3.2.43 \(\int \coth ^5(c+d x) (a+b \tanh ^2(c+d x)) \, dx\) [143]

Optimal. Leaf size=49 \[ -\frac {(a+b) \coth ^2(c+d x)}{2 d}-\frac {a \coth ^4(c+d x)}{4 d}+\frac {(a+b) \log (\sinh (c+d x))}{d} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3710, 12, 3554, 3556} \begin {gather*} -\frac {(a+b) \coth ^2(c+d x)}{2 d}+\frac {(a+b) \log (\sinh (c+d x))}{d}-\frac {a \coth ^4(c+d x)}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3710

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[
(A*b^2 + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*
Tan[e + f*x])^(m + 1)*Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] &&
NeQ[A*b^2 + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \coth ^5(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac {a \coth ^4(c+d x)}{4 d}+\int (a+b) \coth ^3(c+d x) \, dx\\ &=-\frac {a \coth ^4(c+d x)}{4 d}+(a+b) \int \coth ^3(c+d x) \, dx\\ &=-\frac {(a+b) \coth ^2(c+d x)}{2 d}-\frac {a \coth ^4(c+d x)}{4 d}+(a+b) \int \coth (c+d x) \, dx\\ &=-\frac {(a+b) \coth ^2(c+d x)}{2 d}-\frac {a \coth ^4(c+d x)}{4 d}+\frac {(a+b) \log (\sinh (c+d x))}{d}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 51, normalized size = 1.04 \begin {gather*} -\frac {2 (a+b) \coth ^2(c+d x)+a \coth ^4(c+d x)-4 (a+b) (\log (\cosh (c+d x))+\log (\tanh (c+d x)))}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.71, size = 56, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+b \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(56\)
default	\(\frac {a \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+b \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(56\)
risch	\(-a x -b x -\frac {2 a c}{d}-\frac {2 b c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (2 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}-2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+2 a +b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (45) = 90\).
time = 0.29, size = 206, normalized size = 4.20 \begin {gather*} a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + b {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 4*(e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) + e^(-
6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + b*(
x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - e^(-
4*d*x - 4*c) - 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (45) = 90\).
time = 0.37, size = 1216, normalized size = 24.82 \begin {gather*} -\frac {{\left (a + b\right )} d x \cosh \left (d x + c\right )^{8} + 8 \, {\left (a + b\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + {\left (a + b\right )} d x \sinh \left (d x + c\right )^{8} - 2 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )^{6} + 2 \, {\left (14 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{2} - 2 \, {\left (a + b\right )} d x + 2 \, a + b\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (14 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{3} - 3 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (3 \, {\left (a + b\right )} d x - 2 \, a - 2 \, b\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{4} + 3 \, {\left (a + b\right )} d x - 15 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{5} - 5 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, {\left (a + b\right )} d x - 2 \, a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} d x - 2 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (14 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{6} - 15 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )^{4} - 2 \, {\left (a + b\right )} d x + 6 \, {\left (3 \, {\left (a + b\right )} d x - 2 \, a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, a + b\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{8} + 8 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + {\left (a + b\right )} \sinh \left (d x + c\right )^{8} - 4 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a - b\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{4} - 30 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 4 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{6} - 15 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a - b\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{7} - 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{5} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (2 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{7} - 3 \, {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )^{5} + 2 \, {\left (3 \, {\left (a + b\right )} d x - 2 \, a - 2 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (2 \, {\left (a + b\right )} d x - 2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} - 4 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} - 30 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} - 10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 4 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} - 15 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} - 3 \, d \cosh \left (d x + c\right )^{5} + 3 \, d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + b)*d*x*cosh(d*x + c)^8 + 8*(a + b)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + (a + b)*d*x*sinh(d*x + c)^8 - 2*
(2*(a + b)*d*x - 2*a - b)*cosh(d*x + c)^6 + 2*(14*(a + b)*d*x*cosh(d*x + c)^2 - 2*(a + b)*d*x + 2*a + b)*sinh(
d*x + c)^6 + 4*(14*(a + b)*d*x*cosh(d*x + c)^3 - 3*(2*(a + b)*d*x - 2*a - b)*cosh(d*x + c))*sinh(d*x + c)^5 +
2*(3*(a + b)*d*x - 2*a - 2*b)*cosh(d*x + c)^4 + 2*(35*(a + b)*d*x*cosh(d*x + c)^4 + 3*(a + b)*d*x - 15*(2*(a +
b)*d*x - 2*a - b)*cosh(d*x + c)^2 - 2*a - 2*b)*sinh(d*x + c)^4 + 8*(7*(a + b)*d*x*cosh(d*x + c)^5 - 5*(2*(a +
b)*d*x - 2*a - b)*cosh(d*x + c)^3 + (3*(a + b)*d*x - 2*a - 2*b)*cosh(d*x + c))*sinh(d*x + c)^3 + (a + b)*d*x
- 2*(2*(a + b)*d*x - 2*a - b)*cosh(d*x + c)^2 + 2*(14*(a + b)*d*x*cosh(d*x + c)^6 - 15*(2*(a + b)*d*x - 2*a -
b)*cosh(d*x + c)^4 - 2*(a + b)*d*x + 6*(3*(a + b)*d*x - 2*a - 2*b)*cosh(d*x + c)^2 + 2*a + b)*sinh(d*x + c)^2
- ((a + b)*cosh(d*x + c)^8 + 8*(a + b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a + b)*sinh(d*x + c)^8 - 4*(a + b)*cos
h(d*x + c)^6 + 4*(7*(a + b)*cosh(d*x + c)^2 - a - b)*sinh(d*x + c)^6 + 8*(7*(a + b)*cosh(d*x + c)^3 - 3*(a + b
)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(a + b)*cosh(d*x + c)^4 + 2*(35*(a + b)*cosh(d*x + c)^4 - 30*(a + b)*cosh
(d*x + c)^2 + 3*a + 3*b)*sinh(d*x + c)^4 + 8*(7*(a + b)*cosh(d*x + c)^5 - 10*(a + b)*cosh(d*x + c)^3 + 3*(a +
b)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a + b)*cosh(d*x + c)^2 + 4*(7*(a + b)*cosh(d*x + c)^6 - 15*(a + b)*cosh
(d*x + c)^4 + 9*(a + b)*cosh(d*x + c)^2 - a - b)*sinh(d*x + c)^2 + 8*((a + b)*cosh(d*x + c)^7 - 3*(a + b)*cosh
(d*x + c)^5 + 3*(a + b)*cosh(d*x + c)^3 - (a + b)*cosh(d*x + c))*sinh(d*x + c) + a + b)*log(2*sinh(d*x + c)/(c
osh(d*x + c) - sinh(d*x + c))) + 4*(2*(a + b)*d*x*cosh(d*x + c)^7 - 3*(2*(a + b)*d*x - 2*a - b)*cosh(d*x + c)^
5 + 2*(3*(a + b)*d*x - 2*a - 2*b)*cosh(d*x + c)^3 - (2*(a + b)*d*x - 2*a - b)*cosh(d*x + c))*sinh(d*x + c))/(d
*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 4*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d
*x + c)^2 - d)*sinh(d*x + c)^6 + 8*(7*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 6*d*cosh(d*x +
c)^4 + 2*(35*d*cosh(d*x + c)^4 - 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 - 10*d*c
osh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 - 4*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 - 15*d*cosh
(d*x + c)^4 + 9*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - 3*d*cosh(d*x + c)^5 + 3*d*cosh
(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (45) = 90\).
time = 0.47, size = 93, normalized size = 1.90 \begin {gather*} -\frac {{\left (d x + c\right )} {\left (a + b\right )} - {\left (a + b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {2 \, {\left ({\left (2 \, a + b\right )} e^{\left (6 \, d x + 6 \, c\right )} - 2 \, {\left (a + b\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (2 \, a + b\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d*x + c)*(a + b) - (a + b)*log(abs(e^(2*d*x + 2*c) - 1)) + 2*((2*a + b)*e^(6*d*x + 6*c) - 2*(a + b)*e^(4*d*
x + 4*c) + (2*a + b)*e^(2*d*x + 2*c))/(e^(2*d*x + 2*c) - 1)^4)/d

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Mupad [B]
time = 1.25, size = 177, normalized size = 3.61 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )\,\left (a+b\right )}{d}-\frac {2\,\left (2\,a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-x\,\left (a+b\right )-\frac {2\,\left (4\,a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,a}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(exp(2*c)*exp(2*d*x) - 1)*(a + b))/d - (2*(2*a + b))/(d*(exp(2*c + 2*d*x) - 1)) - x*(a + b) - (2*(4*a + b)
)/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*a)/(d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*
c + 6*d*x) - 1)) - (4*a)/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) +
1))

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