3.46 \(\int \frac{\text{csch}^2(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d}+\frac{5 \coth (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{15 \coth (c+d x)}{8 a^3 d}+\frac{\coth (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2} \]

[Out]

(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*d) - (15*Coth[c + d*x])/(8*a^3*d) + Coth[c +
d*x]/(4*a*d*(a + b*Tanh[c + d*x]^2)^2) + (5*Coth[c + d*x])/(8*a^2*d*(a + b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.086946, antiderivative size = 112, 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, 290, 325, 205} \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d}+\frac{5 \coth (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{15 \coth (c+d x)}{8 a^3 d}+\frac{\coth (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*d) - (15*Coth[c + d*x])/(8*a^3*d) + Coth[c +
d*x]/(4*a*d*(a + b*Tanh[c + d*x]^2)^2) + (5*Coth[c + d*x])/(8*a^2*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 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\coth (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=\frac{\coth (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{5 \coth (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 d}\\ &=-\frac{15 \coth (c+d x)}{8 a^3 d}+\frac{\coth (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{5 \coth (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 d}\\ &=-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d}-\frac{15 \coth (c+d x)}{8 a^3 d}+\frac{\coth (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{5 \coth (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.881923, size = 109, normalized size = 0.97 \[ \frac{-15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )-\frac{\sqrt{a} b \sinh (2 (c+d x)) ((9 a+7 b) \cosh (2 (c+d x))+9 a-7 b)}{((a+b) \cosh (2 (c+d x))+a-b)^2}-8 \sqrt{a} \coth (c+d x)}{8 a^{7/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*Sqrt[a]*Coth[c + d*x] - (Sqrt[a]*b*(9*a - 7*b + (9*a
+ 7*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(8*a^(7/2)*d)

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Maple [B]  time = 0.116, size = 816, normalized size = 7.3 \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)^2/(a+b*tanh(d*x+c)^2)^3,x)

[Out]

-1/2/d/a^3*tanh(1/2*d*x+1/2*c)-9/4/d/a^2*b/(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)^2*tanh(1/2*d*x+1/2*c)^7-27/4/d/a^2*b/(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)^2*tanh(1/2*d*x+1/2*c)^5-7/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*t
anh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^5-27/4/d/a^2*b/(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)^2*tanh(1/2*d*x+1/2*c)^3-7/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)^2*tanh(1/2*d*x+1/2*c)^3-9/4/d/a^2*b/(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)^2*tanh(1/2*d*x+1/2*c)+15/8/d/a^2*b/(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))-15/8/d/a^3*b/((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))+15/8/d/a^3*b^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))+15/8/d/a^2*b/(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))+15/8/d/a^3*b/((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))+15/8/d/a^3*b^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))-1/2/d/a^3/tanh(1/2*d*x+1/2*c)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.14165, size = 19301, normalized size = 172.33 \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)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^8 + 32*(8*a^4 + 23*a^3*b + 45*a^2*
b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + 4*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)
*sinh(d*x + c)^8 + 8*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^6 + 8*(16*a^4 + 23*a^3*b - 45*a*b^3
- 30*b^4 + 14*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(14*(
8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^3 + 3*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)
*cosh(d*x + c))*sinh(d*x + c)^5 + 8*(24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4)*cosh(d*x + c)^4 + 8*(3
5*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^4 + 24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b
^3 + 45*b^4 + 15*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*a^4 + 164*a^3*b
+ 292*a^2*b^2 + 220*a*b^3 + 60*b^4 + 32*(7*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^
5 + 5*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^3 + (24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45
*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*(16*a^4 + 41*a^3*b - 55*a*b^3 - 30*b^4)*cosh(d*x + c)^2 + 8*(14*(8*a^
4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^6 + 15*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*co
sh(d*x + c)^4 + 16*a^4 + 41*a^3*b - 55*a*b^3 - 30*b^4 + 6*(24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4)*
cosh(d*x + c)^2)*sinh(d*x + c)^2 - 15*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^10 + 10*(a^4
+ 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 +
b^4)*sinh(d*x + c)^10 + (3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^8 + (3*a^4 + 4*a^3*b -
6*a^2*b^2 - 12*a*b^3 - 5*b^4 + 45*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8
+ 8*(15*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^3 + (3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3
- 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^6 + 2*(105*(a^4
+ 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^4 + a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4 + 14*(3*a^4 + 4*a^
3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*
b^3 + b^4)*cosh(d*x + c)^5 + 14*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 3*(a^4 + 2*
a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c
)^4 + 2*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^6 + 35*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 1
2*a*b^3 - 5*b^4)*cosh(d*x + c)^4 - a^4 - 2*a^2*b^2 - 8*a*b^3 - 5*b^4 + 15*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*
cosh(d*x + c)^2)*sinh(d*x + c)^4 - a^4 - 4*a^3*b - 6*a^2*b^2 - 4*a*b^3 - b^4 + 8*(15*(a^4 + 4*a^3*b + 6*a^2*b^
2 + 4*a*b^3 + b^4)*cosh(d*x + c)^7 + 7*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 5*(a
^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^3 - (a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c))*sinh(d*
x + c)^3 - (3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^2 + (45*(a^4 + 4*a^3*b + 6*a^2*b^2 +
4*a*b^3 + b^4)*cosh(d*x + c)^8 + 28*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 30*(a^
4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^4 - 3*a^4 - 4*a^3*b + 6*a^2*b^2 + 12*a*b^3 + 5*b^4 - 12*(a^4 +
2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^
4)*cosh(d*x + c)^9 + 4*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^7 + 6*(a^4 + 2*a^2*b^2 +
8*a*b^3 + 5*b^4)*cosh(d*x + c)^5 - 4*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^3 - (3*a^4 + 4*a^3*b -
6*a^2*b^2 - 12*a*b^3 - 5*b^4)*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*(2*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^7 + 3*
(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^5 + 2*(24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4
)*cosh(d*x + c)^3 + (16*a^4 + 41*a^3*b - 55*a*b^3 - 30*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 4*a^6*b + 6*
a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^10 + 10*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cos
h(d*x + c)*sinh(d*x + c)^9 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*sinh(d*x + c)^10 + (3*a^7 + 4
*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^8 + (45*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 +
a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d)*sinh(d*x + c)^8 + 2*(a^
7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*
b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)
^7 + 2*(105*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 14*(3*a^7 + 4*a^6*b - 6*a^5*
b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d)*sinh(d*x + c)^6
- 2*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^4 + 4*(63*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^
3 + a^3*b^4)*d*cosh(d*x + c)^5 + 14*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^3 +
3*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^7 + 4*a^6*b + 6*a^5*
b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^6 + 35*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cos
h(d*x + c)^4 + 15*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^2 - (a^7 + 2*a^5*b^2 + 8*a^4*b^3 +
5*a^3*b^4)*d)*sinh(d*x + c)^4 - (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^2 + 8*
(15*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^7 + 7*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*
a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^5 + 5*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^3 - (a^7
+ 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4
*b^3 + a^3*b^4)*d*cosh(d*x + c)^8 + 28*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^
6 + 30*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^4 - 12*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b
^4)*d*cosh(d*x + c)^2 - (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d)*sinh(d*x + c)^2 - (a^7 + 4*a
^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d + 2*(5*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x
+ c)^9 + 4*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^7 + 6*(a^7 + 2*a^5*b^2 + 8*a
^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^5 - 4*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^3 - (3*a^7
+ 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(8*a^4 + 23*a^3*b +
45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^8 + 16*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(
d*x + c)*sinh(d*x + c)^7 + 2*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*sinh(d*x + c)^8 + 4*(16*a^4 +
23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^6 + 4*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4 + 14*(8*a^4 + 23*a^3
*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(14*(8*a^4 + 23*a^3*b + 45*a^2*b^2 +
45*a*b^3 + 15*b^4)*cosh(d*x + c)^3 + 3*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c))*sinh(d*x + c)^5
+ 4*(24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4)*cosh(d*x + c)^4 + 4*(35*(8*a^4 + 23*a^3*b + 45*a^2*b^
2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^4 + 24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4 + 15*(16*a^4 + 23*a
^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*a^4 + 82*a^3*b + 146*a^2*b^2 + 110*a*b^3 + 30*
b^4 + 16*(7*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4)*cosh(d*x + c)^5 + 5*(16*a^4 + 23*a^3*b - 45*a*
b^3 - 30*b^4)*cosh(d*x + c)^3 + (24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4)*cosh(d*x + c))*sinh(d*x +
c)^3 + 4*(16*a^4 + 41*a^3*b - 55*a*b^3 - 30*b^4)*cosh(d*x + c)^2 + 4*(14*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a
*b^3 + 15*b^4)*cosh(d*x + c)^6 + 15*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^4 + 16*a^4 + 41*a^3*
b - 55*a*b^3 - 30*b^4 + 6*(24*a^4 + 32*a^3*b + 5*a^2*b^2 + 50*a*b^3 + 45*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 15*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^10 + 10*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3
+ b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sinh(d*x + c)^10 + (3*a^4 +
4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^8 + (3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4 + 4
5*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(a^4 + 4*a^3*b + 6*a^2*
b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^3 + (3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*
x + c)^7 + 2*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^6 + 2*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3
+ b^4)*cosh(d*x + c)^4 + a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4 + 14*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b
^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^5 + 14*
(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 3*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(
d*x + c))*sinh(d*x + c)^5 - 2*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^4 + 2*(105*(a^4 + 4*a^3*b + 6*
a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^6 + 35*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^4
- a^4 - 2*a^2*b^2 - 8*a*b^3 - 5*b^4 + 15*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4
- a^4 - 4*a^3*b - 6*a^2*b^2 - 4*a*b^3 - b^4 + 8*(15*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)
^7 + 7*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 5*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4
)*cosh(d*x + c)^3 - (a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - (3*a^4 + 4*a^3*b - 6*
a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^2 + (45*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^8
+ 28*(3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 30*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)
*cosh(d*x + c)^4 - 3*a^4 - 4*a^3*b + 6*a^2*b^2 + 12*a*b^3 + 5*b^4 - 12*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cos
h(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^9 + 4*(3*a^4 +
4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*cosh(d*x + c)^7 + 6*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^
5 - 4*(a^4 + 2*a^2*b^2 + 8*a*b^3 + 5*b^4)*cosh(d*x + c)^3 - (3*a^4 + 4*a^3*b - 6*a^2*b^2 - 12*a*b^3 - 5*b^4)*c
osh(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*(2*(8*a^4 + 23*a^3*b + 45*a^2*b^2 + 45*a*b^3 + 15*b^4
)*cosh(d*x + c)^7 + 3*(16*a^4 + 23*a^3*b - 45*a*b^3 - 30*b^4)*cosh(d*x + c)^5 + 2*(24*a^4 + 32*a^3*b + 5*a^2*b
^2 + 50*a*b^3 + 45*b^4)*cosh(d*x + c)^3 + (16*a^4 + 41*a^3*b - 55*a*b^3 - 30*b^4)*cosh(d*x + c))*sinh(d*x + c)
)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^10 + 10*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^
4*b^3 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*sinh(d*
x + c)^10 + (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^8 + (45*(a^7 + 4*a^6*b + 6*
a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^2 + (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d)*s
inh(d*x + c)^8 + 2*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^6 + 8*(15*(a^7 + 4*a^6*b + 6*a^5*
b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^3 + (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d
*x + c))*sinh(d*x + c)^7 + 2*(105*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^4 + 14*(3*
a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b
^4)*d)*sinh(d*x + c)^6 - 2*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^4 + 4*(63*(a^7 + 4*a^6*b
+ 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^5 + 14*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^
4)*d*cosh(d*x + c)^3 + 3*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(
a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^6 + 35*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*
b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^4 + 15*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^2 - (a^7 + 2
*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d)*sinh(d*x + c)^4 - (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*
d*cosh(d*x + c)^2 + 8*(15*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^7 + 7*(3*a^7 + 4*a
^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^5 + 5*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*c
osh(d*x + c)^3 - (a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^7 + 4*a^6
*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d*cosh(d*x + c)^8 + 28*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3
*b^4)*d*cosh(d*x + c)^6 + 30*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^4 - 12*(a^7 + 2*a^5*b^2
+ 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^2 - (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d)*sinh(d
*x + c)^2 - (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d + 2*(5*(a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3
+ a^3*b^4)*d*cosh(d*x + c)^9 + 4*(3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*d*cosh(d*x + c)^7 + 6*
(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*cosh(d*x + c)^5 - 4*(a^7 + 2*a^5*b^2 + 8*a^4*b^3 + 5*a^3*b^4)*d*co
sh(d*x + c)^3 - (3*a^7 + 4*a^6*b - 6*a^5*b^2 - 12*a^4*b^3 - 5*a^3*b^4)*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}^{2}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.91584, size = 474, normalized size = 4.23 \begin{align*} -\frac{\frac{15 \, b \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 )}{\sqrt{a b} a^{3}} - \frac{2 \,{\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 13 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 7 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 13 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 21 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 23 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 25 \, a^{2} b^{2} + 23 \, a b^{3} + 7 \, b^{4}\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} 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 )}^{2}} + \frac{16}{a^{3}{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*(15*b*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^3) - 2*(9*a^3*b*
e^(6*d*x + 6*c) + 3*a^2*b^2*e^(6*d*x + 6*c) - 13*a*b^3*e^(6*d*x + 6*c) - 7*b^4*e^(6*d*x + 6*c) + 27*a^3*b*e^(4
*d*x + 4*c) + 3*a^2*b^2*e^(4*d*x + 4*c) + 13*a*b^3*e^(4*d*x + 4*c) + 21*b^4*e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*
x + 2*c) + 25*a^2*b^2*e^(2*d*x + 2*c) - 23*a*b^3*e^(2*d*x + 2*c) - 21*b^4*e^(2*d*x + 2*c) + 9*a^3*b + 25*a^2*b
^2 + 23*a*b^3 + 7*b^4)/((a^5 + 2*a^4*b + a^3*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)^2) + 16/(a^3*(e^(2*d*x + 2*c) - 1)))/d