3.272 \(\int x^m \sinh ^3(a+b \log (c x^n)) \, dx\)
Optimal. Leaf size=203 \[ \frac{(m+1) x^{m+1} \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-9 b^2 n^2}+\frac{6 b^2 (m+1) n^2 x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right )}{\left ((m+1)^2-9 b^2 n^2\right ) \left ((m+1)^2-b^2 n^2\right )}-\frac{6 b^3 n^3 x^{m+1} \cosh \left (a+b \log \left (c x^n\right )\right )}{\left ((m+1)^2-9 b^2 n^2\right ) \left ((m+1)^2-b^2 n^2\right )}-\frac{3 b n x^{m+1} \sinh ^2\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-9 b^2 n^2} \]
[Out]
(-6*b^3*n^3*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/(((1 + m)^2 - 9*b^2*n^2)*((1 + m)^2 - b^2*n^2)) + (6*b^2*(1 + m)
*n^2*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/(((1 + m)^2 - 9*b^2*n^2)*((1 + m)^2 - b^2*n^2)) - (3*b*n*x^(1 + m)*Cosh
[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^2)/((1 + m)^2 - 9*b^2*n^2) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n
]]^3)/((1 + m)^2 - 9*b^2*n^2)
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Rubi [A] time = 0.0862306, antiderivative size = 197, normalized size of antiderivative =
0.97, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules
used = {5529, 5527} \[ \frac{(m+1) x^{m+1} \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-9 b^2 n^2}+\frac{6 b^2 (m+1) n^2 x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right )}{-10 b^2 (m+1)^2 n^2+9 b^4 n^4+(m+1)^4}-\frac{6 b^3 n^3 x^{m+1} \cosh \left (a+b \log \left (c x^n\right )\right )}{-10 b^2 (m+1)^2 n^2+9 b^4 n^4+(m+1)^4}-\frac{3 b n x^{m+1} \sinh ^2\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-9 b^2 n^2} \]
Antiderivative was successfully verified.
[In]
Int[x^m*Sinh[a + b*Log[c*x^n]]^3,x]
[Out]
(-6*b^3*n^3*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/((1 + m)^4 - 10*b^2*(1 + m)^2*n^2 + 9*b^4*n^4) + (6*b^2*(1 + m)*
n^2*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/((1 + m)^4 - 10*b^2*(1 + m)^2*n^2 + 9*b^4*n^4) - (3*b*n*x^(1 + m)*Cosh[a
+ b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^2)/((1 + m)^2 - 9*b^2*n^2) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]]
^3)/((1 + m)^2 - 9*b^2*n^2)
Rule 5529
Int[((e_.)*(x_))^(m_.)*Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> -Simp[((m + 1)*(e*
x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2), x] + (-Dist[(b^2*d^2*n^2*p*(p - 1)
)/(b^2*d^2*n^2*p^2 - (m + 1)^2), Int[(e*x)^m*Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^
(m + 1)*Cosh[d*(a + b*Log[c*x^n])]*Sinh[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2), x])
/; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]
Rule 5527
Int[((e_.)*(x_))^(m_.)*Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> -Simp[((m + 1)*(e*x)^(m
+ 1)*Sinh[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 - e*(m + 1)^2), x] + Simp[(b*d*n*(e*x)^(m + 1)*Cosh[d*(a + b*
Log[c*x^n])])/(b^2*d^2*e*n^2 - e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1
)^2, 0]
Rubi steps
\begin{align*} \int x^m \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{3 b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac{(1+m) x^{1+m} \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac{\left (6 b^2 n^2\right ) \int x^m \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx}{(1+m)^2-9 b^2 n^2}\\ &=-\frac{6 b^3 n^3 x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^4-10 b^2 (1+m)^2 n^2+9 b^4 n^4}+\frac{6 b^2 (1+m) n^2 x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^4-10 b^2 (1+m)^2 n^2+9 b^4 n^4}-\frac{3 b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac{(1+m) x^{1+m} \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 1.32658, size = 292, normalized size = 1.44 \[ \frac{1}{4} x^{m+1} \left (-\frac{3 \cosh (b n \log (x)) \left ((m+1) \sinh \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-b n \cosh \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{(-b n+m+1) (b n+m+1)}-\frac{3 \sinh (b n \log (x)) \left ((m+1) \cosh \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-b n \sinh \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{(-b n+m+1) (b n+m+1)}+\frac{\cosh (3 b n \log (x)) \left ((m+1) \sinh \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-3 b n \cosh \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{(-3 b n+m+1) (3 b n+m+1)}+\frac{\sinh (3 b n \log (x)) \left ((m+1) \cosh \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-3 b n \sinh \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{(-3 b n+m+1) (3 b n+m+1)}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*Sinh[a + b*Log[c*x^n]]^3,x]
[Out]
(x^(1 + m)*((-3*Cosh[b*n*Log[x]]*(-(b*n*Cosh[a - b*n*Log[x] + b*Log[c*x^n]]) + (1 + m)*Sinh[a - b*n*Log[x] + b
*Log[c*x^n]]))/((1 + m - b*n)*(1 + m + b*n)) - (3*Sinh[b*n*Log[x]]*((1 + m)*Cosh[a - b*n*Log[x] + b*Log[c*x^n]
] - b*n*Sinh[a - b*n*Log[x] + b*Log[c*x^n]]))/((1 + m - b*n)*(1 + m + b*n)) + (Cosh[3*b*n*Log[x]]*(-3*b*n*Cosh
[3*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sinh[3*(a - b*n*Log[x] + b*Log[c*x^n])]))/((1 + m - 3*b*n)*(1 +
m + 3*b*n)) + (Sinh[3*b*n*Log[x]]*((1 + m)*Cosh[3*(a - b*n*Log[x] + b*Log[c*x^n])] - 3*b*n*Sinh[3*(a - b*n*Log
[x] + b*Log[c*x^n])]))/((1 + m - 3*b*n)*(1 + m + 3*b*n))))/4
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*sinh(a+b*ln(c*x^n))^3,x)
[Out]
int(x^m*sinh(a+b*ln(c*x^n))^3,x)
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Maxima [A] time = 1.23049, size = 186, normalized size = 0.92 \begin{align*} \frac{c^{3 \, b} x e^{\left (3 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) + 3 \, a\right )}}{8 \,{\left (3 \, b n + m + 1\right )}} - \frac{3 \, c^{b} x e^{\left (b \log \left (x^{n}\right ) + m \log \left (x\right ) + a\right )}}{8 \,{\left (b n + m + 1\right )}} - \frac{3 \, x e^{\left (-b \log \left (x^{n}\right ) + m \log \left (x\right ) - a\right )}}{8 \,{\left (b c^{b} n - c^{b}{\left (m + 1\right )}\right )}} + \frac{x e^{\left (-3 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) - 3 \, a\right )}}{8 \,{\left (3 \, b c^{3 \, b} n - c^{3 \, b}{\left (m + 1\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*sinh(a+b*log(c*x^n))^3,x, algorithm="maxima")
[Out]
1/8*c^(3*b)*x*e^(3*b*log(x^n) + m*log(x) + 3*a)/(3*b*n + m + 1) - 3/8*c^b*x*e^(b*log(x^n) + m*log(x) + a)/(b*n
+ m + 1) - 3/8*x*e^(-b*log(x^n) + m*log(x) - a)/(b*c^b*n - c^b*(m + 1)) + 1/8*x*e^(-3*b*log(x^n) + m*log(x) -
3*a)/(3*b*c^(3*b)*n - c^(3*b)*(m + 1))
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Fricas [B] time = 2.16783, size = 1542, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*sinh(a+b*log(c*x^n))^3,x, algorithm="fricas")
[Out]
1/4*(3*(b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a)^3*cosh(m*log(x)) - 3*(9*b^3*n^3 - (
b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a)*cosh(m*log(x)) + ((m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3
*m + 1)*x*cosh(m*log(x)) + (m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*sinh(m*log(x)))*sinh(b*n*log(x) + b*l
og(c) + a)^3 + 9*((b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a)*cosh(m*log(x)) + (b^3*n^
3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a)*sinh(m*log(x)))*sinh(b*n*log(x) + b*log(c) + a)^2
+ 3*((m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^2*cosh(m*log(x)) - (m^3 -
9*(b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(m*log(x)) + ((m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(
b*n*log(x) + b*log(c) + a)^2 - (m^3 - 9*(b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x)*sinh(m*log(x)))*sinh(b*n*log(x
) + b*log(c) + a) + 3*((b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a)^3 - (9*b^3*n^3 - (b
*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a))*sinh(m*log(x)))/(9*b^4*n^4 + m^4 + 4*m^3 - 10*(b^2*m^2
+ 2*b^2*m + b^2)*n^2 + 6*m^2 + 4*m + 1)
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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(x**m*sinh(a+b*ln(c*x**n))**3,x)
[Out]
Timed out
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Giac [B] time = 1.44072, size = 4354, normalized size = 21.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*sinh(a+b*log(c*x^n))^3,x, algorithm="giac")
[Out]
3/8*b^3*c^(3*b)*n^3*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*
m^3 + 6*m^2 + 4*m + 1) - 27/8*b^3*c^b*n^3*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 -
10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 1/8*b^2*c^(3*b)*m*n^2*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2
*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 27/8*b^2*c^b*m*n^2*x*x^(b*n)*x^m*e^a/(9*b^
4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/8*b*c^(3*b)*m^2*n*x*x^
(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) -
1/8*b^2*c^(3*b)*n^2*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4
*m^3 + 6*m^2 + 4*m + 1) + 3/8*b*c^b*m^2*n*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 -
10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 27/8*b^2*c^b*n^2*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b
^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 1/8*c^(3*b)*m^3*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 -
10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/4*b*c^(3*b)*m*n*x*x^(3*b*n)*x^
m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/8*c^b*m
^3*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)
+ 3/4*b*c^b*m*n*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^
2 + 4*m + 1) + 3/8*c^(3*b)*m^2*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b
^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/8*b*c^(3*b)*n*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b
^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 27/8*b^3*n^3*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^
2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 3/8*b^3*n^3*x*x^m*e^(-3*a)/((9*b
^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) - 9/8*
c^b*m^2*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m
+ 1) + 3/8*b*c^b*n*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6
*m^2 + 4*m + 1) + 3/8*c^(3*b)*m*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*
b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 27/8*b^2*m*n^2*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 +
m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 1/8*b^2*m*n^2*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*
m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) - 9/8*c^b*m*x*x^(b*n)*
x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 1/8*c^(3*b)
*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m +
1) + 3/8*b*m^2*n*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2
+ 4*m + 1)*c^b*x^(b*n)) - 27/8*b^2*n^2*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2
*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) - 3/8*b*m^2*n*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^
2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 1/8*b^2*n^2*x*x^m*e^(-3*a)/((9*b^4*
n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) - 3/8*c^b
*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) +
3/8*m^3*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)
*c^b*x^(b*n)) + 3/4*b*m*n*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3
+ 6*m^2 + 4*m + 1)*c^b*x^(b*n)) - 1/8*m^3*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 1
0*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) - 3/4*b*m*n*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^
2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 9/8*m^2*x*x^m*e^(-a)/((9*b
^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 3/8*b*n*x*
x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*
n)) - 3/8*m^2*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 +
4*m + 1)*c^(3*b)*x^(3*b*n)) - 3/8*b*n*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^
2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 9/8*m*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2
*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) - 3/8*m*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*
m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 3/8*x*x^m*e^(-a)/((9
*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) - 1/8*x*x^
m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x
^(3*b*n))