3.36 \(\int \frac{\sinh (c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{3 \cosh (c+d x)}{2 d (a+b)^2}-\frac{\cosh (c+d x)}{2 d (a+b) \left (a-b \text{sech}^2(c+d x)+b\right )}-\frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{2 d (a+b)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0746356, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3664, 290, 325, 208} \[ \frac{3 \cosh (c+d x)}{2 d (a+b)^2}-\frac{\cosh (c+d x)}{2 d (a+b) \left (a-b \text{sech}^2(c+d x)+b\right )}-\frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{2 d (a+b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/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 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{\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b-b x^2\right )^2} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text{sech}^2(c+d x)\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b-b x^2\right )} \, dx,x,\text{sech}(c+d x)\right )}{2 (a+b) d}\\ &=\frac{3 \cosh (c+d x)}{2 (a+b)^2 d}-\frac{\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text{sech}^2(c+d x)\right )}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\text{sech}(c+d x)\right )}{2 (a+b)^2 d}\\ &=-\frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{2 (a+b)^{5/2} d}+\frac{3 \cosh (c+d x)}{2 (a+b)^2 d}-\frac{\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text{sech}^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 0.683644, size = 133, normalized size = 1.45 \[ \frac{\frac{2 \cosh (c+d x) \left (1-\frac{b}{(a+b) \cosh (2 (c+d x))+a-b}\right )}{(a+b)^2}-\frac{3 i \sqrt{b} \left (\tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )\right )}{(a+b)^{5/2}}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] +
Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(5/2) + (2*Cosh[c + d*x]*(1 - b/(a - b + (a + b)*Cosh[2*(c + d*x
)])))/(a + b)^2)/(2*d)

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Maple [B]  time = 0.079, size = 167, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({\frac{1}{ \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+2\,{\frac{b}{ \left ( a+b \right ) ^{2}} \left ({\frac{1}{ \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} \left ( -1/2\,{\frac{ \left ( a+2\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{a}}-1/2 \right ) }-3/4\,{\frac{1}{\sqrt{ab+{b}^{2}}}{\it Artanh} \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 ) } \right ) }-{\frac{1}{ \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/(a+b)^2/(tanh(1/2*d*x+1/2*c)+1)+2*b/(a+b)^2*((-1/2*(a+2*b)/a*tanh(1/2*d*x+1/2*c)^2-1/2)/(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)-3/4/(a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d
*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2)))-1/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.58527, size = 5933, normalized size = 64.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(a + b)*cosh(d*x + c)^6 + 12*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a + b)*sinh(d*x + c)^6 + 6*(a
- b)*cosh(d*x + c)^4 + 6*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^4 + 8*(5*(a + b)*cosh(d*x + c)^3 +
3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(a - b)*cosh(d*x + c)^2 + 6*(5*(a + b)*cosh(d*x + c)^4 + 6*(a - b
)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 3*((a + b)*cosh(d*x + c)^5 + 5*(a + b)*cosh(d*x + c)*sinh(d*x + c
)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c
)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + (a + b)*cosh(d*x + c) + (5*(a
+ b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(b/(a + b))*log(((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 + 3*b)*cosh(d*x + c)^2 + 2*
(3*(a + b)*cosh(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (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)) + a + b)/((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)) + 12*((a + b)*cosh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))
*sinh(d*x + c) + 2*a + 2*b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*sinh(d*x + c)^5 + 2*(a^3 + a^2*b - a*
b^2 - b^3)*d*cosh(d*x + c)^3 + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^2 + (a^3 + a^2*b - a*b^2 -
b^3)*d)*sinh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c) + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*d*cosh(d*x + c)^3 + 3*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*(a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*d*cosh(d*x + c)^4 + 6*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*d)*sinh(d*x + c)), 1/2*((a + b)*cosh(d*x + c)^6 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a + b)*sinh
(d*x + c)^6 + 3*(a - b)*cosh(d*x + c)^4 + 3*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^4 + 4*(5*(a + b)
*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c)^2 + 3*(5*(a + b)*cosh(d*
x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 - 3*((a + b)*cosh(d*x + c)^5 + 5*(a + b)*cosh(d*
x + c)*sinh(d*x + c)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^2 +
a - b)*sinh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + (a + b)*cos
h(d*x + c) + (5*(a + b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(-b/(a + b))*a
rctan(1/2*((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 -
3*b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a - 3*b)*sinh(d*x + c))*sqrt(-b/(a + b))/b) + 3*((a + b)*cos
h(d*x + c)^5 + 5*(a + b)*cosh(d*x + c)*sinh(d*x + c)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 +
2*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c
))*sinh(d*x + c)^2 + (a + b)*cosh(d*x + c) + (5*(a + b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*s
inh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-b/(a + b))/b)
+ 6*((a + b)*cosh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)/((a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)*sinh(d*x + c
)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*sinh(d*x + c)^5 + 2*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c)^3 + 2*
(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^2 + (a^3 + a^2*b - a*b^2 - b^3)*d)*sinh(d*x + c)^3 + (a^3 +
3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c) + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^3 + 3*(a^3 +
a^2*b - a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^4 +
6*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d)*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(sinh(d*x+c)/(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [C]  time = 1.69834, size = 6742, normalized size = 73.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(6*(3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(
a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a +
b)))) - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(
a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3 - 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(
2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) +
b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a
+ b)))) + 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) +
b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(
a + b)))) + 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) +
b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*
c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(
2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b
/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2
*c))*sqrt(-a*b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b
/(a + b))))^3 - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b
) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) + (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^
(2*c))*sqrt(-a*b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b
/(a + b)))))*arctan((((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e
^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3
*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a^4*b*e^(2*c) + 3*
a^3*b^2*e^(2*c) + 3*a^2*b^3*e^(2*c) + a*b^4*e^(2*c)) + 6*(3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqr
t(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)
)))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt
(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))
))^3 - 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a +
b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*s
qrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a +
b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))
*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a
+ b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))
^2 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a
+ b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))
))^2 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)
)))^3 + (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c
))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b)))) + (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(
a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arctan(-(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*
e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) - e^(d*x))
/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*sin(
1/2*arccos(-(a - b)/(a + b)))))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c) + 3*a^2*b^3*e^(2*c) + a*b^4*e^(2*c)) + 3*((
2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3
*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt
(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))
^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqr
t(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)
)))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*s
qrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)
)))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))
+ 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b
))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))
^2 - 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a +
b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^
2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(
-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))
)^3 + 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))
))^3 - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a +
b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sq
rt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))
)))*log(2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1
/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^
(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*d*x))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c) + 3*a^2*b^3*e^(2*c) +
a*b^4*e^(2*c)) - 3*((2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c
) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(
a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*
c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-
a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a*b^2*e^(2*c) - (a*b*e^(
2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-
a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(
a + b) + b/(a + b)))) + 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(
-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(
a + b) + b/(a + b))))^2 - 9*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a*b^2*e^(2*c) - (a*b*e^(2*c)
- b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/
(a + b) + b/(a + b))))^3 + 3*(2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arcc
os(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a
/(a + b) + b/(a + b))))^3 - (2*a*b^2*e^(2*c) - (a*b*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + (2*a*b^2*e^(2*c) - (a*b*e^(2
*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a
/(a + b) + b/(a + b)))))*log(-2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4
*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^
3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*d*x))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c)
+ 3*a^2*b^3*e^(2*c) + a*b^4*e^(2*c)) + 4*e^(d*x + 10*c)/(a^2*e^(9*c) + 2*a*b*e^(9*c) + b^2*e^(9*c)) + 4*(a*e^(
4*d*x + 4*c) - b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 4*b*e^(2*d*x + 2*c) + a + b)/((a^2*e^c + 2*a*b*e^c +
b^2*e^c)*(a*e^(5*d*x + 4*c) + b*e^(5*d*x + 4*c) + 2*a*e^(3*d*x + 2*c) - 2*b*e^(3*d*x + 2*c) + a*e^(d*x) + b*e^
(d*x))))/d