3.41 \(\int \frac{\text{csch}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=110 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{7/2} d \sqrt{a-b}}-\frac{\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac{(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac{\coth ^5(c+d x)}{5 a d} \]

[Out]

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

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Rubi [A]  time = 0.135537, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3187, 461, 208} \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{7/2} d \sqrt{a-b}}-\frac{\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac{(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac{\coth ^5(c+d x)}{5 a d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 3187

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + (a + b)*ff^2*x^2)^p)/(1 + ff^2*x^2)^(m/2 + p
+ 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\text{csch}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^6 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^6}+\frac{-2 a-b}{a^2 x^4}+\frac{a^2+a b+b^2}{a^3 x^2}+\frac{b^3}{a^3 \left (-a+(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac{(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac{\coth ^5(c+d x)}{5 a d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{-a+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{a^3 d}\\ &=-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a-b} d}-\frac{\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac{(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac{\coth ^5(c+d x)}{5 a d}\\ \end{align*}

Mathematica [A]  time = 1.45808, size = 155, normalized size = 1.41 \[ -\frac{\text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left (\sqrt{a} \sqrt{a-b} \coth (c+d x) \left (3 a^2 \text{csch}^4(c+d x)+8 a^2-a (4 a+5 b) \text{csch}^2(c+d x)+10 a b+15 b^2\right )+15 b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )\right )}{30 a^{7/2} d \sqrt{a-b} \left (a \text{csch}^2(c+d x)+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

-((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*(15*b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[
a]*Sqrt[a - b]*Coth[c + d*x]*(8*a^2 + 10*a*b + 15*b^2 - a*(4*a + 5*b)*Csch[c + d*x]^2 + 3*a^2*Csch[c + d*x]^4)
))/(30*a^(7/2)*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2))

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Maple [B]  time = 0.074, size = 519, normalized size = 4.7 \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)^6/(a+b*sinh(d*x+c)^2),x)

[Out]

-1/160/d/a*tanh(1/2*d*x+1/2*c)^5+5/96/d/a*tanh(1/2*d*x+1/2*c)^3+1/24/d/a^2*tanh(1/2*d*x+1/2*c)^3*b-5/16/d/a*ta
nh(1/2*d*x+1/2*c)-3/8/d/a^2*tanh(1/2*d*x+1/2*c)*b-1/2/d/a^3*b^2*tanh(1/2*d*x+1/2*c)-1/160/d/a/tanh(1/2*d*x+1/2
*c)^5+5/96/d/a/tanh(1/2*d*x+1/2*c)^3+1/24/d/a^2/tanh(1/2*d*x+1/2*c)^3*b-5/16/d/a/tanh(1/2*d*x+1/2*c)-3/8/d*b/a
^2/tanh(1/2*d*x+1/2*c)-1/2/d/a^3/tanh(1/2*d*x+1/2*c)*b^2-1/d*b^3/a^3/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arct
anh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/d*b^4/a^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))+1/d*b^3/a^3/((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/d*b^4/a^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))

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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)^6/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.78102, size = 10761, normalized size = 97.83 \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)^6/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/30*(60*(a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 480*(a^2*b^2 - a*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + 60*(a^2*b
^2 - a*b^3)*sinh(d*x + c)^8 - 120*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 120*(a^3*b + a^2*b^2 - 2*a*b^3
- 14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 240*(14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^3 - 3*(a^3*
b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 40*(8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)
^4 + 40*(105*(a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3 - 45*(a^3*b + a^2*b^2 - 2
*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*a^4 + 8*a^3*b + 20*a^2*b^2 - 60*a*b^3 + 160*(21*(a^2*b^2 - a*b^3
)*cosh(d*x + c)^5 - 15*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cos
h(d*x + c))*sinh(d*x + c)^3 - 40*(4*a^4 + a^3*b + a^2*b^2 - 6*a*b^3)*cosh(d*x + c)^2 + 40*(42*(a^2*b^2 - a*b^3
)*cosh(d*x + c)^6 - 45*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 - 4*a^4 - a^3*b - a^2*b^2 + 6*a*b^3 + 6*(8*
a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 15*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d
*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 - 5*b^3*cosh(d*x + c)^8 + 10*b^3*cosh(d*x + c)^6 + 5*(9*b^3*cos
h(d*x + c)^2 - b^3)*sinh(d*x + c)^8 + 40*(3*b^3*cosh(d*x + c)^3 - b^3*cosh(d*x + c))*sinh(d*x + c)^7 - 10*b^3*
cosh(d*x + c)^4 + 10*(21*b^3*cosh(d*x + c)^4 - 14*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^6 + 4*(63*b^3*cosh(
d*x + c)^5 - 70*b^3*cosh(d*x + c)^3 + 15*b^3*cosh(d*x + c))*sinh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^2 + 10*(21*b
^3*cosh(d*x + c)^6 - 35*b^3*cosh(d*x + c)^4 + 15*b^3*cosh(d*x + c)^2 - b^3)*sinh(d*x + c)^4 + 40*(3*b^3*cosh(d
*x + c)^7 - 7*b^3*cosh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^3 - b^3*cosh(d*x + c))*sinh(d*x + c)^3 - b^3 + 5*(9*b^
3*cosh(d*x + c)^8 - 28*b^3*cosh(d*x + c)^6 + 30*b^3*cosh(d*x + c)^4 - 12*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x +
c)^2 + 10*(b^3*cosh(d*x + c)^9 - 4*b^3*cosh(d*x + c)^7 + 6*b^3*cosh(d*x + c)^5 - 4*b^3*cosh(d*x + c)^3 + b^3*
cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 +
b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)
^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(b*cosh(d*x
+ c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 +
4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2
+ 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 80*(6*(a^2*
b^2 - a*b^3)*cosh(d*x + c)^7 - 9*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + 2*(8*a^4 - a^3*b + 2*a^2*b^2 -
9*a*b^3)*cosh(d*x + c)^3 - (4*a^4 + a^3*b + a^2*b^2 - 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^5 - a^4*b)*d*
cosh(d*x + c)^10 + 10*(a^5 - a^4*b)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^5 - a^4*b)*d*sinh(d*x + c)^10 - 5*(a^
5 - a^4*b)*d*cosh(d*x + c)^8 + 5*(9*(a^5 - a^4*b)*d*cosh(d*x + c)^2 - (a^5 - a^4*b)*d)*sinh(d*x + c)^8 + 10*(a
^5 - a^4*b)*d*cosh(d*x + c)^6 + 40*(3*(a^5 - a^4*b)*d*cosh(d*x + c)^3 - (a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*
x + c)^7 + 10*(21*(a^5 - a^4*b)*d*cosh(d*x + c)^4 - 14*(a^5 - a^4*b)*d*cosh(d*x + c)^2 + (a^5 - a^4*b)*d)*sinh
(d*x + c)^6 - 10*(a^5 - a^4*b)*d*cosh(d*x + c)^4 + 4*(63*(a^5 - a^4*b)*d*cosh(d*x + c)^5 - 70*(a^5 - a^4*b)*d*
cosh(d*x + c)^3 + 15*(a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*(a^5 - a^4*b)*d*cosh(d*x + c)^6 -
35*(a^5 - a^4*b)*d*cosh(d*x + c)^4 + 15*(a^5 - a^4*b)*d*cosh(d*x + c)^2 - (a^5 - a^4*b)*d)*sinh(d*x + c)^4 +
5*(a^5 - a^4*b)*d*cosh(d*x + c)^2 + 40*(3*(a^5 - a^4*b)*d*cosh(d*x + c)^7 - 7*(a^5 - a^4*b)*d*cosh(d*x + c)^5
+ 5*(a^5 - a^4*b)*d*cosh(d*x + c)^3 - (a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(9*(a^5 - a^4*b)*d*co
sh(d*x + c)^8 - 28*(a^5 - a^4*b)*d*cosh(d*x + c)^6 + 30*(a^5 - a^4*b)*d*cosh(d*x + c)^4 - 12*(a^5 - a^4*b)*d*c
osh(d*x + c)^2 + (a^5 - a^4*b)*d)*sinh(d*x + c)^2 - (a^5 - a^4*b)*d + 10*((a^5 - a^4*b)*d*cosh(d*x + c)^9 - 4*
(a^5 - a^4*b)*d*cosh(d*x + c)^7 + 6*(a^5 - a^4*b)*d*cosh(d*x + c)^5 - 4*(a^5 - a^4*b)*d*cosh(d*x + c)^3 + (a^5
- a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)), -1/15*(30*(a^2*b^2 - a*b^3)*cosh(d*x + c)^8 + 240*(a^2*b^2 - a*b^3)
*cosh(d*x + c)*sinh(d*x + c)^7 + 30*(a^2*b^2 - a*b^3)*sinh(d*x + c)^8 - 60*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*
x + c)^6 - 60*(a^3*b + a^2*b^2 - 2*a*b^3 - 14*(a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 120*(14*(a^
2*b^2 - a*b^3)*cosh(d*x + c)^3 - 3*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 20*(8*a^4 - a^
3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^4 + 20*(105*(a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*a^4 - a^3*b + 2*a^2
*b^2 - 9*a*b^3 - 45*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*a^4 + 4*a^3*b + 10*a^2*b
^2 - 30*a*b^3 + 80*(21*(a^2*b^2 - a*b^3)*cosh(d*x + c)^5 - 15*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (8
*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 20*(4*a^4 + a^3*b + a^2*b^2 - 6*a*b^3)*co
sh(d*x + c)^2 + 20*(42*(a^2*b^2 - a*b^3)*cosh(d*x + c)^6 - 45*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 - 4*
a^4 - a^3*b - a^2*b^2 + 6*a*b^3 + 6*(8*a^4 - a^3*b + 2*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 1
5*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 - 5*b^3*cosh(d*x + c)^8
+ 10*b^3*cosh(d*x + c)^6 + 5*(9*b^3*cosh(d*x + c)^2 - b^3)*sinh(d*x + c)^8 + 40*(3*b^3*cosh(d*x + c)^3 - b^3*c
osh(d*x + c))*sinh(d*x + c)^7 - 10*b^3*cosh(d*x + c)^4 + 10*(21*b^3*cosh(d*x + c)^4 - 14*b^3*cosh(d*x + c)^2 +
b^3)*sinh(d*x + c)^6 + 4*(63*b^3*cosh(d*x + c)^5 - 70*b^3*cosh(d*x + c)^3 + 15*b^3*cosh(d*x + c))*sinh(d*x +
c)^5 + 5*b^3*cosh(d*x + c)^2 + 10*(21*b^3*cosh(d*x + c)^6 - 35*b^3*cosh(d*x + c)^4 + 15*b^3*cosh(d*x + c)^2 -
b^3)*sinh(d*x + c)^4 + 40*(3*b^3*cosh(d*x + c)^7 - 7*b^3*cosh(d*x + c)^5 + 5*b^3*cosh(d*x + c)^3 - b^3*cosh(d*
x + c))*sinh(d*x + c)^3 - b^3 + 5*(9*b^3*cosh(d*x + c)^8 - 28*b^3*cosh(d*x + c)^6 + 30*b^3*cosh(d*x + c)^4 - 1
2*b^3*cosh(d*x + c)^2 + b^3)*sinh(d*x + c)^2 + 10*(b^3*cosh(d*x + c)^9 - 4*b^3*cosh(d*x + c)^7 + 6*b^3*cosh(d*
x + c)^5 - 4*b^3*cosh(d*x + c)^3 + b^3*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(b*cosh(d*x
+ c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b)) + 40*(6*
(a^2*b^2 - a*b^3)*cosh(d*x + c)^7 - 9*(a^3*b + a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + 2*(8*a^4 - a^3*b + 2*a^2*b
^2 - 9*a*b^3)*cosh(d*x + c)^3 - (4*a^4 + a^3*b + a^2*b^2 - 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^5 - a^4*
b)*d*cosh(d*x + c)^10 + 10*(a^5 - a^4*b)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^5 - a^4*b)*d*sinh(d*x + c)^10 -
5*(a^5 - a^4*b)*d*cosh(d*x + c)^8 + 5*(9*(a^5 - a^4*b)*d*cosh(d*x + c)^2 - (a^5 - a^4*b)*d)*sinh(d*x + c)^8 +
10*(a^5 - a^4*b)*d*cosh(d*x + c)^6 + 40*(3*(a^5 - a^4*b)*d*cosh(d*x + c)^3 - (a^5 - a^4*b)*d*cosh(d*x + c))*si
nh(d*x + c)^7 + 10*(21*(a^5 - a^4*b)*d*cosh(d*x + c)^4 - 14*(a^5 - a^4*b)*d*cosh(d*x + c)^2 + (a^5 - a^4*b)*d)
*sinh(d*x + c)^6 - 10*(a^5 - a^4*b)*d*cosh(d*x + c)^4 + 4*(63*(a^5 - a^4*b)*d*cosh(d*x + c)^5 - 70*(a^5 - a^4*
b)*d*cosh(d*x + c)^3 + 15*(a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*(a^5 - a^4*b)*d*cosh(d*x + c
)^6 - 35*(a^5 - a^4*b)*d*cosh(d*x + c)^4 + 15*(a^5 - a^4*b)*d*cosh(d*x + c)^2 - (a^5 - a^4*b)*d)*sinh(d*x + c)
^4 + 5*(a^5 - a^4*b)*d*cosh(d*x + c)^2 + 40*(3*(a^5 - a^4*b)*d*cosh(d*x + c)^7 - 7*(a^5 - a^4*b)*d*cosh(d*x +
c)^5 + 5*(a^5 - a^4*b)*d*cosh(d*x + c)^3 - (a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(9*(a^5 - a^4*b)
*d*cosh(d*x + c)^8 - 28*(a^5 - a^4*b)*d*cosh(d*x + c)^6 + 30*(a^5 - a^4*b)*d*cosh(d*x + c)^4 - 12*(a^5 - a^4*b
)*d*cosh(d*x + c)^2 + (a^5 - a^4*b)*d)*sinh(d*x + c)^2 - (a^5 - a^4*b)*d + 10*((a^5 - a^4*b)*d*cosh(d*x + c)^9
- 4*(a^5 - a^4*b)*d*cosh(d*x + c)^7 + 6*(a^5 - a^4*b)*d*cosh(d*x + c)^5 - 4*(a^5 - a^4*b)*d*cosh(d*x + c)^3 +
(a^5 - a^4*b)*d*cosh(d*x + c))*sinh(d*x + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**6/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.49275, size = 289, normalized size = 2.63 \begin{align*} -\frac{b^{3} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} a^{3} d} - \frac{2 \,{\left (15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 30 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 60 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 70 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 50 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 60 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )}}{15 \, a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

-b^3*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a^3*d) - 2/15*(15*b^2*e^(8*d
*x + 8*c) - 30*a*b*e^(6*d*x + 6*c) - 60*b^2*e^(6*d*x + 6*c) + 80*a^2*e^(4*d*x + 4*c) + 70*a*b*e^(4*d*x + 4*c)
+ 90*b^2*e^(4*d*x + 4*c) - 40*a^2*e^(2*d*x + 2*c) - 50*a*b*e^(2*d*x + 2*c) - 60*b^2*e^(2*d*x + 2*c) + 8*a^2 +
10*a*b + 15*b^2)/(a^3*d*(e^(2*d*x + 2*c) - 1)^5)