### 3.183 $$\int \frac{\tanh ^2(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=85 $-\frac{\tanh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{(a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d (a+b)^2}+\frac{x}{(a+b)^2}$

[Out]

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

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Rubi [A]  time = 0.107958, antiderivative size = 85, 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 = {3670, 471, 522, 206, 205} $-\frac{\tanh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{(a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b} d (a+b)^2}+\frac{x}{(a+b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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]))

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

Mathematica [A]  time = 0.37579, size = 86, normalized size = 1.01 $\frac{\frac{(b-a) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}-\frac{(a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}+2 (c+d x)}{2 d (a+b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.023, size = 162, normalized size = 1.9 \begin{align*}{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a+b \right ) ^{2}}}-{\frac{a\tanh \left ( dx+c \right ) }{2\,d \left ( a+b \right ) ^{2} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{b\tanh \left ( dx+c \right ) }{2\,d \left ( a+b \right ) ^{2} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{a}{2\,d \left ( a+b \right ) ^{2}}\arctan \left ({b\tanh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{b}{2\,d \left ( a+b \right ) ^{2}}\arctan \left ({b\tanh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.14726, size = 4747, normalized size = 55.85 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^2*b + a*b^2)*d*x*cosh(d*x + c)^4 + 16*(a^2*b + a*b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 4*(a^2*b
+ a*b^2)*d*x*sinh(d*x + c)^4 + 4*a^2*b + 4*a*b^2 + 4*(a^2*b + a*b^2)*d*x + 4*(a^2*b - a*b^2 + 2*(a^2*b - a*b^2
)*d*x)*cosh(d*x + c)^2 + 4*(6*(a^2*b + a*b^2)*d*x*cosh(d*x + c)^2 + a^2*b - a*b^2 + 2*(a^2*b - a*b^2)*d*x)*sin
h(d*x + c)^2 + ((a^2 - b^2)*cosh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - b^2)*sinh(d
*x + c)^4 + 2*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - b^2)*cosh(d*x + c)^2 + a^2 - 2*a*b + b^2)*sinh
(d*x + c)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(d*x + c)^3 + (a^2 - 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*s
qrt(-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)*cos
h(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) + 8*(2*(a^2*b + a*b^2)*d*x*cosh(d*x + c)^3 + (a^
2*b - a*b^2 + 2*(a^2*b - a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*
cosh(d*x + c)^4 + 4*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*b + 3*a^3*b
^2 + 3*a^2*b^3 + a*b^4)*d*sinh(d*x + c)^4 + 2*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4)*d*cosh(d*x + c)^2 + 2*(3*(a^
4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^4*b + a^3*b^2 - a^2*b^3 - a*b^4)*d)*sinh(d*x + c)^
2 + (a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d + 4*((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 +
(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*(a^2*b + a*b^2)*d*x*cosh(d*x + c)
^4 + 8*(a^2*b + a*b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(a^2*b + a*b^2)*d*x*sinh(d*x + c)^4 + 2*a^2*b + 2
*a*b^2 + 2*(a^2*b + a*b^2)*d*x + 2*(a^2*b - a*b^2 + 2*(a^2*b - a*b^2)*d*x)*cosh(d*x + c)^2 + 2*(6*(a^2*b + a*b
^2)*d*x*cosh(d*x + c)^2 + a^2*b - a*b^2 + 2*(a^2*b - a*b^2)*d*x)*sinh(d*x + c)^2 - ((a^2 - b^2)*cosh(d*x + c)^
4 + 4*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - b^2)*sinh(d*x + c)^4 + 2*(a^2 - 2*a*b + b^2)*cosh(d*x
+ c)^2 + 2*(3*(a^2 - b^2)*cosh(d*x + c)^2 + a^2 - 2*a*b + b^2)*sinh(d*x + c)^2 + a^2 - b^2 + 4*((a^2 - b^2)*c
osh(d*x + c)^3 + (a^2 - 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*((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)) + 4*(2*(a^2*b +
a*b^2)*d*x*cosh(d*x + c)^3 + (a^2*b - a*b^2 + 2*(a^2*b - a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b +
3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 4*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*cosh(d*x + c)*s
inh(d*x + c)^3 + (a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*sinh(d*x + c)^4 + 2*(a^4*b + a^3*b^2 - a^2*b^3 - a*
b^4)*d*cosh(d*x + c)^2 + 2*(3*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^4*b + a^3*b^2 - a
^2*b^3 - a*b^4)*d)*sinh(d*x + c)^2 + (a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*d + 4*((a^4*b + 3*a^3*b^2 + 3*a^2
*b^3 + a*b^4)*d*cosh(d*x + c)^3 + (a^4*b + a^3*b^2 - a^2*b^3 - a*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [A]  time = 83.2681, size = 2144, normalized size = 25.22 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x/tanh(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((x - tanh(c + d*x)/d)/a**2, Eq(b, 0)), ((x - 1/
(d*tanh(c + d*x)))/b**2, Eq(a, 0)), (x*tanh(c)**2/(a + b*tanh(c)**2)**2, Eq(d, 0)), (4*I*a**(3/2)*b*d*x*sqrt(1
/b)/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/
b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(
1/b)*tanh(c + d*x)**2) - 2*I*a**(3/2)*b*sqrt(1/b)*tanh(c + d*x)/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**
2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**
2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) + 4*I*sqrt(a)*b**2*d*x*sqrt
(1/b)*tanh(c + d*x)**2/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(
5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*s
qrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) - 2*I*sqrt(a)*b**2*sqrt(1/b)*tanh(c + d*x)/(4*I*a**(7/2)*b*d*sqrt(1/
b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt
(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) - a**2
*log(-I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c
+ d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3
*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) + a**2*log(I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))/
(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) +
8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)
*tanh(c + d*x)**2) - a*b*log(-I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))*tanh(c + d*x)**2/(4*I*a**(7/2)*b*d*sqrt(1/b
) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(
1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) + a*b*l
og(-I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c +
d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d
*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) + a*b*log(I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))*tan
h(c + d*x)**2/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2
*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b*
*4*d*sqrt(1/b)*tanh(c + d*x)**2) - a*b*log(I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))/(4*I*a**(7/2)*b*d*sqrt(1/b) +
4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)
*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) + b**2*log(
-I*sqrt(a)*sqrt(1/b) + tanh(c + d*x))*tanh(c + d*x)**2/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(
1/b)*tanh(c + d*x)**2 + 8*I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a
**(3/2)*b**3*d*sqrt(1/b) + 4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2) - b**2*log(I*sqrt(a)*sqrt(1/b) + tan
h(c + d*x))*tanh(c + d*x)**2/(4*I*a**(7/2)*b*d*sqrt(1/b) + 4*I*a**(5/2)*b**2*d*sqrt(1/b)*tanh(c + d*x)**2 + 8*
I*a**(5/2)*b**2*d*sqrt(1/b) + 8*I*a**(3/2)*b**3*d*sqrt(1/b)*tanh(c + d*x)**2 + 4*I*a**(3/2)*b**3*d*sqrt(1/b) +
4*I*sqrt(a)*b**4*d*sqrt(1/b)*tanh(c + d*x)**2), True))

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Giac [B]  time = 1.22123, size = 251, normalized size = 2.95 \begin{align*} -\frac{{\left (a - b\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 )}{2 \,{\left (a^{2} d + 2 \, a b d + b^{2} d\right )} \sqrt{a b}} + \frac{d x + c}{a^{2} d + 2 \, a b d + b^{2} d} + \frac{a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + a + b}{{\left (a^{2} d + 2 \, a b d + b^{2} d\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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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