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

Optimal. Leaf size=161 \[ \frac{b^{3/2} (5 a-4 b) \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{2 a^3 d (a-b)^{3/2}}-\frac{b (a-2 b) \cosh (c+d x)}{2 a^2 d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}+\frac{(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b \cosh ^2(c+d x)-b\right )} \]

[Out]

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

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Rubi [A]  time = 0.272916, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3186, 414, 527, 522, 206, 205} \[ \frac{b^{3/2} (5 a-4 b) \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{2 a^3 d (a-b)^{3/2}}-\frac{b (a-2 b) \cosh (c+d x)}{2 a^2 d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}+\frac{(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b \cosh ^2(c+d x)-b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3186

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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{\text{csch}^3(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{a+b+3 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{2 a d}\\ &=-\frac{(a-2 b) b \cosh (c+d x)}{2 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-2 \left (a^2+2 a b-2 b^2\right )-2 (a-2 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{4 a^2 (a-b) d}\\ &=-\frac{(a-2 b) b \cosh (c+d x)}{2 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )}+\frac{\left ((5 a-4 b) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 (a-b) d}+\frac{(a+4 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}\\ &=\frac{(5 a-4 b) b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{2 a^3 (a-b)^{3/2} d}+\frac{(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac{(a-2 b) b \cosh (c+d x)}{2 a^2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a-b+b \cosh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 1.4059, size = 350, normalized size = 2.17 \[ \frac{\text{csch}^3(c+d x) (2 a+b \cosh (2 (c+d x))-b) \left (\frac{8 a b^2 \coth (c+d x)}{a-b}+\frac{4 b^{3/2} (5 a-4 b) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )}{(a-b)^{3/2}}+\frac{4 b^{3/2} (5 a-4 b) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )}{(a-b)^{3/2}}-a \text{csch}^2\left (\frac{1}{2} (c+d x)\right ) \text{csch}(c+d x) (2 a+b \cosh (2 (c+d x))-b)-a \text{csch}(c+d x) \text{sech}^2\left (\frac{1}{2} (c+d x)\right ) (2 a+b \cosh (2 (c+d x))-b)-4 (a+4 b) \text{csch}(c+d x) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) (2 a+b \cosh (2 (c+d x))-b)\right )}{32 a^3 d \left (a \text{csch}^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^3*((8*a*b^2*Coth[c + d*x])/(a - b) + (4*(5*a - 4*b)*b^(3/2)*Arc
Tan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x])/(a - b
)^(3/2) + (4*(5*a - 4*b)*b^(3/2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]]*(2*a - b + b*Cosh
[2*(c + d*x)])*Csch[c + d*x])/(a - b)^(3/2) - a*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[(c + d*x)/2]^2*Csch[c + d
*x] - 4*(a + 4*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]*Log[Tanh[(c + d*x)/2]] - a*(2*a - b + b*Cosh[2
*(c + d*x)])*Csch[c + d*x]*Sech[(c + d*x)/2]^2))/(32*a^3*d*(b + a*Csch[c + d*x]^2)^2)

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Maple [B]  time = 0.079, size = 415, normalized size = 2.6 \begin{align*}{\frac{1}{8\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-2\,{\frac{\ln \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) b}{d{a}^{3}}}-{\frac{{b}^{2}}{d{a}^{2} \left ( a-b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{-1}}+2\,{\frac{{b}^{3} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{3} \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) \left ( a-b \right ) }}+{\frac{{b}^{2}}{d{a}^{2} \left ( a-b \right ) } \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{-1}}+{\frac{5\,{b}^{2}}{2\,d{a}^{2} \left ( a-b \right ) }\arctan \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,a+4\,b \right ){\frac{1}{\sqrt{ab-{b}^{2}}}}} \right ){\frac{1}{\sqrt{ab-{b}^{2}}}}}-2\,{\frac{{b}^{3}}{d{a}^{3} \left ( a-b \right ) \sqrt{ab-{b}^{2}}}\arctan \left ( 1/4\,{\frac{2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,a+4\,b}{\sqrt{ab-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3/(a+b*sinh(d*x+c)^2)^2,x)

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/a^2-1/8/d/a^2/tanh(1/2*d*x+1/2*c)^2-1/2/d/a^2*ln(tanh(1/2*d*x+1/2*c))-2/d/a^3*ln(t
anh(1/2*d*x+1/2*c))*b-1/d/a^2*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)/(a-b)*tanh(1/2*d*x+1/2*c)^2+2/d*b^3/a^3/(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)/(a-b)*tanh(1/2*d*x+1/2*c)^2+1/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tan
h(1/2*d*x+1/2*c)^2*b+a)/(a-b)+5/2/d/a^2*b^2/(a-b)/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*
b)/(a*b-b^2)^(1/2))-2/d*b^3/a^3/(a-b)/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)
^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (a b e^{\left (7 \, c\right )} - 2 \, b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} +{\left (4 \, a^{2} e^{\left (5 \, c\right )} - 5 \, a b e^{\left (5 \, c\right )} + 2 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} +{\left (4 \, a^{2} e^{\left (3 \, c\right )} - 5 \, a b e^{\left (3 \, c\right )} + 2 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a b e^{c} - 2 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{a^{3} b d - a^{2} b^{2} d +{\left (a^{3} b d e^{\left (8 \, c\right )} - a^{2} b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (a^{4} d e^{\left (6 \, c\right )} - 2 \, a^{3} b d e^{\left (6 \, c\right )} + a^{2} b^{2} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \,{\left (4 \, a^{4} d e^{\left (4 \, c\right )} - 7 \, a^{3} b d e^{\left (4 \, c\right )} + 3 \, a^{2} b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (a^{4} d e^{\left (2 \, c\right )} - 2 \, a^{3} b d e^{\left (2 \, c\right )} + a^{2} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \frac{{\left (a + 4 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, a^{3} d} - \frac{{\left (a + 4 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, a^{3} d} + 8 \, \int \frac{{\left (5 \, a b^{2} e^{\left (3 \, c\right )} - 4 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (5 \, a b^{2} e^{c} - 4 \, b^{3} e^{c}\right )} e^{\left (d x\right )}}{8 \,{\left (a^{4} b - a^{3} b^{2} +{\left (a^{4} b e^{\left (4 \, c\right )} - a^{3} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{5} e^{\left (2 \, c\right )} - 3 \, a^{4} b e^{\left (2 \, c\right )} + a^{3} b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a*b*e^(7*c) - 2*b^2*e^(7*c))*e^(7*d*x) + (4*a^2*e^(5*c) - 5*a*b*e^(5*c) + 2*b^2*e^(5*c))*e^(5*d*x) + (4*a^2
*e^(3*c) - 5*a*b*e^(3*c) + 2*b^2*e^(3*c))*e^(3*d*x) + (a*b*e^c - 2*b^2*e^c)*e^(d*x))/(a^3*b*d - a^2*b^2*d + (a
^3*b*d*e^(8*c) - a^2*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^4*d*e^(6*c) - 2*a^3*b*d*e^(6*c) + a^2*b^2*d*e^(6*c))*e^(6
*d*x) - 2*(4*a^4*d*e^(4*c) - 7*a^3*b*d*e^(4*c) + 3*a^2*b^2*d*e^(4*c))*e^(4*d*x) + 4*(a^4*d*e^(2*c) - 2*a^3*b*d
*e^(2*c) + a^2*b^2*d*e^(2*c))*e^(2*d*x)) + 1/2*(a + 4*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^3*d) - 1/2*(a + 4*b)
*log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) + 8*integrate(1/8*((5*a*b^2*e^(3*c) - 4*b^3*e^(3*c))*e^(3*d*x) - (5*a*b
^2*e^c - 4*b^3*e^c)*e^(d*x))/(a^4*b - a^3*b^2 + (a^4*b*e^(4*c) - a^3*b^2*e^(4*c))*e^(4*d*x) + 2*(2*a^5*e^(2*c)
- 3*a^4*b*e^(2*c) + a^3*b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 3.66935, size = 19038, normalized size = 118.25 \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)^3/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/4*(4*(a^2*b - 2*a*b^2)*cosh(d*x + c)^7 + 28*(a^2*b - 2*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(a^2*b - 2
*a*b^2)*sinh(d*x + c)^7 + 4*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 4*(4*a^3 - 5*a^2*b + 2*a*b^2 + 21*(a
^2*b - 2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(a^2*b - 2*a*b^2)*cosh(d*x + c)^3 + (4*a^3 - 5*a^2*b
+ 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + 4*(35*(a^2*b - 2*a
*b^2)*cosh(d*x + c)^4 + 4*a^3 - 5*a^2*b + 2*a*b^2 + 10*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^3 + 4*(21*(a^2*b - 2*a*b^2)*cosh(d*x + c)^5 + 10*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^3 -
5*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - ((5*a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(5*a*b^2 - 4*b^3)*c
osh(d*x + c)*sinh(d*x + c)^7 + (5*a*b^2 - 4*b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)
^6 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3 + 7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - 4*b
^3)*cosh(d*x + c)^3 + 3*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(20*a^2*b - 31*a*b^2 +
12*b^3)*cosh(d*x + c)^4 + 2*(35*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^4 - 20*a^2*b + 31*a*b^2 - 12*b^3 + 30*(5*a^2*b
- 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b -
9*a*b^2 + 4*b^3)*cosh(d*x + c)^3 - (20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a*b^2 - 4
*b^3 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 4*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(5*a^2*b -
9*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 5*a^2*b - 9*a*b^2 + 4*b^3 - 3*(20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c)^
2)*sinh(d*x + c)^2 + 8*((5*a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (2
0*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c)^3 + (5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(
-b/(a - b))*log((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 - 3*b)*cos
h(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d
*x + c))*sinh(d*x + c) + 4*((a - b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a - b)*sinh(d
*x + c)^3 + (a - b)*cosh(d*x + c) + (3*(a - b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c))*sqrt(-b/(a - b)) + 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)) + 4*(a^2*b - 2*a*b^2)*cosh(d*x + c) - 2*((a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a^2*b + 3*a*b^2
- 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b + 3*a*b^2 - 4*b^3)*sinh(d*x + c)^8 + 4*(a^3 + 2*a^2*b - 7*a*b^
2 + 4*b^3)*cosh(d*x + c)^6 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3 + 7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d
*x + c))*sinh(d*x + c)^5 - 2*(4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a^2*b + 3*a*b^2 -
4*b^3)*cosh(d*x + c)^4 - 4*a^3 - 9*a^2*b + 25*a*b^2 - 12*b^3 + 30*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)
*cosh(d*x + c)^3 - (4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*b + 3*a*b^2 - 4*
b^3 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 4*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 1
5*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3 - 3*(4*a^3 + 9*a^2*b - 2
5*a*b^2 + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(a^3 + 2
*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^3 + (a^3 + 2*a
^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*((a^2*b + 3*a
*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b + 3*a*b^2 -
4*b^3)*sinh(d*x + c)^8 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4
*b^3 + 7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x
+ c)^3 + 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(4*a^3 + 9*a^2*b - 25*a*b^2 +
12*b^3)*cosh(d*x + c)^4 + 2*(35*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^4 - 4*a^3 - 9*a^2*b + 25*a*b^2 - 12*b^
3 + 30*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cos
h(d*x + c)^5 + 10*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^3 - (4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*co
sh(d*x + c))*sinh(d*x + c)^3 + a^2*b + 3*a*b^2 - 4*b^3 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^2 +
4*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + a^3 +
2*a^2*b - 7*a*b^2 + 4*b^3 - 3*(4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^
2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (4*a^3 + 9*a^2*
b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^3 + (a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(c
osh(d*x + c) + sinh(d*x + c) - 1) + 4*(7*(a^2*b - 2*a*b^2)*cosh(d*x + c)^6 + 5*(4*a^3 - 5*a^2*b + 2*a*b^2)*cos
h(d*x + c)^4 + a^2*b - 2*a*b^2 + 3*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^4*b - a^3*b
^2)*d*cosh(d*x + c)^8 + 8*(a^4*b - a^3*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b - a^3*b^2)*d*sinh(d*x + c
)^8 + 4*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 - 2*a^4*
b + a^3*b^2)*d)*sinh(d*x + c)^6 - 2*(4*a^5 - 7*a^4*b + 3*a^3*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^4*b - a^3*b^2)*d
*cosh(d*x + c)^3 + 3*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b - a^3*b^2)*d*co
sh(d*x + c)^4 + 30*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^2 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*d)*sinh(d*x + c
)^4 + 4*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^5 + 10*(a^5 - 2*a
^4*b + a^3*b^2)*d*cosh(d*x + c)^3 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4
*b - a^3*b^2)*d*cosh(d*x + c)^6 + 15*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^4 - 3*(4*a^5 - 7*a^4*b + 3*a^3*
b^2)*d*cosh(d*x + c)^2 + (a^5 - 2*a^4*b + a^3*b^2)*d)*sinh(d*x + c)^2 + (a^4*b - a^3*b^2)*d + 8*((a^4*b - a^3*
b^2)*d*cosh(d*x + c)^7 + 3*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^5 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*d*cosh(
d*x + c)^3 + (a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(a^2*b - 2*a*b^2)*cosh(d*x + c
)^7 + 14*(a^2*b - 2*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a^2*b - 2*a*b^2)*sinh(d*x + c)^7 + 2*(4*a^3 - 5*
a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 2*(4*a^3 - 5*a^2*b + 2*a*b^2 + 21*(a^2*b - 2*a*b^2)*cosh(d*x + c)^2)*sinh(d
*x + c)^5 + 10*(7*(a^2*b - 2*a*b^2)*cosh(d*x + c)^3 + (4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(d*x + c)
^4 + 2*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + 2*(35*(a^2*b - 2*a*b^2)*cosh(d*x + c)^4 + 4*a^3 - 5*a^2*b
+ 2*a*b^2 + 10*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(21*(a^2*b - 2*a*b^2)*cosh(d*
x + c)^5 + 10*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c))*sinh(
d*x + c)^2 - ((5*a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(5*a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (5*a*b^2
- 4*b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3 + 7*(
5*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(5*a^2*b - 9*a*
b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c)^4 + 2*(35*(5*a*b^
2 - 4*b^3)*cosh(d*x + c)^4 - 20*a^2*b + 31*a*b^2 - 12*b^3 + 30*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*si
nh(d*x + c)^4 + 8*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^3 - (20*
a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a*b^2 - 4*b^3 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3)*co
sh(d*x + c)^2 + 4*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 5*a^
2*b - 9*a*b^2 + 4*b^3 - 3*(20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((5*a*b^2 - 4*b^
3)*cosh(d*x + c)^7 + 3*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x +
c)^3 + (5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a - b))*arctan(1/2*sqrt(b/(a - b))*(
cosh(d*x + c) + sinh(d*x + c))) + ((5*a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(5*a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(
d*x + c)^7 + (5*a*b^2 - 4*b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(5*a^2*b -
9*a*b^2 + 4*b^3 + 7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^
3 + 3*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x +
c)^4 + 2*(35*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^4 - 20*a^2*b + 31*a*b^2 - 12*b^3 + 30*(5*a^2*b - 9*a*b^2 + 4*b^3
)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b - 9*a*b^2 + 4*b^3)*c
osh(d*x + c)^3 - (20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a*b^2 - 4*b^3 + 4*(5*a^2*b
- 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 4*(7*(5*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(5*a^2*b - 9*a*b^2 + 4*b^3)*c
osh(d*x + c)^4 + 5*a^2*b - 9*a*b^2 + 4*b^3 - 3*(20*a^2*b - 31*a*b^2 + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 8*((5*a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (20*a^2*b - 31*a*b^2
+ 12*b^3)*cosh(d*x + c)^3 + (5*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a - b))*arctan(
1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - 3*b)*cosh(d*x + c) + (
3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))*sqrt(b/(a - b))/b) + 2*(a^2*b - 2*a*b^2)*cosh(d*x + c) - ((a^2
*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b + 3
*a*b^2 - 4*b^3)*sinh(d*x + c)^8 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(a^3 + 2*a^2*b - 7*a
*b^2 + 4*b^3 + 7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*c
osh(d*x + c)^3 + 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(4*a^3 + 9*a^2*b - 25*
a*b^2 + 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^4 - 4*a^3 - 9*a^2*b + 25*a*b^2
- 12*b^3 + 30*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^2*b + 3*a*b^2 - 4*
b^3)*cosh(d*x + c)^5 + 10*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^3 - (4*a^3 + 9*a^2*b - 25*a*b^2 + 12
*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*b + 3*a*b^2 - 4*b^3 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x
+ c)^2 + 4*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^4
+ a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3 - 3*(4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 8*((a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (4*a^3
+ 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^3 + (a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c
))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a^2*b + 3*a*b^2 -
4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b + 3*a*b^2 - 4*b^3)*sinh(d*x + c)^8 + 4*(a^3 + 2*a^2*b - 7*a*b^2
+ 4*b^3)*cosh(d*x + c)^6 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3 + 7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*s
inh(d*x + c)^6 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^5 - 2*(4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a^2*b + 3*a*b^2 - 4*
b^3)*cosh(d*x + c)^4 - 4*a^3 - 9*a^2*b + 25*a*b^2 - 12*b^3 + 30*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c
)^2)*sinh(d*x + c)^4 + 8*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*c
osh(d*x + c)^3 - (4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*b + 3*a*b^2 - 4*b^
3 + 4*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 4*(7*(a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*
(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3 - 3*(4*a^3 + 9*a^2*b - 25*
a*b^2 + 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^2*b + 3*a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(a^3 + 2*a
^2*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c)^5 - (4*a^3 + 9*a^2*b - 25*a*b^2 + 12*b^3)*cosh(d*x + c)^3 + (a^3 + 2*a^2
*b - 7*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*(a^2*b - 2*a
*b^2)*cosh(d*x + c)^6 + 5*(4*a^3 - 5*a^2*b + 2*a*b^2)*cosh(d*x + c)^4 + a^2*b - 2*a*b^2 + 3*(4*a^3 - 5*a^2*b +
2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^4*b - a^3*b^2)*d*cosh(d*x + c)^8 + 8*(a^4*b - a^3*b^2)*d*cosh(d*
x + c)*sinh(d*x + c)^7 + (a^4*b - a^3*b^2)*d*sinh(d*x + c)^8 + 4*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^6 +
4*(7*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 - 2*a^4*b + a^3*b^2)*d)*sinh(d*x + c)^6 - 2*(4*a^5 - 7*a^4*b
+ 3*a^3*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^3 + 3*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^4 + 30*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d
*x + c)^2 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*d)*sinh(d*x + c)^4 + 4*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^2 +
8*(7*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^5 + 10*(a^5 - 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^3 - (4*a^5 - 7*a^4*b
+ 3*a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b - a^3*b^2)*d*cosh(d*x + c)^6 + 15*(a^5 - 2*a^4*b +
a^3*b^2)*d*cosh(d*x + c)^4 - 3*(4*a^5 - 7*a^4*b + 3*a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 - 2*a^4*b + a^3*b^2)*d)
*sinh(d*x + c)^2 + (a^4*b - a^3*b^2)*d + 8*((a^4*b - a^3*b^2)*d*cosh(d*x + c)^7 + 3*(a^5 - 2*a^4*b + a^3*b^2)*
d*cosh(d*x + c)^5 - (4*a^5 - 7*a^4*b + 3*a^3*b^2)*d*cosh(d*x + c)^3 + (a^5 - 2*a^4*b + a^3*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)**3/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError