\(\int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) [76]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-i \text {Int}\left (\frac {i \sinh (c+d x)}{a+b \tanh ^3(c+d x)},x\right ) \]

[Out]

-I*Unintegrable(I*sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[In]

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

[Out]

(-I)*Defer[Int][(I*Sinh[c + d*x])/(a + b*Tanh[c + d*x]^3), x]

Rubi steps \begin{align*} \text {integral}& = -\left (i \int \frac {i \sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(409\) vs. \(2(31)=62\).

Time = 0.72 (sec) , antiderivative size = 409, normalized size of antiderivative = 19.48 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {6 a \cosh (c+d x)+b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {2 a c+b c+2 a d x+b d x+4 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 a c \text {$\#$1}^4-b c \text {$\#$1}^4+2 a d x \text {$\#$1}^4-b d x \text {$\#$1}^4+4 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\&\right ]-6 b \sinh (c+d x)}{6 (a-b) (a+b) d} \]

[In]

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

[Out]

(6*a*Cosh[c + d*x] + b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (2*a*c
+ b*c + 2*a*d*x + b*d*x + 4*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*
x)/2]*#1] + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*
a*c*#1^4 - b*c*#1^4 + 2*a*d*x*#1^4 - b*d*x*#1^4 + 4*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d
*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1
- Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ] - 6*b*Sinh[c + d*x])/
(6*(a - b)*(a + b)*d)

Maple [N/A] (verified)

Time = 1.62 (sec) , antiderivative size = 159, normalized size of antiderivative = 7.57

method	result	size
derivativedivides	\(\frac {-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} a -2 \textit {\_R}^{3} b +6 \textit {\_R}^{2} a -2 \textit {\_R} b +a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a -b \right ) \left (a +b \right )}+\frac {4}{\left (4 a -4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\)	\(159\)
default	\(\frac {-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} a -2 \textit {\_R}^{3} b +6 \textit {\_R}^{2} a -2 \textit {\_R} b +a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a -b \right ) \left (a +b \right )}+\frac {4}{\left (4 a -4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\)	\(159\)
risch	\(\text {Expression too large to display}\)	\(1209\)

[In]

int(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(-4/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)-1)+1/3*b/(a-b)/(a+b)*sum((_R^4*a-2*_R^3*b+6*_R^2*a-2*_R*b+a)/(_R^5*a+2*
_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))+4/(4*a-4*b)/(
1+tanh(1/2*d*x+1/2*c)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.74 (sec) , antiderivative size = 40923, normalized size of antiderivative = 1948.71 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 250, normalized size of antiderivative = 11.90 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 3.42 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 83.41 (sec) , antiderivative size = 4474, normalized size of antiderivative = 213.05 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

exp(- c - d*x)/(2*(a*d - b*d)) + symsum(log((81920*a^2*b^5*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b
^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d
^2*z^2 - b^2, z, k)) + 221184*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8
*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^2*b^8*d^3 - 353894
4*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*
z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b^7*d^3 + 1990656*root(2187*a^6*b^2*d^6*z^6
- 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 +
81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^4*b^6*d^3 + 3538944*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729
*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z,
k)^3*a^5*b^5*d^3 - 2211840*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d
^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^6*b^4*d^3 + 7962624*
root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^
4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^3*b^9*d^5 + 15925248*root(2187*a^6*b^2*d^6*z^6 -
2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 8
1*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^8*d^5 - 7962624*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*
a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z,
k)^5*a^5*b^7*d^5 - 31850496*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d
^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^6*b^6*d^5 - 7962624*
root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^
4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*b^5*d^5 + 15925248*root(2187*a^6*b^2*d^6*z^6 -
2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 8
1*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^8*b^4*d^5 + 7962624*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*
a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z,
k)^5*a^9*b^3*d^5 + 98304*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*
z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)*a^2*b^6*d - 98304*root(2187
*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a
^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)*a^3*b^5*d + 24576*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^
6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^
2 - b^2, z, k)*a^4*b^4*d + 8192*a*b^6*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*
b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k))
+ 368640*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b
^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^2*b^7*d^2*exp(d*x)*exp(root(2187*a^6*b^
2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*
d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) - 2285568*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^
2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)
^2*a^3*b^6*d^2*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d
^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) - 5013504*root(2187*a^6
*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b
^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^4*b^5*d^2*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b
^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d
^2*z^2 - b^2, z, k)) - 368640*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8
*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^5*b^4*d^2*exp(d*x)
*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d
^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) + 8626176*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4
*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2
*d^2*z^2 - b^2, z, k)^4*a^3*b^8*d^4*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^
6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) +
40476672*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b
^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^4*b^7*d^4*exp(d*x)*exp(root(2187*a^6*b^
2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*
d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) + 70336512*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a
^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k
)^4*a^5*b^6*d^4*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*
d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) + 54411264*root(2187*a
^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2
*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^6*b^5*d^4*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4
*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2
*d^2*z^2 - b^2, z, k)) + 16588800*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729
*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^7*b^4*d^4*exp(
d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b
^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) + 663552*root(2187*a^6*b^2*d^6*z^6 - 2187*
a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*
b^2*d^2*z^2 - b^2, z, k)^4*a^8*b^3*d^4*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2
*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k))
)/(10*a*b^11 - 10*a^11*b - a^12 + b^12 + 44*a^2*b^10 + 110*a^3*b^9 + 165*a^4*b^8 + 132*a^5*b^7 - 132*a^7*b^5 -
165*a^8*b^4 - 110*a^9*b^3 - 44*a^10*b^2))*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*
z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k), k, 1, 6)
+ exp(c + d*x)/(2*(a*d + b*d))