11*a^
3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)
*cosh(d*x + c)^3 + (4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(4*a^4 + 3*a^3*
b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^3 + 4*(35*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 -
4*a^4 - 3*a^3*b + 10*a^2*b^2 + 9*a*b^3 + 10*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + 4*(21*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^5 + 10*(4*a^4 + 3*a^3*b - 10*a^2*b^2 -
9*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - ((4*a^
3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^8 + 8*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(
d*x + c)^7 + (4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*sinh(d*x + c)^8 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cos
h(d*x + c)^6 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(4*a^3 + 3*a^2*b - 4*a*b^2 -
3*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 +
11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^4 + 12*a^3 + a^2*b + 6*a*b^2 + 9*b^3 + 30*(4*a^3 + 3*a^2*b - 4*a*b
^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 10
*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(
d*x + c)^3 + 4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^2 + 4*(
7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)
^4 + 4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 3*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 8*((4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^7 + 3*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x +
c)^5 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^3 + (4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)
)*sinh(d*x + c))*sqrt(-a^2 - a*b)*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*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*si
nh(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)*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)) - 4*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c) + 4*(7*(4*
a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^6 + 5*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x +
c)^4 - 4*a^4 - 11*a^3*b - 10*a^2*b^2 - 3*a*b^3 - 3*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^8 + 8*(a^7 + 4*a^6*b + 6*a^5
*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4
)*d*sinh(d*x + c)^8 + 4*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 4*a^6*b + 6*a^5*
b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3
*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^
4*b^3 + a^3*b^4)*d*cosh(d*x + c)^3 + 3*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(35*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 30*(a^7 + 2*a^6*b - 2*a^4*b^3 -
a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(a^7
+ 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d
*cosh(d*x + c)^5 + 10*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 +
4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b
^4)*d*cosh(d*x + c)^6 + 15*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7 + 4*a^6*b + 2*a^
5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^2 +
(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d + 8*((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*c
osh(d*x + c)^7 + 3*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^5 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*
a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^3 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c))
, 1/8*(2*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^7 + 14*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b
^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*sinh(d*x + c)^7 + 2*(4*a^4 + 3
*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^5 + 2*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3 + 21*(4*a^4 + 11*a^
3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)
*cosh(d*x + c)^3 + (4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(4*a^4 + 3*a^3*
b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^3 + 2*(35*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 -
4*a^4 - 3*a^3*b + 10*a^2*b^2 + 9*a*b^3 + 10*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + 2*(21*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^5 + 10*(4*a^4 + 3*a^3*b - 10*a^2*b^2 -
9*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((4*a^
3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^8 + 8*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(
d*x + c)^7 + (4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*sinh(d*x + c)^8 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cos
h(d*x + c)^6 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(4*a^3 + 3*a^2*b - 4*a*b^2 -
3*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 +
11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^4 + 12*a^3 + a^2*b + 6*a*b^2 + 9*b^3 + 30*(4*a^3 + 3*a^2*b - 4*a*b
^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 10
*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(
d*x + c)^3 + 4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^2 + 4*(
7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)
^4 + 4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 3*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 8*((4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^7 + 3*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x +
c)^5 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^3 + (4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)
)*sinh(d*x + c))*sqrt(a^2 + a*b)*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))/sq
rt(a^2 + a*b)) + ((4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^8 + 8*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b
^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*sinh(d*x + c)^8 + 4*(4*a^3 + 3*a^2*b
- 4*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 7*(4*a^3 + 11*a^2*b + 10*a*b^2 +
3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(4*a^
3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x
+ c)^4 + 2*(35*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^4 + 12*a^3 + a^2*b + 6*a*b^2 + 9*b^3 + 30*(
4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^
3)*cosh(d*x + c)^5 + 10*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^
3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3 + 4*(4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3
)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(4*a^3 + 3*a^2*b - 4*a*b^2
- 3*b^3)*cosh(d*x + c)^4 + 4*a^3 + 3*a^2*b - 4*a*b^2 - 3*b^3 + 3*(12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 + 11*a^2*b + 10*a*b^2 + 3*b^3)*cosh(d*x + c)^7 + 3*(4*a^3 + 3*a^2*b - 4*a*
b^2 - 3*b^3)*cosh(d*x + c)^5 + (12*a^3 + a^2*b + 6*a*b^2 + 9*b^3)*cosh(d*x + c)^3 + (4*a^3 + 3*a^2*b - 4*a*b^2
- 3*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x +
c))/a) - 2*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cosh(d*x + c) + 2*(7*(4*a^4 + 11*a^3*b + 10*a^2*b^2 + 3*
a*b^3)*cosh(d*x + c)^6 + 5*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^4 - 4*a^4 - 11*a^3*b - 10*a^
2*b^2 - 3*a*b^3 - 3*(4*a^4 + 3*a^3*b - 10*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^7 + 4*a^6*b +
6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^8 + 8*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*co
sh(d*x + c)*sinh(d*x + c)^7 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*sinh(d*x + c)^8 + 4*(a^7 + 2
*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cos
h(d*x + c)^2 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a
^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)
^3 + 3*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7 + 4*a^6*b + 6*a^5*b
^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 30*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + (3*
a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4
)*d*cosh(d*x + c)^2 + 8*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^5 + 10*(a^7 + 2*a
^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d
*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^6 + 15*(a^7
+ 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d
*cosh(d*x + c)^2 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a
^4*b^3 + a^3*b^4)*d + 8*((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^7 + 3*(a^7 + 2*a^6*
b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^5 + (3*a^7 + 4*a^6*b + 2*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x
+ c)^3 + (a^7 + 2*a^6*b - 2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [B]  time = 0.99, size = 1495, normalized size = 11.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((2*(a^3*e^(2*c) + a^2*b*e^(2*c))^2*(8*a^2*b + 6*a*b^2 - (4*a^2 - a*b - 3*b^2)*sqrt(-a*b))*sqrt(a^2 - b^2
+ 2*sqrt(-a*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c)) - (4*a^6 + 7*a^5*b - a^4*b^2 - 7*a^3*b^3 - 3*a^2*b^4 + 2*(
4*a^5 + 11*a^4*b + 10*a^3*b^2 + 3*a^2*b^3)*sqrt(-a*b))*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*abs(-a^3*e^(2*c)
- a^2*b*e^(2*c))*abs(a*e^(2*c) + b*e^(2*c))*e^(2*c) - (8*a^8*b + 14*a^7*b^2 - 2*a^6*b^3 - 14*a^5*b^4 - 6*a^4*
b^5 - (4*a^8 + 3*a^7*b - 8*a^6*b^2 - 6*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*sqrt(-a*b))*sqrt(a^2 - b^2 + 2*sqrt(-a
*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c))*e^(4*c))*arctan(e^(d*x)/sqrt((a^4*e^(2*c) - a^2*b^2*e^(2*c) - sqrt((a^
4*e^(2*c) - a^2*b^2*e^(2*c))^2 - (a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c))*(a^4 + 2*a^3*b + a^2*b^2)))
/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c))))*e^(-4*c)/((a^11 + 5*a^10*b + 9*a^9*b^2 + 5*a^8*b^3 - 5*a^
7*b^4 - 9*a^6*b^5 - 5*a^5*b^6 - a^4*b^7 + 2*(a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5
+ a^4*b^6)*sqrt(-a*b))*abs(-a^3*e^(2*c) - a^2*b*e^(2*c))) + (2*(16*a^3*b - 4*a^2*b^2 - 12*a*b^3 + (4*a^3 - 21
*a^2*b - 14*a*b^2 + 3*b^3)*sqrt(-a*b))*(a^3*e^(2*c) + a^2*b*e^(2*c))^2*abs(a*e^(2*c) + b*e^(2*c)) - (4*a^7 - 1
3*a^6*b - 52*a^5*b^2 - 46*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5 - 4*(4*a^6 + 7*a^5*b - a^4*b^2 - 7*a^3*b^3 - 3*a^2*b
^4)*sqrt(-a*b))*abs(-a^3*e^(2*c) - a^2*b*e^(2*c))*abs(a*e^(2*c) + b*e^(2*c))*e^(2*c) - (16*a^9*b + 12*a^8*b^2
- 32*a^7*b^3 - 24*a^6*b^4 + 16*a^5*b^5 + 12*a^4*b^6 + (4*a^9 - 17*a^8*b - 39*a^7*b^2 + 6*a^6*b^3 + 38*a^5*b^4
+ 11*a^4*b^5 - 3*a^3*b^6)*sqrt(-a*b))*abs(a*e^(2*c) + b*e^(2*c))*e^(4*c))*arctan(e^(d*x)/sqrt((a^4*e^(2*c) - a
^2*b^2*e^(2*c) + sqrt((a^4*e^(2*c) - a^2*b^2*e^(2*c))^2 - (a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c))*(a
^4 + 2*a^3*b + a^2*b^2)))/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c))))*e^(-4*c)/((a^10 + 4*a^9*b + 5*a^
8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6 - 2*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*sq
rt(-a*b))*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))*abs(-a^3*e^(2*c) - a^2*b*e^(2*c))) - 2*(4*a^2*e^(7*d*x + 7*c)
+ 7*a*b*e^(7*d*x + 7*c) + 3*b^2*e^(7*d*x + 7*c) + 4*a^2*e^(5*d*x + 5*c) - a*b*e^(5*d*x + 5*c) - 9*b^2*e^(5*d*
x + 5*c) - 4*a^2*e^(3*d*x + 3*c) + a*b*e^(3*d*x + 3*c) + 9*b^2*e^(3*d*x + 3*c) - 4*a^2*e^(d*x + c) - 7*a*b*e^(
d*x + c) - 3*b^2*e^(d*x + c))/((a^3 + a^2*b)*(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)^2))/d

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maple [B]  time = 0.49, size = 1226, normalized size = 9.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,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)^2/(a+b)*tanh(1/2*d*x+1/2*
c)^7-5/4/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)^2/a/(a+b)*tanh(1/2*
d*x+1/2*c)^7*b-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)^2/(a+b)*tan
h(1/2*d*x+1/2*c)^5-13/4/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)^2/a/
(a+b)*tanh(1/2*d*x+1/2*c)^5*b-3/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)^2/a^2/(a+b)*tanh(1/2*d*x+1/2*c)^5*b^2+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)^2/(a+b)*tanh(1/2*d*x+1/2*c)^3+13/4/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)^2/a/(a+b)*tanh(1/2*d*x+1/2*c)^3*b+3/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)^2/a^2/(a+b)*tanh(1/2*d*x+1/2*c)^3*b^2+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)^2/(a+b)*tanh(1/2*d*x+1/2*c)+5/4/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)^2/a/(a+b)*tanh(1/2*d*x+1/2*c)*b-1/2/d/a/(a+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/a/(a+b)/(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))*b
+1/2/d/a/(a+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/a/(a+b)/(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))*b-3/8/d/a^2/(a+b)*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))+3/8/d/a^2/(a+b)/(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))*b^2+3/8/d/a^2/(a+b)*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))+3/8/d/a^2/(a+b)/(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))*b^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (4 \, a^{2} e^{\left (7 \, c\right )} + 7 \, a b e^{\left (7 \, c\right )} + 3 \, b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (4 \, a^{2} e^{\left (5 \, c\right )} - a b e^{\left (5 \, c\right )} - 9 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} - {\left (4 \, a^{2} e^{\left (3 \, c\right )} - a b e^{\left (3 \, c\right )} - 9 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (4 \, a^{2} e^{c} + 7 \, a b e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a^{5} d + 3 \, a^{4} b d + 3 \, a^{3} b^{2} d + a^{2} b^{3} d + {\left (a^{5} d e^{\left (8 \, c\right )} + 3 \, a^{4} b d e^{\left (8 \, c\right )} + 3 \, a^{3} b^{2} d e^{\left (8 \, c\right )} + a^{2} b^{3} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (a^{5} d e^{\left (6 \, c\right )} + a^{4} b d e^{\left (6 \, c\right )} - a^{3} b^{2} d e^{\left (6 \, c\right )} - a^{2} b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \, {\left (3 \, a^{5} d e^{\left (4 \, c\right )} + a^{4} b d e^{\left (4 \, c\right )} + a^{3} b^{2} d e^{\left (4 \, c\right )} + 3 \, a^{2} b^{3} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{5} d e^{\left (2 \, c\right )} + a^{4} b d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )} - a^{2} b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + 8 \, \int \frac {{\left (4 \, a e^{\left (3 \, c\right )} + 3 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (4 \, a e^{c} + 3 \, b e^{c}\right )} e^{\left (d x\right )}}{32 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} e^{\left (4 \, c\right )} + 2 \, a^{3} b e^{\left (4 \, c\right )} + a^{2} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{4} e^{\left (2 \, c\right )} - a^{2} b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((4*a^2*e^(7*c) + 7*a*b*e^(7*c) + 3*b^2*e^(7*c))*e^(7*d*x) + (4*a^2*e^(5*c) - a*b*e^(5*c) - 9*b^2*e^(5*c))
*e^(5*d*x) - (4*a^2*e^(3*c) - a*b*e^(3*c) - 9*b^2*e^(3*c))*e^(3*d*x) - (4*a^2*e^c + 7*a*b*e^c + 3*b^2*e^c)*e^(
d*x))/(a^5*d + 3*a^4*b*d + 3*a^3*b^2*d + a^2*b^3*d + (a^5*d*e^(8*c) + 3*a^4*b*d*e^(8*c) + 3*a^3*b^2*d*e^(8*c)
+ a^2*b^3*d*e^(8*c))*e^(8*d*x) + 4*(a^5*d*e^(6*c) + a^4*b*d*e^(6*c) - a^3*b^2*d*e^(6*c) - a^2*b^3*d*e^(6*c))*e
^(6*d*x) + 2*(3*a^5*d*e^(4*c) + a^4*b*d*e^(4*c) + a^3*b^2*d*e^(4*c) + 3*a^2*b^3*d*e^(4*c))*e^(4*d*x) + 4*(a^5*
d*e^(2*c) + a^4*b*d*e^(2*c) - a^3*b^2*d*e^(2*c) - a^2*b^3*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/32*((4*a*e^(3*
c) + 3*b*e^(3*c))*e^(3*d*x) + (4*a*e^c + 3*b*e^c)*e^(d*x))/(a^4 + 2*a^3*b + a^2*b^2 + (a^4*e^(4*c) + 2*a^3*b*e
^(4*c) + a^2*b^2*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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