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

Optimal. Leaf size=137 \[ -\frac {\left (3 a^2-6 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} \sqrt {b} d (a+b)^3}-\frac {(3 a-b) \tanh (c+d x)}{8 a d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\tanh (c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x}{(a+b)^3} \]

[Out]

x/(a+b)^3-1/8*(3*a^2-6*a*b-b^2)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(3/2)/(a+b)^3/d/b^(1/2)-1/4*tanh(d*x+c)/
(a+b)/d/(a+b*tanh(d*x+c)^2)^2-1/8*(3*a-b)*tanh(d*x+c)/a/(a+b)^2/d/(a+b*tanh(d*x+c)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3670, 471, 527, 522, 206, 205} \[ -\frac {\left (3 a^2-6 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} \sqrt {b} d (a+b)^3}-\frac {(3 a-b) \tanh (c+d x)}{8 a d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\tanh (c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {x}{(a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a + b)^3 - ((3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(3/2)*Sqrt[b]*(a + b)^3*d)
- Tanh[c + d*x]/(4*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((3*a - b)*Tanh[c + d*x])/(8*a*(a + b)^2*d*(a + b*Ta
nh[c + d*x]^2))

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 206

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

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, 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 527

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

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\tanh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {1+3 x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d}\\ &=-\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(3 a-b) \tanh (c+d x)}{8 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-5 a-b+(-3 a+b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a (a+b)^2 d}\\ &=-\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(3 a-b) \tanh (c+d x)}{8 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^3 d}-\frac {\left (3 a^2-6 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a (a+b)^3 d}\\ &=\frac {x}{(a+b)^3}-\frac {\left (3 a^2-6 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} \sqrt {b} (a+b)^3 d}-\frac {\tanh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(3 a-b) \tanh (c+d x)}{8 a (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.17, size = 137, normalized size = 1.00 \[ \frac {\frac {\left (-3 a^2+6 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {(5 a-b) (a+b) \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}-\frac {4 b (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}+8 (c+d x)}{8 d (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(c + d*x) + ((-3*a^2 + 6*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - (4*b*(a +
b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2 - ((5*a - b)*(a + b)*Sinh[2*(c + d*x)])/(a*(a - b
+ (a + b)*Cosh[2*(c + d*x)])))/(8*(a + b)^3*d)

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 7791, normalized size = 56.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 128*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x +
c)*sinh(d*x + c)^7 + 16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 + 4*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^
3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^6 + 4*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 112*(a^4*
b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 + 16*(a^4*b - a^2*b^3)*d*x)*sinh(d*x + c)^6 + 8*(112*(a^4*b + 2*a
^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^3 + 3*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)
*cosh(d*x + c))*sinh(d*x + c)^5 + 20*a^4*b + 36*a^3*b^2 + 12*a^2*b^3 - 4*a*b^4 + 4*(15*a^4*b - 13*a^3*b^2 + 17
*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c)^4 + 4*(280*(a^4*b + 2*a^3*b^2 + a^
2*b^3)*d*x*cosh(d*x + c)^4 + 15*a^4*b - 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*b^2 + 3*a^2*b^3
)*d*x + 15*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 + 16*(56*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 + 5*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 1
6*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^4*b - 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*
b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x + 4*(15*a^4*b + a^
3*b^2 - 11*a^2*b^3 + 3*a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^2 + 4*(112*(a^4*b + 2*a^3*b^2 + a^2*b^3
)*d*x*cosh(d*x + c)^6 + 15*a^4*b + a^3*b^2 - 11*a^2*b^3 + 3*a*b^4 + 15*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^
4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^4 + 16*(a^4*b - a^2*b^3)*d*x + 6*(15*a^4*b - 13*a^3*b^2 + 17*a^2*b
^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^4 - 10*a^2*b^
2 - 8*a*b^3 - b^4)*cosh(d*x + c)^8 + 8*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (3
*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*sinh(d*x + c)^8 + 4*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x
+ c)^6 + 4*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4 + 7*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^3 + 3*(3*a^4 - 6*a^3*b - 4*a^2*b
^2 + 6*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*cosh
(d*x + c)^4 + 2*(35*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^4 + 9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*
a*b^3 - 3*b^4 + 30*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^4 - 10
*a^2*b^2 - 8*a*b^3 - b^4 + 8*(7*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^5 + 10*(3*a^4 - 6*a^3*b - 4
*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^3 + (9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*cosh(d*x + c))*
sinh(d*x + c)^3 + 4*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^2 + 4*(7*(3*a^4 - 10*a^2*b^2 -
8*a*b^3 - b^4)*cosh(d*x + c)^6 + 15*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^4 + 3*a^4 - 6
*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4 + 3*(9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sin
h(d*x + c)^2 + 8*((3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^7 + 3*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*
b^3 + b^4)*cosh(d*x + c)^5 + (9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (3*a^4 - 6*a
^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b)*log(((a^2 + 2*a*b + b^2)*cosh(d*x +
c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b
^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^
2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*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)*sqrt(-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*(16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^7 + 3*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^
3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^5 + 2*(15*a^4*b - 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3
*a^4*b - 2*a^3*b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^4*b + a^3*b^2 - 11*a^2*b^3 + 3*a*b^4 + 16*(a^4*b
- a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*
b^6)*d*cosh(d*x + c)^8 + 8*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)
*sinh(d*x + c)^7 + (a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*sinh(d*x + c)^8 + 4*(
a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^6 + 4*(7*(a^7*b + 5*a^6*b^2 +
10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 + (a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4
- 3*a^3*b^5 - a^2*b^6)*d)*sinh(d*x + c)^6 + 2*(3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^
2*b^6)*d*cosh(d*x + c)^4 + 8*(7*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x
+ c)^3 + 3*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(35*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^4 + 30*(a^7*b + 3
*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^2 + (3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 +
6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*d)*sinh(d*x + c)^4 + 4*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*
b^5 - a^2*b^6)*d*cosh(d*x + c)^2 + 8*(7*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*
cosh(d*x + c)^5 + 10*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^3 + (3*
a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^
7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^6 + 15*(a^7*b + 3*a^6*b^2 + 2
*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^4 + 3*(3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4
+ 7*a^3*b^5 + 3*a^2*b^6)*d*cosh(d*x + c)^2 + (a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6
)*d)*sinh(d*x + c)^2 + (a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d + 8*((a^7*b + 5*a
^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^7 + 3*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 -
2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^5 + (3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5
+ 3*a^2*b^6)*d*cosh(d*x + c)^3 + (a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x
+ c))*sinh(d*x + c)), 1/8*(8*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 64*(a^4*b + 2*a^3*b^2 + a^2*
b^3)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 8*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 + 2*(5*a^4*b - 5*
a^3*b^2 - 9*a^2*b^3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^6 + 2*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 +
a*b^4 + 112*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 + 16*(a^4*b - a^2*b^3)*d*x)*sinh(d*x + c)^6 + 4
*(112*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^3 + 3*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 16*(a^4
*b - a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 10*a^4*b + 18*a^3*b^2 + 6*a^2*b^3 - 2*a*b^4 + 2*(15*a^4*b
- 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c)^4 + 2*(280*(a^4*b
+ 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^4 + 15*a^4*b - 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^
3*b^2 + 3*a^2*b^3)*d*x + 15*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)
^2)*sinh(d*x + c)^4 + 8*(56*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 + 5*(5*a^4*b - 5*a^3*b^2 - 9*a^2
*b^3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^4*b - 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(
3*a^4*b - 2*a^3*b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x + 2
*(15*a^4*b + a^3*b^2 - 11*a^2*b^3 + 3*a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^2 + 2*(112*(a^4*b + 2*a^
3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^6 + 15*a^4*b + a^3*b^2 - 11*a^2*b^3 + 3*a*b^4 + 15*(5*a^4*b - 5*a^3*b^2 - 9
*a^2*b^3 + a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh(d*x + c)^4 + 16*(a^4*b - a^2*b^3)*d*x + 6*(15*a^4*b - 13*a^3
*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*b^2 + 3*a^2*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*
a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^8 + 8*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)*sinh(
d*x + c)^7 + (3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*sinh(d*x + c)^8 + 4*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 +
b^4)*cosh(d*x + c)^6 + 4*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4 + 7*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^3 + 3*(3*a^4 - 6*
a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^
3 - 3*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^4 + 9*a^4 - 24*a^3*b + 1
8*a^2*b^2 - 16*a*b^3 - 3*b^4 + 30*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 + 3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4 + 8*(7*(3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^5 + 10*(3*a^
4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^3 + (9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*
cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^2 + 4*(7*(3*a^4
- 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^6 + 15*(3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c
)^4 + 3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4 + 3*(9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*cosh(
d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 - 10*a^2*b^2 - 8*a*b^3 - b^4)*cosh(d*x + c)^7 + 3*(3*a^4 - 6*a^3*b - 4
*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c)^5 + (9*a^4 - 24*a^3*b + 18*a^2*b^2 - 16*a*b^3 - 3*b^4)*cosh(d*x + c)^3
+ (3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*((a + b)*c
osh(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)/(a*b)) + 4
*(16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^7 + 3*(5*a^4*b - 5*a^3*b^2 - 9*a^2*b^3 + a*b^4 + 16*(a^4*
b - a^2*b^3)*d*x)*cosh(d*x + c)^5 + 2*(15*a^4*b - 13*a^3*b^2 + 17*a^2*b^3 - 3*a*b^4 + 8*(3*a^4*b - 2*a^3*b^2 +
3*a^2*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^4*b + a^3*b^2 - 11*a^2*b^3 + 3*a*b^4 + 16*(a^4*b - a^2*b^3)*d*x)*cosh
(d*x + c))*sinh(d*x + c))/((a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)
^8 + 8*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (
a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*sinh(d*x + c)^8 + 4*(a^7*b + 3*a^6*b^2 +
2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^6 + 4*(7*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4
*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 + (a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b
^6)*d)*sinh(d*x + c)^6 + 2*(3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*d*cosh(d*x +
c)^4 + 8*(7*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^3 + 3*(a^7*b +
3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7*b + 5*
a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^4 + 30*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3
- 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^2 + (3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b
^5 + 3*a^2*b^6)*d)*sinh(d*x + c)^4 + 4*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cos
h(d*x + c)^2 + 8*(7*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^5 + 10
*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^3 + (3*a^7*b + 7*a^6*b^2 +
6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7*b + 5*a^6*b^2 + 10
*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^6 + 15*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4
- 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^4 + 3*(3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2
*b^6)*d*cosh(d*x + c)^2 + (a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d)*sinh(d*x + c)^2
+ (a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d + 8*((a^7*b + 5*a^6*b^2 + 10*a^5*b^3
+ 10*a^4*b^4 + 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^7 + 3*(a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b
^5 - a^2*b^6)*d*cosh(d*x + c)^5 + (3*a^7*b + 7*a^6*b^2 + 6*a^5*b^3 + 6*a^4*b^4 + 7*a^3*b^5 + 3*a^2*b^6)*d*cosh
(d*x + c)^3 + (a^7*b + 3*a^6*b^2 + 2*a^5*b^3 - 2*a^4*b^4 - 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)
)]

________________________________________________________________________________________

giac [B]  time = 0.44, size = 388, normalized size = 2.83 \[ -\frac {\frac {{\left (3 \, a^{2} - 6 \, a b - b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a b}} - \frac {8 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, {\left (5 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 5 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 9 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 13 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 17 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 11 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((3*a^2 - 6*a*b - b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^4 + 3*a^
3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)) - 8*(d*x + c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 2*(5*a^3*e^(6*d*x + 6*c) -
5*a^2*b*e^(6*d*x + 6*c) - 9*a*b^2*e^(6*d*x + 6*c) + b^3*e^(6*d*x + 6*c) + 15*a^3*e^(4*d*x + 4*c) - 13*a^2*b*e
^(4*d*x + 4*c) + 17*a*b^2*e^(4*d*x + 4*c) - 3*b^3*e^(4*d*x + 4*c) + 15*a^3*e^(2*d*x + 2*c) + a^2*b*e^(2*d*x +
2*c) - 11*a*b^2*e^(2*d*x + 2*c) + 3*b^3*e^(2*d*x + 2*c) + 5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)/((a^4 + 3*a^3*b + 3
*a^2*b^2 + a*b^3)*(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

________________________________________________________________________________________

maple [B]  time = 0.11, size = 340, normalized size = 2.48 \[ -\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 d \left (a +b \right )^{3}}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2 d \left (a +b \right )^{3}}-\frac {3 a b \left (\tanh ^{3}\left (d x +c \right )\right )}{8 d \left (a +b \right )^{3} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{2}}-\frac {\left (\tanh ^{3}\left (d x +c \right )\right ) b^{2}}{4 d \left (a +b \right )^{3} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {b^{3} \left (\tanh ^{3}\left (d x +c \right )\right )}{8 d \left (a +b \right )^{3} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{2} a}-\frac {5 a^{2} \tanh \left (d x +c \right )}{8 d \left (a +b \right )^{3} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{2}}-\frac {3 a b \tanh \left (d x +c \right )}{4 d \left (a +b \right )^{3} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{2}}-\frac {\tanh \left (d x +c \right ) b^{2}}{8 d \left (a +b \right )^{3} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )^{2}}-\frac {3 a \arctan \left (\frac {\tanh \left (d x +c \right ) b}{\sqrt {a b}}\right )}{8 d \left (a +b \right )^{3} \sqrt {a b}}+\frac {3 b \arctan \left (\frac {\tanh \left (d x +c \right ) b}{\sqrt {a b}}\right )}{4 d \left (a +b \right )^{3} \sqrt {a b}}+\frac {\arctan \left (\frac {\tanh \left (d x +c \right ) b}{\sqrt {a b}}\right ) b^{2}}{8 d \left (a +b \right )^{3} a \sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.90, size = 1472, normalized size = 10.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(3*a^3 - 21*a^2*b - 11*a*b^2 - 3*b^3)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^5 + 3*
a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) + 1/32*(3*a^3 - 21*a^2*b - 11*a*b^2 - 3*b^3)*arctan(1/2*((a + b)*e^(
-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) + 1/16*(5*a^4 - 18*a^2*b
^2 - 16*a*b^3 - 3*b^4 + (5*a^4 - 46*a^3*b - 40*a^2*b^2 + 14*a*b^3 + 3*b^4)*e^(6*d*x + 6*c) + (15*a^4 - 104*a^3
*b + 58*a^2*b^2 - 24*a*b^3 - 9*b^4)*e^(4*d*x + 4*c) + (15*a^4 - 58*a^3*b - 56*a^2*b^2 + 26*a*b^3 + 9*b^4)*e^(2
*d*x + 2*c))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 1
0*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*e^(8*d*x + 8*c) + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*
b^5)*e^(6*d*x + 6*c) + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*e^(4*d*x + 4*c) + 4
*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(2*d*x + 2*c))*d) - 1/16*(5*a^4 - 18*a^2*b^2
- 16*a*b^3 - 3*b^4 + (15*a^4 - 58*a^3*b - 56*a^2*b^2 + 26*a*b^3 + 9*b^4)*e^(-2*d*x - 2*c) + (15*a^4 - 104*a^3*
b + 58*a^2*b^2 - 24*a*b^3 - 9*b^4)*e^(-4*d*x - 4*c) + (5*a^4 - 46*a^3*b - 40*a^2*b^2 + 14*a*b^3 + 3*b^4)*e^(-6
*d*x - 6*c))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 -
2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3
*a^2*b^5)*e^(-4*d*x - 4*c) + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-6*d*x - 6*c)
+ (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*e^(-8*d*x - 8*c))*d) - 1/8*(5*a^3 + 13*a^2*b
+ 11*a*b^2 + 3*b^3 + (15*a^3 + 13*a^2*b - 11*a*b^2 - 9*b^3)*e^(-2*d*x - 2*c) + (15*a^3 - a^2*b + 9*a*b^2 + 9*
b^3)*e^(-4*d*x - 4*c) + (5*a^3 - a^2*b - 9*a*b^2 - 3*b^3)*e^(-6*d*x - 6*c))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^
3*b^3 + a^2*b^4 + 4*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*e^(-2*d*x - 2*c) + 2*(3*a^6 + 4*a^5*b + 2*a^4*b^2 +
4*a^3*b^3 + 3*a^2*b^4)*e^(-4*d*x - 4*c) + 4*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*e^(-6*d*x - 6*c) + (a^6 + 4*
a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*e^(-8*d*x - 8*c))*d) + 1/4*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^
(2*d*x + 2*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/4*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-
4*d*x - 4*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 3/16*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)
/sqrt(a*b))/(sqrt(a*b)*a^2*d)

________________________________________________________________________________________

mupad [B]  time = 3.47, size = 255, normalized size = 1.86 \[ \frac {\frac {a^2\,x}{\left (a+b\right )\,\left (a^2+2\,a\,b+b^2\right )}-\frac {\mathrm {tanh}\left (c+d\,x\right )\,\left (5\,a+b\right )}{8\,d\,\left (a^2+2\,a\,b+b^2\right )}+\frac {b^2\,x\,{\mathrm {tanh}\left (c+d\,x\right )}^4}{a^3+3\,a^2\,b+3\,a\,b^2+b^3}+\frac {2\,a\,b\,x\,{\mathrm {tanh}\left (c+d\,x\right )}^2}{a^3+3\,a^2\,b+3\,a\,b^2+b^3}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (3\,a\,b-b^2\right )}{8\,a\,d\,\left (a^2+2\,a\,b+b^2\right )}}{a^2+2\,a\,b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^4}+\frac {\mathrm {atan}\left (\frac {b\,\mathrm {tanh}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )\,\left (-3\,a^2+6\,a\,b+b^2\right )}{\sqrt {a\,b}\,\left (8\,a^4\,d+a\,b\,\left (24\,d\,a^2+24\,d\,a\,b+8\,d\,b^2\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^2*x)/((a + b)*(2*a*b + a^2 + b^2)) - (tanh(c + d*x)*(5*a + b))/(8*d*(2*a*b + a^2 + b^2)) + (b^2*x*tanh(c +
d*x)^4)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (2*a*b*x*tanh(c + d*x)^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (tanh(c
+ d*x)^3*(3*a*b - b^2))/(8*a*d*(2*a*b + a^2 + b^2)))/(a^2 + b^2*tanh(c + d*x)^4 + 2*a*b*tanh(c + d*x)^2) + (a
tan((b*tanh(c + d*x))/(a*b)^(1/2))*(6*a*b - 3*a^2 + b^2))/((a*b)^(1/2)*(8*a^4*d + a*b*(24*a^2*d + 8*b^2*d + 24
*a*b*d)))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________