3.4.18 \(\int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [318]

Optimal. Leaf size=77 \[ \frac {(a-b)^2 \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a-2 b) \sinh (c+d x)}{b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]

[Out]

-(a-2*b)*sinh(d*x+c)/b^2/d+1/3*sinh(d*x+c)^3/b/d+(a-b)^2*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/b^(5/2)/d/a^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 77, 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 = {3269, 398, 211} \begin {gather*} \frac {(a-b)^2 \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a-2 b) \sinh (c+d x)}{b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)^2*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a - 2*b)*Sinh[c + d*x])/(b^2*d) +
Sinh[c + d*x]^3/(3*b*d)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3269

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

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 79, normalized size = 1.03 \begin {gather*} \frac {-\frac {12 (a-b)^2 \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )}{\sqrt {a}}+3 \sqrt {b} (-4 a+7 b) \sinh (c+d x)+b^{3/2} \sinh (3 (c+d x))}{12 b^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*(a - b)^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + 3*Sqrt[b]*(-4*a + 7*b)*Sinh[c + d*x] + b^(3
/2)*Sinh[3*(c + d*x)])/(12*b^(5/2)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(67)=134\).
time = 1.78, size = 318, normalized size = 4.13

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (a^{2}-2 a b +b^{2}\right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 b -a}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 b -a}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(318\)
default	\(\frac {\frac {2 a \left (a^{2}-2 a b +b^{2}\right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 b -a}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 b -a}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(318\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{d x +c}}{2 b^{2} d}+\frac {7 \,{\mathrm e}^{d x +c}}{8 b d}+\frac {{\mathrm e}^{-d x -c} a}{2 b^{2} d}-\frac {7 \,{\mathrm e}^{-d x -c}}{8 b d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{\sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{\sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}\)	\(347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*(4*a*e^(4*c) - 7*b*e^(4*c))*e^(4*d*x) - 3*(4*a*e^(2*c) - 7*b*e^(2*c))*e^(2*d*x) - b*e^(6*d*x + 6*c) +
b)*e^(-3*d*x - 3*c)/(b^2*d) + 1/32*integrate(64*((a^2*e^(3*c) - 2*a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (a^2
*e^c - 2*a*b*e^c + b^2*e^c)*e^(d*x))/(b^3*e^(4*d*x + 4*c) + b^3 + 2*(2*a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x))
, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (67) = 134\).
time = 0.45, size = 1490, normalized size = 19.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(a*b^2*cosh(d*x + c)^6 + 6*a*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + a*b^2*sinh(d*x + c)^6 - 3*(4*a^2*b - 7*
a*b^2)*cosh(d*x + c)^4 + 3*(5*a*b^2*cosh(d*x + c)^2 - 4*a^2*b + 7*a*b^2)*sinh(d*x + c)^4 + 4*(5*a*b^2*cosh(d*x
+ c)^3 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a*b^2 + 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2
+ 3*(5*a*b^2*cosh(d*x + c)^4 + 4*a^2*b - 7*a*b^2 - 6*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 1
2*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b
+ b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^3)*sqrt(-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 + 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) - 4*(cosh(d*x + c)
^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c)
)*sqrt(-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)*cos
h(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)) + 6*(a*b^2*cosh(d*x + c)^5 - 2*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^3 + (4*a^2*b - 7*a*b
^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^3 + 3*a*b^3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*b^3
*d*cosh(d*x + c)*sinh(d*x + c)^2 + a*b^3*d*sinh(d*x + c)^3), 1/24*(a*b^2*cosh(d*x + c)^6 + 6*a*b^2*cosh(d*x +
c)*sinh(d*x + c)^5 + a*b^2*sinh(d*x + c)^6 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^4 + 3*(5*a*b^2*cosh(d*x + c)^
2 - 4*a^2*b + 7*a*b^2)*sinh(d*x + c)^4 + 4*(5*a*b^2*cosh(d*x + c)^3 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c))*sin
h(d*x + c)^3 - a*b^2 + 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2 + 3*(5*a*b^2*cosh(d*x + c)^4 + 4*a^2*b - 7*a*b^2
- 6*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 -
2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 - 2*a*
b + b^2)*sinh(d*x + c)^3)*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) + 24*((a^2 - 2*a*b
+ b^2)*cosh(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)*cosh(d*x
+ c)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^3)*sqrt(a*b)*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cos
h(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - b)*sin
h(d*x + c))*sqrt(a*b)/(a*b)) + 6*(a*b^2*cosh(d*x + c)^5 - 2*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^3 + (4*a^2*b - 7
*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^3 + 3*a*b^3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*a
*b^3*d*cosh(d*x + c)*sinh(d*x + c)^2 + a*b^3*d*sinh(d*x + c)^3)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**5/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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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(cosh(d*x+c)^5/(a+b*sinh(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 = 1.28, size = 668, normalized size = 8.68 \begin {gather*} \frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,{\left (a-b\right )}^2\,\sqrt {a\,b^5\,d^2}}{2\,a\,b^2\,d\,\sqrt {{\left (a-b\right )}^4}}\right )-2\,\mathrm {atan}\left (\frac {a\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a\,b^5\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}-6\,a^2\,b^4\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}+6\,a^3\,b^3\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}-2\,a^4\,b^2\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}\right )}{a^2\,b^{11}\,d^2\,{\left (a-b\right )}^2}+\frac {2\,\left (a^5\,\sqrt {a\,b^5\,d^2}-b^5\,\sqrt {a\,b^5\,d^2}+5\,a\,b^4\,\sqrt {a\,b^5\,d^2}-5\,a^4\,b\,\sqrt {a\,b^5\,d^2}-10\,a^2\,b^3\,\sqrt {a\,b^5\,d^2}+10\,a^3\,b^2\,\sqrt {a\,b^5\,d^2}\right )}{a^2\,b^8\,d\,\sqrt {{\left (a-b\right )}^4}\,\sqrt {a\,b^5\,d^2}}\right )\,\sqrt {a\,b^5\,d^2}}{4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3}-\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (a^5\,\sqrt {a\,b^5\,d^2}-b^5\,\sqrt {a\,b^5\,d^2}+5\,a\,b^4\,\sqrt {a\,b^5\,d^2}-5\,a^4\,b\,\sqrt {a\,b^5\,d^2}-10\,a^2\,b^3\,\sqrt {a\,b^5\,d^2}+10\,a^3\,b^2\,\sqrt {a\,b^5\,d^2}\right )}{a\,b^2\,d\,\sqrt {{\left (a-b\right )}^4}\,\left (4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3\right )}\right )\right )\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}}{2\,\sqrt {a\,b^5\,d^2}}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a-7\,b\right )}{8\,b^2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a-7\,b\right )}{8\,b^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*atan((exp(d*x)*exp(c)*(a - b)^2*(a*b^5*d^2)^(1/2))/(2*a*b^2*d*((a - b)^4)^(1/2))) - 2*atan((a*b^6*exp(d*x)
*exp(c)*((4*(2*a*b^5*d*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2) - 6*a^2*b^4*d*(a^4 - 4*a^3*b - 4*a*b^
3 + b^4 + 6*a^2*b^2)^(1/2) + 6*a^3*b^3*d*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2) - 2*a^4*b^2*d*(a^4
- 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2)))/(a^2*b^11*d^2*(a - b)^2) + (2*(a^5*(a*b^5*d^2)^(1/2) - b^5*(a*b
^5*d^2)^(1/2) + 5*a*b^4*(a*b^5*d^2)^(1/2) - 5*a^4*b*(a*b^5*d^2)^(1/2) - 10*a^2*b^3*(a*b^5*d^2)^(1/2) + 10*a^3*
b^2*(a*b^5*d^2)^(1/2)))/(a^2*b^8*d*((a - b)^4)^(1/2)*(a*b^5*d^2)^(1/2)))*(a*b^5*d^2)^(1/2))/(12*a*b^2 - 12*a^2
*b + 4*a^3 - 4*b^3) - (2*exp(3*c)*exp(3*d*x)*(a^5*(a*b^5*d^2)^(1/2) - b^5*(a*b^5*d^2)^(1/2) + 5*a*b^4*(a*b^5*d
^2)^(1/2) - 5*a^4*b*(a*b^5*d^2)^(1/2) - 10*a^2*b^3*(a*b^5*d^2)^(1/2) + 10*a^3*b^2*(a*b^5*d^2)^(1/2)))/(a*b^2*d
*((a - b)^4)^(1/2)*(12*a*b^2 - 12*a^2*b + 4*a^3 - 4*b^3))))*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2))
/(2*(a*b^5*d^2)^(1/2)) - exp(- 3*c - 3*d*x)/(24*b*d) + exp(3*c + 3*d*x)/(24*b*d) - (exp(c + d*x)*(4*a - 7*b))/
(8*b^2*d) + (exp(- c - d*x)*(4*a - 7*b))/(8*b^2*d)

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