### 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 - ((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))

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Rubi [A]  time = 0.157022, antiderivative size = 137, normalized size of antiderivative = 1., 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 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 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 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 )^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 = 1.06931, size = 137, normalized size = 1. $\frac{\frac{\left (-3 a^2+6 a b+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}-\frac{(5 a-b) (a+b) \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}-\frac{4 b (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}+8 (c+d x)}{8 d (a+b)^3}$

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)

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Maple [B]  time = 0.026, size = 340, normalized size = 2.5 \begin{align*}{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a+b \right ) ^{3}}}-{\frac{3\,ab \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{b}^{2}}{4\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}a}}-{\frac{5\,{a}^{2}\tanh \left ( dx+c \right ) }{8\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{3\,ab\tanh \left ( dx+c \right ) }{4\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{\tanh \left ( dx+c \right ){b}^{2}}{8\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{3\,a}{8\,d \left ( a+b \right ) ^{3}}\arctan \left ({b\tanh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,b}{4\,d \left ( a+b \right ) ^{3}}\arctan \left ({b\tanh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}}{8\,d \left ( a+b \right ) ^{3}a}\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 ) ^{3}}} \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)^3,x)

[Out]

1/2/d/(a+b)^3*ln(tanh(d*x+c)+1)-3/8/d/(a+b)^3*a/(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+b)^3*a^2/(a+b*t
anh(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+b)^3*a/(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^2-1/2/d/(a
+b)^3*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.33037, size = 17383, normalized size = 126.88 \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)^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)
)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.26422, size = 539, normalized size = 3.93 \begin{align*} -\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 )}{8 \,{\left (a^{4} d + 3 \, a^{3} b d + 3 \, a^{2} b^{2} d + a b^{3} d\right )} \sqrt{a b}} + \frac{d x + c}{a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d} + \frac{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}}{4 \,{\left (a^{4} d + 3 \, a^{3} b d + 3 \, a^{2} b^{2} d + a b^{3} 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 )}^{2}} \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)^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*d + 3*a
^3*b*d + 3*a^2*b^2*d + a*b^3*d)*sqrt(a*b)) + (d*x + c)/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) + 1/4*(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*d + 3*a^3*b*d + 3*a^2*b^2*d + a*b^3*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)^2)