### 3.33 $$\int \frac{\sinh ^4(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=192 $\frac{3 x \left (a^2-6 a b+b^2\right )}{8 (a+b)^4}+\frac{3 b (3 a-b) \tanh (c+d x)}{8 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 d (a+b)^4}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{(5 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}$

[Out]

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

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Rubi [A]  time = 0.252353, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.261, Rules used = {3663, 470, 527, 522, 206, 205} $\frac{3 x \left (a^2-6 a b+b^2\right )}{8 (a+b)^4}+\frac{3 b (3 a-b) \tanh (c+d x)}{8 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 d (a+b)^4}+\frac{\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{(5 a-b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3663

Int[sin[(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^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

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{\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^3 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{a+(4 a-b) x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d}\\ &=-\frac{(5 a-b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 a (a-b)+3 (5 a-b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d}\\ &=-\frac{(5 a-b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 (3 a-b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{6 a^2 (a-3 b)-6 a (3 a-b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{16 a (a+b)^3 d}\\ &=-\frac{(5 a-b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 (3 a-b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{(3 a (a-b) b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^4 d}+\frac{\left (3 \left (a^2-6 a b+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^4 d}\\ &=\frac{3 \left (a^2-6 a b+b^2\right ) x}{8 (a+b)^4}+\frac{3 \sqrt{a} (a-b) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 (a+b)^4 d}-\frac{(5 a-b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 (3 a-b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.933154, size = 132, normalized size = 0.69 $\frac{12 \left (a^2-6 a b+b^2\right ) (c+d x)+(a+b)^2 \sinh (4 (c+d x))-8 (a-b) (a+b) \sinh (2 (c+d x))+48 \sqrt{a} \sqrt{b} (a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )+\frac{16 a b (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{32 d (a+b)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(12*(a^2 - 6*a*b + b^2)*(c + d*x) + 48*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*(a
- b)*(a + b)*Sinh[2*(c + d*x)] + (16*a*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + (a +
b)^2*Sinh[4*(c + d*x)])/(32*(a + b)^4*d)

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Maple [B]  time = 0.119, size = 1246, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x)

[Out]

1/d*a^2*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x
+1/2*c)^3+1/d*a*b^2/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*ta
nh(1/2*d*x+1/2*c)^3+1/d*a^2*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)
^2*b+a)*tanh(1/2*d*x+1/2*c)+1/d*a*b^2/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*
x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)+3/2/d*a^2*b/(a+b)^4/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2
*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-3/2/d*a^2*b/(a+b)^4/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arcta
n(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-3/2/d*a*b^2/(a+b)^4/((2*(b*(a+b))^(1/2)-a-2*b)*a)
^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+3/2/d*a*b^2/(a+b)^4/((2*(b*(a+b))^(1
/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-3/2/d*a^3*b/(a+b)^4/(b*(
a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/
2))+3/2/d*a*b^3/(a+b)^4/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*
(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+3/2/d*a*b^3/(a+b)^4/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arcta
n(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-3/2/d*a^3*b/(a+b)^4/(b*(a+b))^(1/2)/((2*(b*(a+b))
^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-9/4/d/(a+b)^4*ln(tan
h(1/2*d*x+1/2*c)+1)*a*b+1/2/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^3+1/2/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)+1)^3-1/4/d/
(a+b)^2/(tanh(1/2*d*x+1/2*c)+1)^4+1/4/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^4-7/8/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1
)^2*b-3/8/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)*a+5/8/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)*b+3/8/d/(a+b)^4*ln(tanh(1/
2*d*x+1/2*c)+1)*a^2+3/8/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c)+1)*b^2-3/8/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c)-1)*a^2-3/
8/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c)-1)*b^2-1/8/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^2*a+7/8/d/(a+b)^3/(tanh(1/2*d*
x+1/2*c)-1)^2*b-3/8/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)*a+5/8/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)*b+1/8/d/(a+b)^3/
(tanh(1/2*d*x+1/2*c)+1)^2*a+9/4/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c)-1)*a*b

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.30391, size = 17565, normalized size = 91.48 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/64*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^12 + 12*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sin
h(d*x + c)^11 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^12 - 6*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)
^10 - 6*(a^3 + a^2*b - a*b^2 - b^3 - 11*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20
*(11*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 - 3*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x +
c)^9 - (15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^8 + (49
5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - 15*a^3 + 19*a^2*b + 19*a*b^2 - 15*b^3 + 24*(a^3 - 5*a^2*b
- 5*a*b^2 + b^3)*d*x - 270*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*cosh(d*x + c)^5 - 90*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^3 - (15*a^3 - 19*a^2*b - 19*a
*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 - 16*(4*a^2*b - 4*a*b^2
- 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^6 + 4*(231*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x +
c)^6 - 315*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^4 - 16*a^2*b + 16*a*b^2 + 12*(a^3 - 7*a^2*b + 7*a*b^2 - b
^3)*d*x - 7*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*
sinh(d*x + c)^6 + 8*(99*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 - 189*(a^3 + a^2*b - a*b^2 - b^3)*cosh
(d*x + c)^5 - 7*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)
^3 - 12*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + (15*a^3 -
83*a^2*b - 83*a*b^2 + 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + (495*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*cosh(d*x + c)^8 - 1260*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^6 - 70*(15*a^3 - 19*a^2*b -
19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + 15*a^3 - 83*a^2*b - 83*a*b^2 + 1
5*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x - 240*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*
x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(55*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 - 180*(a^3 + a^2*b
- a*b^2 - b^3)*cosh(d*x + c)^7 - 14*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^
3)*d*x)*cosh(d*x + c)^5 - 80*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^3 + (15
*a^3 - 83*a^2*b - 83*a*b^2 + 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 -
a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 6*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^2 + 2*(33*(a^3 + 3*a^2*b + 3*a*b^
2 + b^3)*cosh(d*x + c)^10 - 135*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^8 - 14*(15*a^3 - 19*a^2*b - 19*a*b^2
+ 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^6 - 120*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*
b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^4 + 3*a^3 + 3*a^2*b - 3*a*b^2 - 3*b^3 + 3*(15*a^3 - 83*a^2*b - 83*a*b^2
+ 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 48*((a^2 - b^2)*cosh(d*x
+ c)^8 + 8*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 - b^2)*sinh(d*x + c)^8 + 2*(a^2 - 2*a*b + b^2)*co
sh(d*x + c)^6 + 2*(14*(a^2 - b^2)*cosh(d*x + c)^2 + a^2 - 2*a*b + b^2)*sinh(d*x + c)^6 + 4*(14*(a^2 - b^2)*cos
h(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^2 - b^2)*cosh(d*x + c)^4 + (70*(a^2 -
b^2)*cosh(d*x + c)^4 + 30*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^4 + 4*(14*(a^2 - b^2
)*cosh(d*x + c)^5 + 10*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(1
4*(a^2 - b^2)*cosh(d*x + c)^6 + 15*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^4 + 3*(a^2 - b^2)*cosh(d*x + c)^2)*sinh(d
*x + c)^2 + 4*(2*(a^2 - b^2)*cosh(d*x + c)^7 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^5 + (a^2 - b^2)*cosh(d*x +
c)^3)*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)) + 4*(3*(a^3 + 3*a^2*b + 3*
a*b^2 + b^3)*cosh(d*x + c)^11 - 15*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^9 - 2*(15*a^3 - 19*a^2*b - 19*a*b
^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 - 24*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2
*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^5 + (15*a^3 - 83*a^2*b - 83*a*b^2 + 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^
2 + b^3)*d*x)*cosh(d*x + c)^3 + 3*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + 5*a^4*b +
10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^8 + 8*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*
b^4 + b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*sinh(
d*x + c)^8 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c)^6 + 2*(14*(a^5 + 5*a^4*
b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^2 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*
b^4 - b^5)*d)*sinh(d*x + c)^6 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^4 +
4*(14*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^3 + 3*(a^5 + 3*a^4*b + 2*a^3*b
^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + (70*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^
3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^4 + 30*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x +
c)^2 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d)*sinh(d*x + c)^4 + 4*(14*(a^5 + 5*a^4*b +
10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^5 + 10*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b
^4 - b^5)*d*cosh(d*x + c)^3 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c))*sinh(
d*x + c)^3 + 2*(14*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^6 + 15*(a^5 + 3*a
^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c)^4 + 3*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 +
5*a*b^4 + b^5)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 +
b^5)*d*cosh(d*x + c)^7 + 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c)^5 + (a^5 +
5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^3)*sinh(d*x + c)), 1/64*((a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^12 + 12*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^11 + (a^3 + 3
*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^12 - 6*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^10 - 6*(a^3 + a^2*b - a
*b^2 - b^3 - 11*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(a^3 + 3*a^2*b + 3*
a*b^2 + b^3)*cosh(d*x + c)^3 - 3*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^9 - (15*a^3 - 19*a^2
*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^8 + (495*(a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^4 - 15*a^3 + 19*a^2*b + 19*a*b^2 - 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x - 2
70*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d
*x + c)^5 - 90*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^3 - (15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3
- 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 - 16*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*
a*b^2 - b^3)*d*x)*cosh(d*x + c)^6 + 4*(231*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 - 315*(a^3 + a^2*b
- a*b^2 - b^3)*cosh(d*x + c)^4 - 16*a^2*b + 16*a*b^2 + 12*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x - 7*(15*a^3 - 19
*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(99*
(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 - 189*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^5 - 7*(15*a^3
- 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 12*(4*a^2*b - 4*a*b
^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + (15*a^3 - 83*a^2*b - 83*a*b^2 + 1
5*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + (495*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*
x + c)^8 - 1260*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^6 - 70*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(
a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + 15*a^3 - 83*a^2*b - 83*a*b^2 + 15*b^3 + 24*(a^3 - 5*a^2*
b - 5*a*b^2 + b^3)*d*x - 240*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^2)*sinh
(d*x + c)^4 + 4*(55*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 - 180*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x
+ c)^7 - 14*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5
- 80*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^3 + (15*a^3 - 83*a^2*b - 83*a*b
^2 + 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 3*a^2*b - 3*a*b^2
- b^3 + 6*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^2 + 2*(33*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^1
0 - 135*(a^3 + a^2*b - a*b^2 - b^3)*cosh(d*x + c)^8 - 14*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*
a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^6 - 120*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*
cosh(d*x + c)^4 + 3*a^3 + 3*a^2*b - 3*a*b^2 - 3*b^3 + 3*(15*a^3 - 83*a^2*b - 83*a*b^2 + 15*b^3 + 24*(a^3 - 5*a
^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 96*((a^2 - b^2)*cosh(d*x + c)^8 + 8*(a^2 - b^2)*
cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 - b^2)*sinh(d*x + c)^8 + 2*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^6 + 2*(14*(a
^2 - b^2)*cosh(d*x + c)^2 + a^2 - 2*a*b + b^2)*sinh(d*x + c)^6 + 4*(14*(a^2 - b^2)*cosh(d*x + c)^3 + 3*(a^2 -
2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^2 - b^2)*cosh(d*x + c)^4 + (70*(a^2 - b^2)*cosh(d*x + c)^4 +
30*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^4 + 4*(14*(a^2 - b^2)*cosh(d*x + c)^5 + 10*(
a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(14*(a^2 - b^2)*cosh(d*x +
c)^6 + 15*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^4 + 3*(a^2 - b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(a^2 -
b^2)*cosh(d*x + c)^7 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^5 + (a^2 - b^2)*cosh(d*x + c)^3)*sinh(d*x + c))*sqr
t(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*(3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^11 - 15*(a^3 + a^2*b - a*b^2 - b
^3)*cosh(d*x + c)^9 - 2*(15*a^3 - 19*a^2*b - 19*a*b^2 + 15*b^3 - 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(
d*x + c)^7 - 24*(4*a^2*b - 4*a*b^2 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*d*x)*cosh(d*x + c)^5 + (15*a^3 - 83*a^2
*b - 83*a*b^2 + 15*b^3 + 24*(a^3 - 5*a^2*b - 5*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 3*(a^3 + a^2*b - a*b^2 - b^
3)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^8
+ 8*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5 + 5*a^4*b
+ 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*sinh(d*x + c)^8 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a
*b^4 - b^5)*d*cosh(d*x + c)^6 + 2*(14*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c
)^2 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d)*sinh(d*x + c)^6 + (a^5 + 5*a^4*b + 10*a^3*b^2
+ 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^4 + 4*(14*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 +
b^5)*d*cosh(d*x + c)^3 + 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c))*sinh(d*x +
c)^5 + (70*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^4 + 30*(a^5 + 3*a^4*b +
2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c)^2 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4
+ b^5)*d)*sinh(d*x + c)^4 + 4*(14*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^5
+ 10*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c)^3 + (a^5 + 5*a^4*b + 10*a^3*b^2 +
10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(14*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3
+ 5*a*b^4 + b^5)*d*cosh(d*x + c)^6 + 15*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x +
c)^4 + 3*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(
a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(d*x + c)^7 + 3*(a^5 + 3*a^4*b + 2*a^3*b^2 - 2*
a^2*b^3 - 3*a*b^4 - b^5)*d*cosh(d*x + c)^5 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*cosh(
d*x + c)^3)*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(sinh(d*x+c)**4/(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 3.3925, size = 709, normalized size = 3.69 \begin{align*} \frac{\frac{24 \,{\left (a^{2} - 6 \, a b + b^{2}\right )} d x}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{96 \,{\left (a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )}\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 ) e^{\left (-2 \, c\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt{a b}} - \frac{{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 108 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x\right )}}{a^{4} e^{\left (4 \, c\right )} + 4 \, a^{3} b e^{\left (4 \, c\right )} + 6 \, a^{2} b^{2} e^{\left (4 \, c\right )} + 4 \, a b^{3} e^{\left (4 \, c\right )} + b^{4} e^{\left (4 \, c\right )}} + \frac{a^{2} e^{\left (4 \, d x + 28 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 28 \, c\right )} + b^{2} e^{\left (4 \, d x + 28 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 26 \, c\right )} + 8 \, b^{2} e^{\left (2 \, d x + 26 \, c\right )}}{a^{4} e^{\left (24 \, c\right )} + 4 \, a^{3} b e^{\left (24 \, c\right )} + 6 \, a^{2} b^{2} e^{\left (24 \, c\right )} + 4 \, a b^{3} e^{\left (24 \, c\right )} + b^{4} e^{\left (24 \, c\right )}} - \frac{64 \,{\left (a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} b + a b^{2}\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\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 )}}}{64 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/64*(24*(a^2 - 6*a*b + b^2)*d*x/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 96*(a^2*b*e^(2*c) - a*b^2*e^(2*
c))*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))*e^(-2*c)/((a^4 + 4*a^3*b + 6*a^2*b^2
+ 4*a*b^3 + b^4)*sqrt(a*b)) - (18*a^2*e^(4*d*x + 4*c) - 108*a*b*e^(4*d*x + 4*c) + 18*b^2*e^(4*d*x + 4*c) - 8*
a^2*e^(2*d*x + 2*c) + 8*b^2*e^(2*d*x + 2*c) + a^2 + 2*a*b + b^2)*e^(-4*d*x)/(a^4*e^(4*c) + 4*a^3*b*e^(4*c) + 6
*a^2*b^2*e^(4*c) + 4*a*b^3*e^(4*c) + b^4*e^(4*c)) + (a^2*e^(4*d*x + 28*c) + 2*a*b*e^(4*d*x + 28*c) + b^2*e^(4*
d*x + 28*c) - 8*a^2*e^(2*d*x + 26*c) + 8*b^2*e^(2*d*x + 26*c))/(a^4*e^(24*c) + 4*a^3*b*e^(24*c) + 6*a^2*b^2*e^
(24*c) + 4*a*b^3*e^(24*c) + b^4*e^(24*c)) - 64*(a^2*b*e^(2*d*x + 2*c) - a*b^2*e^(2*d*x + 2*c) + a^2*b + a*b^2)
/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(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)))/d