3.2.38 \(\int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [138]

Optimal. Leaf size=70 \[ -\frac {(a+2 b) \log (\cosh (c+d x))}{b^2 d}+\frac {(a+b)^2 \log \left (b+a \cosh ^2(c+d x)\right )}{2 a b^2 d}-\frac {\text {sech}^2(c+d x)}{2 b d} \]

[Out]

-(a+2*b)*ln(cosh(d*x+c))/b^2/d+1/2*(a+b)^2*ln(b+a*cosh(d*x+c)^2)/a/b^2/d-1/2*sech(d*x+c)^2/b/d

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Rubi [A]
time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 457, 90} \begin {gather*} \frac {(a+b)^2 \log \left (a \cosh ^2(c+d x)+b\right )}{2 a b^2 d}-\frac {(a+2 b) \log (\cosh (c+d x))}{b^2 d}-\frac {\text {sech}^2(c+d x)}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]

[Out]

-(((a + 2*b)*Log[Cosh[c + d*x]])/(b^2*d)) + ((a + b)^2*Log[b + a*Cosh[c + d*x]^2])/(2*a*b^2*d) - Sech[c + d*x]
^2/(2*b*d)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4223

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

Rubi steps

\begin {align*} \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^3 \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(1-x)^2}{x^2 (b+a x)} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{b x^2}+\frac {-a-2 b}{b^2 x}+\frac {(a+b)^2}{b^2 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {(a+2 b) \log (\cosh (c+d x))}{b^2 d}+\frac {(a+b)^2 \log \left (b+a \cosh ^2(c+d x)\right )}{2 a b^2 d}-\frac {\text {sech}^2(c+d x)}{2 b d}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 98, normalized size = 1.40 \begin {gather*} -\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (2 a (a+2 b) \log (\cosh (c+d x))-(a+b)^2 \log \left (a+b+a \sinh ^2(c+d x)\right )+a b \text {sech}^2(c+d x)\right )}{4 a b^2 d \left (a+b \text {sech}^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]

[Out]

-1/4*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(2*a*(a + 2*b)*Log[Cosh[c + d*x]] - (a + b)^2*Log[a + b
+ a*Sinh[c + d*x]^2] + a*b*Sech[c + d*x]^2))/(a*b^2*d*(a + b*Sech[c + d*x]^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(66)=132\).
time = 1.91, size = 183, normalized size = 2.61

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 \left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{b^{2} a}-\frac {\left (2 b +a \right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 b}{\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 b}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{b^{2}}}{d}\)	\(183\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 \left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{b^{2} a}-\frac {\left (2 b +a \right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 b}{\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {2 b}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{b^{2}}}{d}\)	\(183\)
risch	\(-\frac {x}{a}-\frac {2 c}{a d}-\frac {2 \,{\mathrm e}^{2 d x +2 c}}{b d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 b^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{b d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d}-\frac {2 \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{b d}-\frac {\ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a}{b^{2} d}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.48, size = 131, normalized size = 1.87 \begin {gather*} \frac {d x + c}{a d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + b e^{\left (-4 \, d x - 4 \, c\right )} + b\right )} d} - \frac {{\left (a + 2 \, b\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a b^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

(d*x + c)/(a*d) - 2*e^(-2*d*x - 2*c)/((2*b*e^(-2*d*x - 2*c) + b*e^(-4*d*x - 4*c) + b)*d) - (a + 2*b)*log(e^(-2
*d*x - 2*c) + 1)/(b^2*d) + 1/2*(a^2 + 2*a*b + b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/
(a*b^2*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (66) = 132\).
time = 0.44, size = 736, normalized size = 10.51 \begin {gather*} -\frac {2 \, b^{2} d x \cosh \left (d x + c\right )^{4} + 8 \, b^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, b^{2} d x \sinh \left (d x + c\right )^{4} + 2 \, b^{2} d x + 4 \, {\left (b^{2} d x + a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, b^{2} d x \cosh \left (d x + c\right )^{2} + b^{2} d x + a b\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) + 2 \, {\left ({\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + 4 \, {\left ({\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (b^{2} d x \cosh \left (d x + c\right )^{3} + {\left (b^{2} d x + a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a b^{2} d \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} d \cosh \left (d x + c\right )^{2} + a b^{2} d + 2 \, {\left (3 \, a b^{2} d \cosh \left (d x + c\right )^{2} + a b^{2} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a b^{2} d \cosh \left (d x + c\right )^{3} + a b^{2} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

-1/2*(2*b^2*d*x*cosh(d*x + c)^4 + 8*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*b^2*d*x*sinh(d*x + c)^4 + 2*b^2*
d*x + 4*(b^2*d*x + a*b)*cosh(d*x + c)^2 + 4*(3*b^2*d*x*cosh(d*x + c)^2 + b^2*d*x + a*b)*sinh(d*x + c)^2 - ((a^
2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*s
inh(d*x + c)^4 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*
b + b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*co
sh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(
d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 2*((a^2 + 2*a*b)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b)*cosh(d*x + c)*
sinh(d*x + c)^3 + (a^2 + 2*a*b)*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b)*cosh(d*
x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 2*a*b + 4*((a^2 + 2*a*b)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(
d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(b^2*d*x*cosh(d*x + c)^3 + (
b^2*d*x + a*b)*cosh(d*x + c))*sinh(d*x + c))/(a*b^2*d*cosh(d*x + c)^4 + 4*a*b^2*d*cosh(d*x + c)*sinh(d*x + c)^
3 + a*b^2*d*sinh(d*x + c)^4 + 2*a*b^2*d*cosh(d*x + c)^2 + a*b^2*d + 2*(3*a*b^2*d*cosh(d*x + c)^2 + a*b^2*d)*si
nh(d*x + c)^2 + 4*(a*b^2*d*cosh(d*x + c)^3 + a*b^2*d*cosh(d*x + c))*sinh(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{5}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)**5/(a+b*sech(d*x+c)**2),x)

[Out]

Integral(tanh(c + d*x)**5/(a + b*sech(c + d*x)**2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 1.81, size = 421, normalized size = 6.01 \begin {gather*} \frac {2}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {x}{a}-\frac {\ln \left (39\,a\,b^7+243\,a^7\,b+27\,a^8+2\,b^8+289\,a^2\,b^6+1017\,a^3\,b^5+1791\,a^4\,b^4+1701\,a^5\,b^3+891\,a^6\,b^2+27\,a^8\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,b^8\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+39\,a\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+243\,a^7\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+289\,a^2\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1017\,a^3\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1791\,a^4\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1701\,a^5\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+891\,a^6\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a+2\,b\right )}{b^2\,d}+\frac {\ln \left (a\,b^2+6\,a^2\,b+3\,a^3+6\,a^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,a^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+26\,a\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+24\,a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+6\,a^2\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,a\,b^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)^5/(a + b/cosh(c + d*x)^2),x)

[Out]

2/(b*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - 2/(b*d*(exp(2*c + 2*d*x) + 1)) - x/a - (log(39*a*b^7 + 2
43*a^7*b + 27*a^8 + 2*b^8 + 289*a^2*b^6 + 1017*a^3*b^5 + 1791*a^4*b^4 + 1701*a^5*b^3 + 891*a^6*b^2 + 27*a^8*ex
p(2*c)*exp(2*d*x) + 2*b^8*exp(2*c)*exp(2*d*x) + 39*a*b^7*exp(2*c)*exp(2*d*x) + 243*a^7*b*exp(2*c)*exp(2*d*x) +
289*a^2*b^6*exp(2*c)*exp(2*d*x) + 1017*a^3*b^5*exp(2*c)*exp(2*d*x) + 1791*a^4*b^4*exp(2*c)*exp(2*d*x) + 1701*
a^5*b^3*exp(2*c)*exp(2*d*x) + 891*a^6*b^2*exp(2*c)*exp(2*d*x))*(a + 2*b))/(b^2*d) + (log(a*b^2 + 6*a^2*b + 3*a
^3 + 6*a^3*exp(2*c)*exp(2*d*x) + 3*a^3*exp(4*c)*exp(4*d*x) + 4*b^3*exp(2*c)*exp(2*d*x) + 26*a*b^2*exp(2*c)*exp
(2*d*x) + 24*a^2*b*exp(2*c)*exp(2*d*x) + a*b^2*exp(4*c)*exp(4*d*x) + 6*a^2*b*exp(4*c)*exp(4*d*x))*(2*a*b + a^2
+ b^2))/(2*a*b^2*d)

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