### 3.192 $$\int \frac{\tanh ^4(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx$$

Optimal. Leaf size=137 $-\frac{\left (a^2+6 a b-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} b^{3/2} d (a+b)^3}-\frac{(a+5 b) \tanh (c+d x)}{8 b d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{a \tanh (c+d x)}{4 b d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{x}{(a+b)^3}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.195489, 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, 470, 527, 522, 206, 205} $-\frac{\left (a^2+6 a b-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} b^{3/2} d (a+b)^3}-\frac{(a+5 b) \tanh (c+d x)}{8 b d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{a \tanh (c+d x)}{4 b d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{x}{(a+b)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

x/(a + b)^3 - ((a^2 + 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*b^(3/2)*(a + b)^3*d)
+ (a*Tanh[c + d*x])/(4*b*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((a + 5*b)*Tanh[c + d*x])/(8*b*(a + b)^2*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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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 ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a \tanh (c+d x)}{4 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{a+(-a-4 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 b (a+b) d}\\ &=\frac{a \tanh (c+d x)}{4 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac{(a+5 b) \tanh (c+d x)}{8 b (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-a (a-3 b)+a (a+5 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a b (a+b)^2 d}\\ &=\frac{a \tanh (c+d x)}{4 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac{(a+5 b) \tanh (c+d x)}{8 b (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 (a^2+6 a b-3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 b (a+b)^3 d}\\ &=\frac{x}{(a+b)^3}-\frac{\left (a^2+6 a b-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} b^{3/2} (a+b)^3 d}+\frac{a \tanh (c+d x)}{4 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac{(a+5 b) \tanh (c+d x)}{8 b (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.06016, size = 135, normalized size = 0.99 $\frac{-\frac{\left (a^2+6 a b-3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{(a-5 b) (a+b) \sinh (2 (c+d x))}{b ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{4 a (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}+8 (c+d x)}{8 d (a+b)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.029, 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{{a}^{2} \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{3\,ab \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{5\, \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{b}^{2}}{8\,d \left ( a+b \right ) ^{3} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{{a}^{3}\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}b}}-{\frac{{a}^{2}\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{3\,ab\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{{a}^{2}}{8\,d \left ( a+b \right ) ^{3}b}\arctan \left ({b\tanh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,a}{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{3\,b}{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{\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)^4/(a+b*tanh(d*x+c)^2)^3,x)

[Out]

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

________________________________________________________________________________________

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)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.19345, 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)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.25725, size = 533, normalized size = 3.89 \begin{align*} -\frac{{\left (a^{2} + 6 \, a b - 3 \, 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^{3} b d + 3 \, a^{2} b^{2} d + 3 \, a b^{3} d + b^{4} 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{a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 5 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 17 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 13 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 15 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 11 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}}{4 \,{\left (a^{3} b d + 3 \, a^{2} b^{2} d + 3 \, a b^{3} d + b^{4} 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)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

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