3.40 \(\int \frac{\text{csch}^4(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{(a+2 b) \coth (c+d x)}{a^3 d}+\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{7/2} d}-\frac{\coth ^3(c+d x)}{3 a^2 d} \]

[Out]

(Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*d) + ((a + 2*b)*Coth[c + d*x])/(a^3*d
) - Coth[c + d*x]^3/(3*a^2*d) + (b*(a + b)*Tanh[c + d*x])/(2*a^3*d*(a + b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.153873, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3663, 456, 1261, 205} \[ \frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{(a+2 b) \coth (c+d x)}{a^3 d}+\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{7/2} d}-\frac{\coth ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*d) + ((a + 2*b)*Coth[c + d*x])/(a^3*d
) - Coth[c + d*x]^3/(3*a^2*d) + (b*(a + b)*Tanh[c + d*x])/(2*a^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 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -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]

Rubi steps

\begin{align*} \int \frac{\text{csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{x^4 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{b \operatorname{Subst}\left (\int \frac{-\frac{2}{a b}+\frac{2 (a+b) x^2}{a^2 b}-\frac{(a+b) x^4}{a^3}}{x^4 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{b \operatorname{Subst}\left (\int \left (-\frac{2}{a^2 b x^4}+\frac{2 (a+2 b)}{a^3 b x^2}+\frac{-3 a-5 b}{a^3 \left (a+b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac{(a+2 b) \coth (c+d x)}{a^3 d}-\frac{\coth ^3(c+d x)}{3 a^2 d}+\frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{(b (3 a+5 b)) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^3 d}\\ &=\frac{\sqrt{b} (3 a+5 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{7/2} d}+\frac{(a+2 b) \coth (c+d x)}{a^3 d}-\frac{\coth ^3(c+d x)}{3 a^2 d}+\frac{b (a+b) \tanh (c+d x)}{2 a^3 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.774897, size = 114, normalized size = 1.01 \[ \frac{3 \sqrt{b} (3 a+5 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )+\frac{3 \sqrt{a} b (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}+2 \sqrt{a} \coth (c+d x) \left (-a \text{csch}^2(c+d x)+2 a+6 b\right )}{6 a^{7/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + 2*Sqrt[a]*Coth[c + d*x]*(2*a + 6*b - a*Csch[c
+ d*x]^2) + (3*Sqrt[a]*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(6*a^(7/2)*d)

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Maple [B]  time = 0.119, size = 1012, normalized size = 9. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24/d/a^2*tanh(1/2*d*x+1/2*c)^3+3/8/d/a^2*tanh(1/2*d*x+1/2*c)+1/d/a^3*tanh(1/2*d*x+1/2*c)*b+1/d*b/a^2/(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^3*b^2/(t
anh(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*b/a^2/
(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^3*b^
2/(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/(b
*(a+b))^(1/2)/a/((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))*b+3/2/d*b/a^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))-4/d*b^2/a^2/(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/(b*(a+b))^(1/2)/a/((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))*b-3/2/d*b/a^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))-4/d*b^2/a^2/(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))+5/2/d/a^3*b^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))-5/2/d/a^3*b^3/(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))-5
/2/d/a^3*b^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))-5/2/d/a^3*b^3/(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))-1/24/d/a^2/tanh(1/2*d*x+1/2*c)^3+3/8/d/a^2/tanh(1/2*d*x+1/2*c)+1/d/a^3/tanh(1/2*d*x
+1/2*c)*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(csch(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 = 2.78602, size = 12585, normalized size = 111.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(12*(3*a*b + 5*b^2)*cosh(d*x + c)^8 + 96*(3*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 12*(3*a*b + 5*b
^2)*sinh(d*x + c)^8 - 24*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^6 + 24*(14*(3*a*b + 5*b^2)*cosh(d*x + c)^2 - 2*a
^2 - a*b - 10*b^2)*sinh(d*x + c)^6 + 48*(14*(3*a*b + 5*b^2)*cosh(d*x + c)^3 - 3*(2*a^2 + a*b + 10*b^2)*cosh(d*
x + c))*sinh(d*x + c)^5 - 8*(10*a^2 - 2*a*b - 45*b^2)*cosh(d*x + c)^4 + 8*(105*(3*a*b + 5*b^2)*cosh(d*x + c)^4
- 45*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^2 - 10*a^2 + 2*a*b + 45*b^2)*sinh(d*x + c)^4 + 32*(21*(3*a*b + 5*b^
2)*cosh(d*x + c)^5 - 15*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^3 - (10*a^2 - 2*a*b - 45*b^2)*cosh(d*x + c))*sinh
(d*x + c)^3 - 8*(2*a^2 + 13*a*b + 30*b^2)*cosh(d*x + c)^2 + 8*(42*(3*a*b + 5*b^2)*cosh(d*x + c)^6 - 45*(2*a^2
+ a*b + 10*b^2)*cosh(d*x + c)^4 - 6*(10*a^2 - 2*a*b - 45*b^2)*cosh(d*x + c)^2 - 2*a^2 - 13*a*b - 30*b^2)*sinh(
d*x + c)^2 + 3*((3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^10 + 10*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x +
c)^9 + (3*a^2 + 8*a*b + 5*b^2)*sinh(d*x + c)^10 - (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^8 + (45*(3*a^2 + 8*
a*b + 5*b^2)*cosh(d*x + c)^2 - 3*a^2 - 20*a*b - 25*b^2)*sinh(d*x + c)^8 + 8*(15*(3*a^2 + 8*a*b + 5*b^2)*cosh(d
*x + c)^3 - (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x +
c)^6 + 2*(105*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^4 - 14*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^2 - 3*a^2 +
10*a*b + 25*b^2)*sinh(d*x + c)^6 + 4*(63*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^5 - 14*(3*a^2 + 20*a*b + 25*b^
2)*cosh(d*x + c)^3 - 3*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 - 10*a*b - 25*b^2)*
cosh(d*x + c)^4 + 2*(105*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^6 - 35*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^
4 - 15*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b - 25*b^2)*sinh(d*x + c)^4 + 8*(15*(3*a^2 + 8
*a*b + 5*b^2)*cosh(d*x + c)^7 - 7*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^5 - 5*(3*a^2 - 10*a*b - 25*b^2)*cosh
(d*x + c)^3 + (3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x +
c)^2 + (45*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^8 - 28*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^6 - 30*(3*a^2
- 10*a*b - 25*b^2)*cosh(d*x + c)^4 + 12*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^2 + 3*a^2 + 20*a*b + 25*b^2)*s
inh(d*x + c)^2 - 3*a^2 - 8*a*b - 5*b^2 + 2*(5*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^9 - 4*(3*a^2 + 20*a*b + 25
*b^2)*cosh(d*x + c)^7 - 6*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^5 + 4*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c
)^3 + (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*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^2 + a*b)*cosh(d*x
+ c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - a*b)*sqrt(-b/a))/((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)) + 16*a^2 + 76*a*b + 60*b^2 + 16*(6*(3*a*b + 5*b^2)*cosh(d*x + c)^7 - 9*(2*a^2 +
a*b + 10*b^2)*cosh(d*x + c)^5 - 2*(10*a^2 - 2*a*b - 45*b^2)*cosh(d*x + c)^3 - (2*a^2 + 13*a*b + 30*b^2)*cosh(
d*x + c))*sinh(d*x + c))/((a^4 + a^3*b)*d*cosh(d*x + c)^10 + 10*(a^4 + a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^9
+ (a^4 + a^3*b)*d*sinh(d*x + c)^10 - (a^4 + 5*a^3*b)*d*cosh(d*x + c)^8 + (45*(a^4 + a^3*b)*d*cosh(d*x + c)^2 -
(a^4 + 5*a^3*b)*d)*sinh(d*x + c)^8 - 2*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^6 + 8*(15*(a^4 + a^3*b)*d*cosh(d*x + c
)^3 - (a^4 + 5*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^4 + a^3*b)*d*cosh(d*x + c)^4 - 14*(a^4 + 5*
a^3*b)*d*cosh(d*x + c)^2 - (a^4 - 5*a^3*b)*d)*sinh(d*x + c)^6 + 2*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^4 + 4*(63*(a
^4 + a^3*b)*d*cosh(d*x + c)^5 - 14*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^3 - 3*(a^4 - 5*a^3*b)*d*cosh(d*x + c))*sinh
(d*x + c)^5 + 2*(105*(a^4 + a^3*b)*d*cosh(d*x + c)^6 - 35*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^4 - 15*(a^4 - 5*a^3*
b)*d*cosh(d*x + c)^2 + (a^4 - 5*a^3*b)*d)*sinh(d*x + c)^4 + (a^4 + 5*a^3*b)*d*cosh(d*x + c)^2 + 8*(15*(a^4 + a
^3*b)*d*cosh(d*x + c)^7 - 7*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^5 - 5*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^3 + (a^4 - 5
*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^4 + a^3*b)*d*cosh(d*x + c)^8 - 28*(a^4 + 5*a^3*b)*d*cosh(d*x
+ c)^6 - 30*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^4 + 12*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^2 + (a^4 + 5*a^3*b)*d)*sin
h(d*x + c)^2 - (a^4 + a^3*b)*d + 2*(5*(a^4 + a^3*b)*d*cosh(d*x + c)^9 - 4*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^7 -
6*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^5 + 4*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^3 + (a^4 + 5*a^3*b)*d*cosh(d*x + c))*s
inh(d*x + c)), 1/6*(6*(3*a*b + 5*b^2)*cosh(d*x + c)^8 + 48*(3*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 6*(
3*a*b + 5*b^2)*sinh(d*x + c)^8 - 12*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^6 + 12*(14*(3*a*b + 5*b^2)*cosh(d*x +
c)^2 - 2*a^2 - a*b - 10*b^2)*sinh(d*x + c)^6 + 24*(14*(3*a*b + 5*b^2)*cosh(d*x + c)^3 - 3*(2*a^2 + a*b + 10*b
^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(10*a^2 - 2*a*b - 45*b^2)*cosh(d*x + c)^4 + 4*(105*(3*a*b + 5*b^2)*cosh
(d*x + c)^4 - 45*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^2 - 10*a^2 + 2*a*b + 45*b^2)*sinh(d*x + c)^4 + 16*(21*(3
*a*b + 5*b^2)*cosh(d*x + c)^5 - 15*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^3 - (10*a^2 - 2*a*b - 45*b^2)*cosh(d*x
+ c))*sinh(d*x + c)^3 - 4*(2*a^2 + 13*a*b + 30*b^2)*cosh(d*x + c)^2 + 4*(42*(3*a*b + 5*b^2)*cosh(d*x + c)^6 -
45*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^4 - 6*(10*a^2 - 2*a*b - 45*b^2)*cosh(d*x + c)^2 - 2*a^2 - 13*a*b - 30
*b^2)*sinh(d*x + c)^2 + 3*((3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^10 + 10*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)
*sinh(d*x + c)^9 + (3*a^2 + 8*a*b + 5*b^2)*sinh(d*x + c)^10 - (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^8 + (45*
(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^2 - 3*a^2 - 20*a*b - 25*b^2)*sinh(d*x + c)^8 + 8*(15*(3*a^2 + 8*a*b + 5*
b^2)*cosh(d*x + c)^3 - (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(3*a^2 - 10*a*b - 25*b^2)*
cosh(d*x + c)^6 + 2*(105*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^4 - 14*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^
2 - 3*a^2 + 10*a*b + 25*b^2)*sinh(d*x + c)^6 + 4*(63*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^5 - 14*(3*a^2 + 20*
a*b + 25*b^2)*cosh(d*x + c)^3 - 3*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 - 10*a*b
- 25*b^2)*cosh(d*x + c)^4 + 2*(105*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^6 - 35*(3*a^2 + 20*a*b + 25*b^2)*cos
h(d*x + c)^4 - 15*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b - 25*b^2)*sinh(d*x + c)^4 + 8*(15
*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^7 - 7*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^5 - 5*(3*a^2 - 10*a*b - 2
5*b^2)*cosh(d*x + c)^3 + (3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (3*a^2 + 20*a*b + 25*b^2)*
cosh(d*x + c)^2 + (45*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^8 - 28*(3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c)^6 -
30*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^4 + 12*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^2 + 3*a^2 + 20*a*b
+ 25*b^2)*sinh(d*x + c)^2 - 3*a^2 - 8*a*b - 5*b^2 + 2*(5*(3*a^2 + 8*a*b + 5*b^2)*cosh(d*x + c)^9 - 4*(3*a^2 +
20*a*b + 25*b^2)*cosh(d*x + c)^7 - 6*(3*a^2 - 10*a*b - 25*b^2)*cosh(d*x + c)^5 + 4*(3*a^2 - 10*a*b - 25*b^2)*c
osh(d*x + c)^3 + (3*a^2 + 20*a*b + 25*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*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(b/a)/b) + 8*a^2 + 38*
a*b + 30*b^2 + 8*(6*(3*a*b + 5*b^2)*cosh(d*x + c)^7 - 9*(2*a^2 + a*b + 10*b^2)*cosh(d*x + c)^5 - 2*(10*a^2 - 2
*a*b - 45*b^2)*cosh(d*x + c)^3 - (2*a^2 + 13*a*b + 30*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + a^3*b)*d*cosh
(d*x + c)^10 + 10*(a^4 + a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^4 + a^3*b)*d*sinh(d*x + c)^10 - (a^4 + 5*
a^3*b)*d*cosh(d*x + c)^8 + (45*(a^4 + a^3*b)*d*cosh(d*x + c)^2 - (a^4 + 5*a^3*b)*d)*sinh(d*x + c)^8 - 2*(a^4 -
5*a^3*b)*d*cosh(d*x + c)^6 + 8*(15*(a^4 + a^3*b)*d*cosh(d*x + c)^3 - (a^4 + 5*a^3*b)*d*cosh(d*x + c))*sinh(d*
x + c)^7 + 2*(105*(a^4 + a^3*b)*d*cosh(d*x + c)^4 - 14*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^2 - (a^4 - 5*a^3*b)*d)*
sinh(d*x + c)^6 + 2*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^4 + 4*(63*(a^4 + a^3*b)*d*cosh(d*x + c)^5 - 14*(a^4 + 5*a^
3*b)*d*cosh(d*x + c)^3 - 3*(a^4 - 5*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^4 + a^3*b)*d*cosh(d*x
+ c)^6 - 35*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^4 - 15*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^2 + (a^4 - 5*a^3*b)*d)*sinh
(d*x + c)^4 + (a^4 + 5*a^3*b)*d*cosh(d*x + c)^2 + 8*(15*(a^4 + a^3*b)*d*cosh(d*x + c)^7 - 7*(a^4 + 5*a^3*b)*d*
cosh(d*x + c)^5 - 5*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^3 + (a^4 - 5*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45
*(a^4 + a^3*b)*d*cosh(d*x + c)^8 - 28*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^6 - 30*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^4
+ 12*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^2 + (a^4 + 5*a^3*b)*d)*sinh(d*x + c)^2 - (a^4 + a^3*b)*d + 2*(5*(a^4 + a
^3*b)*d*cosh(d*x + c)^9 - 4*(a^4 + 5*a^3*b)*d*cosh(d*x + c)^7 - 6*(a^4 - 5*a^3*b)*d*cosh(d*x + c)^5 + 4*(a^4 -
5*a^3*b)*d*cosh(d*x + c)^3 + (a^4 + 5*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.61048, size = 298, normalized size = 2.64 \begin{align*} \frac{\frac{3 \,{\left (3 \, a b e^{\left (2 \, c\right )} + 5 \, 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 )}}{\sqrt{a b} a^{3}} - \frac{6 \,{\left (a b e^{\left (2 \, d x + 2 \, c\right )} - b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a b + b^{2}\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 )} a^{3}} + \frac{8 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + 3 \, b\right )}}{a^{3}{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(3*a*b*e^(2*c) + 5*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)/(sqrt(a*b)*a^3) - 6*(a*b*e^(2*d*x + 2*c) - b^2*e^(2*d*x + 2*c) + a*b + b^2)/((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)*a^3) + 8*(3*b*e^(4*d*x + 4*c) - 3*a*e^(2*d*
x + 2*c) - 6*b*e^(2*d*x + 2*c) + a + 3*b)/(a^3*(e^(2*d*x + 2*c) - 1)^3))/d