3.1.14 \(\int \frac {\sinh ^4(x)}{a+b \cosh ^2(x)} \, dx\) [14]

Optimal. Leaf size=59 \[ -\frac {(2 a+3 b) x}{2 b^2}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} b^2}+\frac {\cosh (x) \sinh (x)}{2 b} \]

[Out]

-1/2*(2*a+3*b)*x/b^2+1/2*cosh(x)*sinh(x)/b+(a+b)^(3/2)*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/b^2/a^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3270, 425, 536, 212, 214} \begin {gather*} -\frac {x (2 a+3 b)}{2 b^2}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} b^2}+\frac {\sinh (x) \cosh (x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^4/(a + b*Cosh[x]^2),x]

[Out]

-1/2*((2*a + 3*b)*x)/b^2 + ((a + b)^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b^2) + (Cosh[x]*Sin
h[x])/(2*b)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 3270

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

Rubi steps

\begin {align*} \int \frac {\sinh ^4(x)}{a+b \cosh ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-(a+b) x^2\right )} \, dx,x,\coth (x)\right )\\ &=\frac {\cosh (x) \sinh (x)}{2 b}-\frac {\text {Subst}\left (\int \frac {-a-2 b+(-a-b) x^2}{\left (1-x^2\right ) \left (a+(-a-b) x^2\right )} \, dx,x,\coth (x)\right )}{2 b}\\ &=\frac {\cosh (x) \sinh (x)}{2 b}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{a+(-a-b) x^2} \, dx,x,\coth (x)\right )}{b^2}-\frac {(2 a+3 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (x)\right )}{2 b^2}\\ &=-\frac {(2 a+3 b) x}{2 b^2}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} b^2}+\frac {\cosh (x) \sinh (x)}{2 b}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 52, normalized size = 0.88 \begin {gather*} \frac {-4 a x-6 b x+\frac {4 (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a}}+b \sinh (2 x)}{4 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^4/(a + b*Cosh[x]^2),x]

[Out]

(-4*a*x - 6*b*x + (4*(a + b)^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/Sqrt[a] + b*Sinh[2*x])/(4*b^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(47)=94\).
time = 0.74, size = 185, normalized size = 3.14

method	result	size
default	\(\frac {2 \left (a^{2}+2 a b +b^{2}\right ) \left (\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{b^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\left (2 a +3 b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\left (-2 a -3 b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b^{2}}\)	\(185\)
risch	\(-\frac {a x}{b^{2}}-\frac {3 x}{2 b}+\frac {{\mathrm e}^{2 x}}{8 b}-\frac {{\mathrm e}^{-2 x}}{8 b}+\frac {\sqrt {a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {a \left (a +b \right )}-2 a -b}{b}\right )}{2 b^{2}}+\frac {\sqrt {a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {a \left (a +b \right )}-2 a -b}{b}\right )}{2 a b}-\frac {\sqrt {a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {a \left (a +b \right )}+2 a +b}{b}\right )}{2 b^{2}}-\frac {\sqrt {a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {a \left (a +b \right )}+2 a +b}{b}\right )}{2 a b}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^4/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)

[Out]

2/b^2*(a^2+2*a*b+b^2)*(1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2+2*tanh(1/2*x)*a^(1/2)+(a+b)^(1/2))
-1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2-2*tanh(1/2*x)*a^(1/2)+(a+b)^(1/2)))+1/2/b/(tanh(1/2*x)-1
)^2+1/2/b/(tanh(1/2*x)-1)+1/2*(2*a+3*b)/b^2*ln(tanh(1/2*x)-1)-1/2/b/(tanh(1/2*x)+1)^2+1/2/b/(tanh(1/2*x)+1)+1/
2/b^2*(-2*a-3*b)*ln(tanh(1/2*x)+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (47) = 94\).
time = 0.49, size = 348, normalized size = 5.90 \begin {gather*} \frac {{\left (2 \, a + b\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{4 \, \sqrt {{\left (a + b\right )} a} b} - \frac {3 \, \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{16 \, \sqrt {{\left (a + b\right )} a}} - \frac {{\left (2 \, a + b\right )} x}{b^{2}} - \frac {x}{b} + \frac {e^{\left (2 \, x\right )}}{8 \, b} - \frac {e^{\left (-2 \, x\right )}}{8 \, b} + \frac {{\left (2 \, a + b\right )} \log \left (b e^{\left (4 \, x\right )} + 2 \, {\left (2 \, a + b\right )} e^{\left (2 \, x\right )} + b\right )}{8 \, b^{2}} - \frac {{\left (2 \, a + b\right )} \log \left (2 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + b e^{\left (-4 \, x\right )} + b\right )}{8 \, b^{2}} + \frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{32 \, \sqrt {{\left (a + b\right )} a} b^{2}} - \frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{32 \, \sqrt {{\left (a + b\right )} a} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^4/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

1/4*(2*a + b)*log((b*e^(2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(2*x) + 2*a + b + 2*sqrt((a + b)*a)))/(sqrt((
a + b)*a)*b) - 3/16*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(-2*x) + 2*a + b + 2*sqrt((a + b)*a)))
/sqrt((a + b)*a) - (2*a + b)*x/b^2 - x/b + 1/8*e^(2*x)/b - 1/8*e^(-2*x)/b + 1/8*(2*a + b)*log(b*e^(4*x) + 2*(2
*a + b)*e^(2*x) + b)/b^2 - 1/8*(2*a + b)*log(2*(2*a + b)*e^(-2*x) + b*e^(-4*x) + b)/b^2 + 1/32*(8*a^2 + 8*a*b
+ b^2)*log((b*e^(2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(2*x) + 2*a + b + 2*sqrt((a + b)*a)))/(sqrt((a + b)*
a)*b^2) - 1/32*(8*a^2 + 8*a*b + b^2)*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(-2*x) + 2*a + b + 2*
sqrt((a + b)*a)))/(sqrt((a + b)*a)*b^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (47) = 94\).
time = 0.40, size = 568, normalized size = 9.63 \begin {gather*} \left [\frac {b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} - 4 \, {\left (2 \, a + 3 \, b\right )} x \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} - 2 \, {\left (2 \, a + 3 \, b\right )} x\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2}\right )} \sqrt {\frac {a + b}{a}} \log \left (\frac {b^{2} \cosh \left (x\right )^{4} + 4 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{2} \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + 8 \, a^{2} + 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \, {\left (a b \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) \sinh \left (x\right ) + a b \sinh \left (x\right )^{2} + 2 \, a^{2} + a b\right )} \sqrt {\frac {a + b}{a}}}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right ) + 4 \, {\left (b \cosh \left (x\right )^{3} - 2 \, {\left (2 \, a + 3 \, b\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right ) - b}{8 \, {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )}}, \frac {b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} - 4 \, {\left (2 \, a + 3 \, b\right )} x \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} - 2 \, {\left (2 \, a + 3 \, b\right )} x\right )} \sinh \left (x\right )^{2} + 8 \, {\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2}\right )} \sqrt {-\frac {a + b}{a}} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, a + b\right )} \sqrt {-\frac {a + b}{a}}}{2 \, {\left (a + b\right )}}\right ) + 4 \, {\left (b \cosh \left (x\right )^{3} - 2 \, {\left (2 \, a + 3 \, b\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right ) - b}{8 \, {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^4/(a+b*cosh(x)^2),x, algorithm="fricas")

[Out]

[1/8*(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 4*(2*a + 3*b)*x*cosh(x)^2 + 2*(3*b*cosh(x)^2 - 2*(2*
a + 3*b)*x)*sinh(x)^2 + 4*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2)*sqrt((a + b)/a)*
log((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*(2*a*b + b^2)*cosh(x)^2 + 2*(3*b^2*cosh(x)^2
+ 2*a*b + b^2)*sinh(x)^2 + 8*a^2 + 8*a*b + b^2 + 4*(b^2*cosh(x)^3 + (2*a*b + b^2)*cosh(x))*sinh(x) - 4*(a*b*co
sh(x)^2 + 2*a*b*cosh(x)*sinh(x) + a*b*sinh(x)^2 + 2*a^2 + a*b)*sqrt((a + b)/a))/(b*cosh(x)^4 + 4*b*cosh(x)*sin
h(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a +
b)*cosh(x))*sinh(x) + b)) + 4*(b*cosh(x)^3 - 2*(2*a + 3*b)*x*cosh(x))*sinh(x) - b)/(b^2*cosh(x)^2 + 2*b^2*cos
h(x)*sinh(x) + b^2*sinh(x)^2), 1/8*(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 4*(2*a + 3*b)*x*cosh(x
)^2 + 2*(3*b*cosh(x)^2 - 2*(2*a + 3*b)*x)*sinh(x)^2 + 8*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a +
b)*sinh(x)^2)*sqrt(-(a + b)/a)*arctan(1/2*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*a + b)*sqrt(-(a
+ b)/a)/(a + b)) + 4*(b*cosh(x)^3 - 2*(2*a + 3*b)*x*cosh(x))*sinh(x) - b)/(b^2*cosh(x)^2 + 2*b^2*cosh(x)*sinh
(x) + b^2*sinh(x)^2)]

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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(sinh(x)**4/(a+b*cosh(x)**2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (47) = 94\).
time = 0.40, size = 103, normalized size = 1.75 \begin {gather*} -\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} + \frac {e^{\left (2 \, x\right )}}{8 \, b} + \frac {{\left (4 \, a e^{\left (2 \, x\right )} + 6 \, b e^{\left (2 \, x\right )} - b\right )} e^{\left (-2 \, x\right )}}{8 \, b^{2}} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^4/(a+b*cosh(x)^2),x, algorithm="giac")

[Out]

-1/2*(2*a + 3*b)*x/b^2 + 1/8*e^(2*x)/b + 1/8*(4*a*e^(2*x) + 6*b*e^(2*x) - b)*e^(-2*x)/b^2 + (a^2 + 2*a*b + b^2
)*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/(sqrt(-a^2 - a*b)*b^2)

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Mupad [B]
time = 1.27, size = 146, normalized size = 2.47 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{8\,b}-\frac {{\mathrm {e}}^{-2\,x}}{8\,b}-\frac {x\,\left (2\,a+3\,b\right )}{2\,b^2}+\frac {\ln \left (-\frac {4\,{\mathrm {e}}^{2\,x}\,{\left (a+b\right )}^2}{b^3}-\frac {2\,{\left (a+b\right )}^{3/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,b^3}\right )\,{\left (a+b\right )}^{3/2}}{2\,\sqrt {a}\,b^2}-\frac {\ln \left (\frac {2\,{\left (a+b\right )}^{3/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,b^3}-\frac {4\,{\mathrm {e}}^{2\,x}\,{\left (a+b\right )}^2}{b^3}\right )\,{\left (a+b\right )}^{3/2}}{2\,\sqrt {a}\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^4/(a + b*cosh(x)^2),x)

[Out]

exp(2*x)/(8*b) - exp(-2*x)/(8*b) - (x*(2*a + 3*b))/(2*b^2) + (log(- (4*exp(2*x)*(a + b)^2)/b^3 - (2*(a + b)^(3
/2)*(b + 2*a*exp(2*x) + b*exp(2*x)))/(a^(1/2)*b^3))*(a + b)^(3/2))/(2*a^(1/2)*b^2) - (log((2*(a + b)^(3/2)*(b
+ 2*a*exp(2*x) + b*exp(2*x)))/(a^(1/2)*b^3) - (4*exp(2*x)*(a + b)^2)/b^3)*(a + b)^(3/2))/(2*a^(1/2)*b^2)

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