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

Optimal. Leaf size=237 \[ -\frac{\sqrt{b} \left (6 \sqrt{a}-5 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\sqrt{b} \left (6 \sqrt{a}+5 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}+\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 d (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac{\coth (c+d x)}{a^2 d} \]

[Out]

-((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a
] - Sqrt[b])^(3/2)*d) + ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/
4)])/(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Coth[c + d*x]/(a^2*d) + (b*Tanh[c + d*x]*(a - (a + b)*Tanh[c +
d*x]^2))/(4*a^2*(a - b)*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))

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Rubi [A]  time = 0.533692, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3217, 1334, 1664, 1166, 208} \[ -\frac{\sqrt{b} \left (6 \sqrt{a}-5 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\sqrt{b} \left (6 \sqrt{a}+5 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}+\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 d (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac{\coth (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a
] - Sqrt[b])^(3/2)*d) + ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/
4)])/(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Coth[c + d*x]/(a^2*d) + (b*Tanh[c + d*x]*(a - (a + b)*Tanh[c +
d*x]^2))/(4*a^2*(a - b)*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 1334

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d +
e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*
f - 2*a*g)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*
x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x
^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x],
x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^4}{x^2 \left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-8 a b+\frac{2 a (8 a-7 b) b x^2}{a-b}-\frac{2 b \left (4 a^2-a b-b^2\right ) x^4}{a-b}}{x^2 \left (a-2 a x^2+(a-b) x^4\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b d}\\ &=\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{8 b}{x^2}+\frac{2 b^2 \left (a-(7 a-5 b) x^2\right )}{(a-b) \left (a-2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{8 a^2 b d}\\ &=-\frac{\coth (c+d x)}{a^2 d}+\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac{b \operatorname{Subst}\left (\int \frac{a+(-7 a+5 b) x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{4 a^2 (a-b) d}\\ &=-\frac{\coth (c+d x)}{a^2 d}+\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac{\left (\left (7 a+\frac{2 \sqrt{a} (3 a-2 b)}{\sqrt{b}}-5 b\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{-a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b) d}-\frac{\left (b \left (-7 a+\frac{2 \sqrt{a} (3 a-2 b)}{\sqrt{b}}+5 b\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b) d}\\ &=-\frac{\left (6 \sqrt{a}-5 \sqrt{b}\right ) \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt{a}-\sqrt{b}\right )^{3/2} d}+\frac{\left (6 \sqrt{a}+5 \sqrt{b}\right ) \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt{a}+\sqrt{b}\right )^{3/2} d}-\frac{\coth (c+d x)}{a^2 d}+\frac{b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.87298, size = 272, normalized size = 1.15 \[ \frac{\frac{\left (6 a \sqrt{b}+5 \sqrt{a} b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\left (\sqrt{a}+\sqrt{b}\right ) \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left (6 a \sqrt{b}-5 \sqrt{a} b\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tanh (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\left (\sqrt{a}-\sqrt{b}\right ) \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{4 \sqrt{a} b \sinh (2 (c+d x)) (2 a-b \cosh (2 (c+d x))+b)}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}-8 \sqrt{a} \coth (c+d x)}{8 a^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.09, size = 765, normalized size = 3.2 \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*sinh(d*x+c)^4)^2,x)

[Out]

-1/2/d/a^2*tanh(1/2*d*x+1/2*c)+1/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^
4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)*b/a/(a-b)*tanh(1/2*d*x+1/2*c)^7-1/2/d/(tanh(1/2*d*
x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2
*c)^2*a+a)/a/(a-b)*tanh(1/2*d*x+1/2*c)^5*b-2/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*ta
nh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/(a-b)*tanh(1/2*d*x+1/2*c)^5-1/2/
d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*ta
nh(1/2*d*x+1/2*c)^2*a+a)/a/(a-b)*tanh(1/2*d*x+1/2*c)^3*b-2/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1
/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/(a-b)*tanh(1/2*d*x
+1/2*c)^3+1/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x
+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)*b/a/(a-b)*tanh(1/2*d*x+1/2*c)-1/16/d*b/a^2/(a-b)*sum((-_R^6*a+(27*a-20*
b)*_R^4+(-27*a+20*b)*_R^2+a)/(_R^7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_
Z^8-4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+a))-1/2/d/a^2/tanh(1/2*d*x+1/2*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(4*a*b - 5*b^2 + (6*a*b*e^(8*c) - 5*b^2*e^(8*c))*e^(8*d*x) - 2*(13*a*b*e^(6*c) - 10*b^2*e^(6*c))*e^(6*d*x)
- 2*(32*a^2*e^(4*c) - 47*a*b*e^(4*c) + 15*b^2*e^(4*c))*e^(4*d*x) - 2*(7*a*b*e^(2*c) - 10*b^2*e^(2*c))*e^(2*d*
x))/(a^3*b*d - a^2*b^2*d - (a^3*b*d*e^(10*c) - a^2*b^2*d*e^(10*c))*e^(10*d*x) + 5*(a^3*b*d*e^(8*c) - a^2*b^2*d
*e^(8*c))*e^(8*d*x) + 2*(8*a^4*d*e^(6*c) - 13*a^3*b*d*e^(6*c) + 5*a^2*b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^4*d*e^
(4*c) - 13*a^3*b*d*e^(4*c) + 5*a^2*b^2*d*e^(4*c))*e^(4*d*x) - 5*(a^3*b*d*e^(2*c) - a^2*b^2*d*e^(2*c))*e^(2*d*x
)) - 4*integrate(1/4*((6*a*b*e^(6*c) - 5*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a*b*e^(4*c) - 5*b^2*e^(4*c))*e^(4*d*x)
+ (6*a*b*e^(2*c) - 5*b^2*e^(2*c))*e^(2*d*x))/(a^3*b - a^2*b^2 + (a^3*b*e^(8*c) - a^2*b^2*e^(8*c))*e^(8*d*x) -
4*(a^3*b*e^(6*c) - a^2*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^4*e^(4*c) - 11*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c))*e^(4*
d*x) - 4*(a^3*b*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 4.28504, size = 20254, normalized size = 85.46 \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*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(8*(6*a*b - 5*b^2)*cosh(d*x + c)^8 + 64*(6*a*b - 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + 8*(6*a*b - 5*b^2
)*sinh(d*x + c)^8 - 16*(13*a*b - 10*b^2)*cosh(d*x + c)^6 + 16*(14*(6*a*b - 5*b^2)*cosh(d*x + c)^2 - 13*a*b + 1
0*b^2)*sinh(d*x + c)^6 + 32*(14*(6*a*b - 5*b^2)*cosh(d*x + c)^3 - 3*(13*a*b - 10*b^2)*cosh(d*x + c))*sinh(d*x
+ c)^5 - 16*(32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c)^4 + 16*(35*(6*a*b - 5*b^2)*cosh(d*x + c)^4 - 15*(13*a*b -
10*b^2)*cosh(d*x + c)^2 - 32*a^2 + 47*a*b - 15*b^2)*sinh(d*x + c)^4 + 64*(7*(6*a*b - 5*b^2)*cosh(d*x + c)^5 -
5*(13*a*b - 10*b^2)*cosh(d*x + c)^3 - (32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*(7*a*b -
10*b^2)*cosh(d*x + c)^2 + 16*(14*(6*a*b - 5*b^2)*cosh(d*x + c)^6 - 15*(13*a*b - 10*b^2)*cosh(d*x + c)^4 - 6*(
32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c)^2 - 7*a*b + 10*b^2)*sinh(d*x + c)^2 + ((a^3*b - a^2*b^2)*d*cosh(d*x +
c)^10 + 10*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b
- a^2*b^2)*d*cosh(d*x + c)^8 + 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^
8 - 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b -
a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*
d*cosh(d*x + c)^2 - (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6 + 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cos
h(d*x + c)^4 + 4*(63*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 -
13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^
3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5
*a^2*b^2)*d)*sinh(d*x + c)^4 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)
^7 - 35*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 5*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + (8*a^4 - 13
*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 - 140*(a^3*b -
a^2*b^2)*d*cosh(d*x + c)^6 - 30*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 12*(8*a^4 - 13*a^3*b + 5*a^
2*b^2)*d*cosh(d*x + c)^2 + 5*(a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 - (a^3*b - a^2*b^2)*d + 2*(5*(a^3*b - a^2*b^
2)*d*cosh(d*x + c)^9 - 20*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 6*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x +
c)^5 + 4*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)
)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b
^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a
^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*
a*b^4 - 625*b^5 + 2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*sqrt((2304*a^4*b^3 - 662
4*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 -
6*a^10*b^5 + a^9*b^6)*d^4)) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)^2 - 2*(1728*
a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)*sinh(d*x + c) - (1728*a^3*b^2 - 3684*a^2*b^3 + 26
25*a*b^4 - 625*b^5)*sinh(d*x + c)^2 + 2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((
2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b
^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d)*sqr
t(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 +
625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b
+ 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cosh(d*x + c)^10 + 1
0*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b - a^2*b^
2)*d*cosh(d*x + c)^8 + 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^8 - 2*(8*
a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*
d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*d*cosh(d*
x + c)^2 - (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6 + 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c
)^4 + 4*(63*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 - 13*a^3*b
+ 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^3*b - a^2
*b^2)*d*cosh(d*x + c)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)
*d)*sinh(d*x + c)^4 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 35*(
a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 5*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + (8*a^4 - 13*a^3*b +
5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 - 140*(a^3*b - a^2*b^2)*
d*cosh(d*x + c)^6 - 30*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 12*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*
cosh(d*x + c)^2 + 5*(a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 - (a^3*b - a^2*b^2)*d + 2*(5*(a^3*b - a^2*b^2)*d*cosh
(d*x + c)^9 - 20*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 6*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^5 + 4*
(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-(
(a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*
b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47
*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 6
25*b^5 + 2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4
+ 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b
^5 + a^9*b^6)*d^4)) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)^2 - 2*(1728*a^3*b^2 -
3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)*sinh(d*x + c) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4
- 625*b^5)*sinh(d*x + c)^2 - 2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*
b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a
^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d)*sqrt(-((a^7
- 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/
((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^
2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cosh(d*x + c)^10 + 10*(a^3*b
- a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b - a^2*b^2)*d*cosh
(d*x + c)^8 + 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^8 - 2*(8*a^4 - 13*
a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*d*cosh(d*
x + c))*sinh(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2
- (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6 + 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 4*(
63*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 - 13*a^3*b + 5*a^2*
b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^3*b - a^2*b^2)*d*c
osh(d*x + c)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(
d*x + c)^4 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 35*(a^3*b - a
^2*b^2)*d*cosh(d*x + c)^5 - 5*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + (8*a^4 - 13*a^3*b + 5*a^2*b^2
)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 - 140*(a^3*b - a^2*b^2)*d*cosh(d*
x + c)^6 - 30*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 12*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x
+ c)^2 + 5*(a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 - (a^3*b - a^2*b^2)*d + 2*(5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)
^9 - 20*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 6*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^5 + 4*(8*a^4 -
13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^7 - 3*a
^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^1
5 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 1
5*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5 - 2
*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^
2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b
^6)*d^4)) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)^2 - 2*(1728*a^3*b^2 - 3684*a^2*
b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)*sinh(d*x + c) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)
*sinh(d*x + c)^2 + 2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624
*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 -
6*a^10*b^5 + a^9*b^6)*d^4)) - 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d)*sqrt(((a^7 - 3*a^6*b +
3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*
a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)
/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) + ((a^3*b - a^2*b^2)*d*cosh(d*x + c)^10 + 10*(a^3*b - a^2*b^2)*
d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b - a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8
+ 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^8 - 2*(8*a^4 - 13*a^3*b + 5*a
^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*d*cosh(d*x + c))*sin
h(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (8*a^4 -
13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6 + 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 4*(63*(a^3*b -
a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^3*b - a^2*b^2)*d*cosh(d*x + c
)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^4
+ 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 35*(a^3*b - a^2*b^2)*d*c
osh(d*x + c)^5 - 5*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*
x + c))*sinh(d*x + c)^3 + (45*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 - 140*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 -
30*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 12*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + 5*
(a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 - (a^3*b - a^2*b^2)*d + 2*(5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^9 - 20*(a^
3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 6*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^5 + 4*(8*a^4 - 13*a^3*b +
5*a^2*b^2)*d*cosh(d*x + c)^3 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^7 - 3*a^6*b + 3*a^
5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*
b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^
7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5 - 2*(36*a^9 -
133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 345
0*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) -
(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cosh(d*x + c)^2 - 2*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*
a*b^4 - 625*b^5)*cosh(d*x + c)*sinh(d*x + c) - (1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*sinh(d*x +
c)^2 - 2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 +
7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5
+ a^9*b^6)*d^4)) - 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2
- a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15
*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*
a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) + 32*a*b - 40*b^2 + 32*(2*(6*a*b - 5*b^2)*cosh(d*x + c)^7 - 3*(13*a*b - 10
*b^2)*cosh(d*x + c)^5 - 2*(32*a^2 - 47*a*b + 15*b^2)*cosh(d*x + c)^3 - (7*a*b - 10*b^2)*cosh(d*x + c))*sinh(d*
x + c))/((a^3*b - a^2*b^2)*d*cosh(d*x + c)^10 + 10*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3*b
- a^2*b^2)*d*sinh(d*x + c)^10 - 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 + 5*(9*(a^3*b - a^2*b^2)*d*cosh(d*x + c)
^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^8 - 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^6 + 40*(3*(a^3*b
- a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^3*b - a^2*b^2)*d
*cosh(d*x + c)^4 - 70*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^6
+ 2*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^4 + 4*(63*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 70*(a^3*b -
a^2*b^2)*d*cosh(d*x + c)^3 - 3*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^3*
b - a^2*b^2)*d*cosh(d*x + c)^6 - 175*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 15*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d
*cosh(d*x + c)^2 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d)*sinh(d*x + c)^4 + 5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 +
8*(15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 35*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 5*(8*a^4 - 13*a^3*b + 5*
a^2*b^2)*d*cosh(d*x + c)^3 + (8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^3*b - a^
2*b^2)*d*cosh(d*x + c)^8 - 140*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 - 30*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(
d*x + c)^4 + 12*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^2 + 5*(a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 - (a
^3*b - a^2*b^2)*d + 2*(5*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^9 - 20*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 6*(8*a
^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^5 + 4*(8*a^4 - 13*a^3*b + 5*a^2*b^2)*d*cosh(d*x + c)^3 + 5*(a^3*b -
a^2*b^2)*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)**2/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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Giac [A]  time = 5.79865, size = 321, normalized size = 1.35 \begin{align*} -\frac{6 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 5 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 26 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 20 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 64 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 94 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 30 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 14 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 20 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a b - 5 \, b^{2}}{2 \,{\left (a^{3} d - a^{2} b d\right )}{\left (b e^{\left (10 \, d x + 10 \, c\right )} - 5 \, b e^{\left (8 \, d x + 8 \, c\right )} - 16 \, a e^{\left (6 \, d x + 6 \, c\right )} + 10 \, b e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a e^{\left (4 \, d x + 4 \, c\right )} - 10 \, b e^{\left (4 \, d x + 4 \, c\right )} + 5 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*a*b*e^(8*d*x + 8*c) - 5*b^2*e^(8*d*x + 8*c) - 26*a*b*e^(6*d*x + 6*c) + 20*b^2*e^(6*d*x + 6*c) - 64*a^2
*e^(4*d*x + 4*c) + 94*a*b*e^(4*d*x + 4*c) - 30*b^2*e^(4*d*x + 4*c) - 14*a*b*e^(2*d*x + 2*c) + 20*b^2*e^(2*d*x
+ 2*c) + 4*a*b - 5*b^2)/((a^3*d - a^2*b*d)*(b*e^(10*d*x + 10*c) - 5*b*e^(8*d*x + 8*c) - 16*a*e^(6*d*x + 6*c) +
10*b*e^(6*d*x + 6*c) + 16*a*e^(4*d*x + 4*c) - 10*b*e^(4*d*x + 4*c) + 5*b*e^(2*d*x + 2*c) - b))