3.14 \(\int \cosh ^m(a+b x) \sinh ^5(a+b x) \, dx\)
Optimal. Leaf size=59 \[ \frac {\cosh ^{m+1}(a+b x)}{b (m+1)}-\frac {2 \cosh ^{m+3}(a+b x)}{b (m+3)}+\frac {\cosh ^{m+5}(a+b x)}{b (m+5)} \]
[Out]
cosh(b*x+a)^(1+m)/b/(1+m)-2*cosh(b*x+a)^(3+m)/b/(3+m)+cosh(b*x+a)^(5+m)/b/(5+m)
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used =
{2565, 270} \[ \frac {\cosh ^{m+1}(a+b x)}{b (m+1)}-\frac {2 \cosh ^{m+3}(a+b x)}{b (m+3)}+\frac {\cosh ^{m+5}(a+b x)}{b (m+5)} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[a + b*x]^m*Sinh[a + b*x]^5,x]
[Out]
Cosh[a + b*x]^(1 + m)/(b*(1 + m)) - (2*Cosh[a + b*x]^(3 + m))/(b*(3 + m)) + Cosh[a + b*x]^(5 + m)/(b*(5 + m))
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rule 2565
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&& !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Rubi steps
\begin {align*} \int \cosh ^m(a+b x) \sinh ^5(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^m \left (1-x^2\right )^2 \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^m-2 x^{2+m}+x^{4+m}\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\cosh ^{1+m}(a+b x)}{b (1+m)}-\frac {2 \cosh ^{3+m}(a+b x)}{b (3+m)}+\frac {\cosh ^{5+m}(a+b x)}{b (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 77, normalized size = 1.31 \[ \frac {\cosh ^{m+1}(a+b x) \left (-4 \left (m^2+8 m+7\right ) \cosh (2 (a+b x))+\left (m^2+4 m+3\right ) \cosh (4 (a+b x))+3 m^2+28 m+89\right )}{8 b (m+1) (m+3) (m+5)} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[a + b*x]^m*Sinh[a + b*x]^5,x]
[Out]
(Cosh[a + b*x]^(1 + m)*(89 + 28*m + 3*m^2 - 4*(7 + 8*m + m^2)*Cosh[2*(a + b*x)] + (3 + 4*m + m^2)*Cosh[4*(a +
b*x)]))/(8*b*(1 + m)*(3 + m)*(5 + m))
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fricas [B] time = 0.42, size = 407, normalized size = 6.90 \[ \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} \cosh \left (b x + a\right )^{5} + 5 \, {\left (m^{2} + 4 \, m + 3\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - {\left (3 \, m^{2} + 28 \, m + 25\right )} \cosh \left (b x + a\right )^{3} + {\left (10 \, {\left (m^{2} + 4 \, m + 3\right )} \cosh \left (b x + a\right )^{3} - 3 \, {\left (3 \, m^{2} + 28 \, m + 25\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (m^{2} + 12 \, m + 75\right )} \cosh \left (b x + a\right )\right )} \cosh \left (m \log \left (\cosh \left (b x + a\right )\right )\right ) + {\left ({\left (m^{2} + 4 \, m + 3\right )} \cosh \left (b x + a\right )^{5} + 5 \, {\left (m^{2} + 4 \, m + 3\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - {\left (3 \, m^{2} + 28 \, m + 25\right )} \cosh \left (b x + a\right )^{3} + {\left (10 \, {\left (m^{2} + 4 \, m + 3\right )} \cosh \left (b x + a\right )^{3} - 3 \, {\left (3 \, m^{2} + 28 \, m + 25\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (m^{2} + 12 \, m + 75\right )} \cosh \left (b x + a\right )\right )} \sinh \left (m \log \left (\cosh \left (b x + a\right )\right )\right )}{16 \, {\left ({\left (b m^{3} + 9 \, b m^{2} + 23 \, b m + 15 \, b\right )} \cosh \left (b x + a\right )^{6} - 3 \, {\left (b m^{3} + 9 \, b m^{2} + 23 \, b m + 15 \, b\right )} \cosh \left (b x + a\right )^{4} \sinh \left (b x + a\right )^{2} + 3 \, {\left (b m^{3} + 9 \, b m^{2} + 23 \, b m + 15 \, b\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (b m^{3} + 9 \, b m^{2} + 23 \, b m + 15 \, b\right )} \sinh \left (b x + a\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)^m*sinh(b*x+a)^5,x, algorithm="fricas")
[Out]
1/16*(((m^2 + 4*m + 3)*cosh(b*x + a)^5 + 5*(m^2 + 4*m + 3)*cosh(b*x + a)*sinh(b*x + a)^4 - (3*m^2 + 28*m + 25)
*cosh(b*x + a)^3 + (10*(m^2 + 4*m + 3)*cosh(b*x + a)^3 - 3*(3*m^2 + 28*m + 25)*cosh(b*x + a))*sinh(b*x + a)^2
+ 2*(m^2 + 12*m + 75)*cosh(b*x + a))*cosh(m*log(cosh(b*x + a))) + ((m^2 + 4*m + 3)*cosh(b*x + a)^5 + 5*(m^2 +
4*m + 3)*cosh(b*x + a)*sinh(b*x + a)^4 - (3*m^2 + 28*m + 25)*cosh(b*x + a)^3 + (10*(m^2 + 4*m + 3)*cosh(b*x +
a)^3 - 3*(3*m^2 + 28*m + 25)*cosh(b*x + a))*sinh(b*x + a)^2 + 2*(m^2 + 12*m + 75)*cosh(b*x + a))*sinh(m*log(co
sh(b*x + a))))/((b*m^3 + 9*b*m^2 + 23*b*m + 15*b)*cosh(b*x + a)^6 - 3*(b*m^3 + 9*b*m^2 + 23*b*m + 15*b)*cosh(b
*x + a)^4*sinh(b*x + a)^2 + 3*(b*m^3 + 9*b*m^2 + 23*b*m + 15*b)*cosh(b*x + a)^2*sinh(b*x + a)^4 - (b*m^3 + 9*b
*m^2 + 23*b*m + 15*b)*sinh(b*x + a)^6)
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giac [B] time = 0.31, size = 721, normalized size = 12.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)^m*sinh(b*x+a)^5,x, algorithm="giac")
[Out]
1/32*(m^2*e^(11*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 11*a) - 3*m^2*e^(9*b*x + m*log(1/2*(e^(2
*b*x + 2*a) + 1)*e^(-b*x - a)) + 9*a) + 2*m^2*e^(7*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 7*a)
+ 2*m^2*e^(5*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 5*a) - 3*m^2*e^(3*b*x + m*log(1/2*(e^(2*b*x
+ 2*a) + 1)*e^(-b*x - a)) + 3*a) + m^2*e^(b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + a) + 4*m*e^(1
1*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 11*a) - 28*m*e^(9*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1
)*e^(-b*x - a)) + 9*a) + 24*m*e^(7*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 7*a) + 24*m*e^(5*b*x
+ m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 5*a) - 28*m*e^(3*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b
*x - a)) + 3*a) + 4*m*e^(b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + a) + 3*e^(11*b*x + m*log(1/2*(e
^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 11*a) - 25*e^(9*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 9*a)
+ 150*e^(7*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 7*a) + 150*e^(5*b*x + m*log(1/2*(e^(2*b*x +
2*a) + 1)*e^(-b*x - a)) + 5*a) - 25*e^(3*b*x + m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + 3*a) + 3*e^(b*x
+ m*log(1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)) + a))/(b*m^3*e^(6*b*x + 6*a) + 9*b*m^2*e^(6*b*x + 6*a) + 23*b
*m*e^(6*b*x + 6*a) + 15*b*e^(6*b*x + 6*a))
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maple [F] time = 0.98, size = 0, normalized size = 0.00 \[ \int \left (\cosh ^{m}\left (b x +a \right )\right ) \left (\sinh ^{5}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(b*x+a)^m*sinh(b*x+a)^5,x)
[Out]
int(cosh(b*x+a)^m*sinh(b*x+a)^5,x)
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maxima [B] time = 0.50, size = 558, normalized size = 9.46 \[ \frac {m^{2} e^{\left ({\left (b x + a\right )} m + 5 \, b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) + 5 \, a\right )}}{32 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} + \frac {m e^{\left ({\left (b x + a\right )} m + 5 \, b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) + 5 \, a\right )}}{8 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} - \frac {{\left (3 \, m^{2} + 28 \, m + 25\right )} e^{\left ({\left (b x + a\right )} m + 3 \, b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) + 3 \, a\right )}}{32 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} + \frac {{\left (m^{2} + 12 \, m + 75\right )} e^{\left ({\left (b x + a\right )} m + b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) + a\right )}}{16 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} + \frac {{\left (m^{2} + 12 \, m + 75\right )} e^{\left ({\left (b x + a\right )} m - b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) - a\right )}}{16 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} - \frac {{\left (3 \, m^{2} + 28 \, m + 25\right )} e^{\left ({\left (b x + a\right )} m - 3 \, b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) - 3 \, a\right )}}{32 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} + \frac {{\left (m^{2} + 4 \, m + 3\right )} e^{\left ({\left (b x + a\right )} m - 5 \, b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) - 5 \, a\right )}}{32 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} + \frac {3 \, e^{\left ({\left (b x + a\right )} m + 5 \, b x + m \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right ) + 5 \, a\right )}}{32 \, {\left (2^{m} m^{3} + 9 \cdot 2^{m} m^{2} + 23 \cdot 2^{m} m + 15 \cdot 2^{m}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)^m*sinh(b*x+a)^5,x, algorithm="maxima")
[Out]
1/32*m^2*e^((b*x + a)*m + 5*b*x + m*log(e^(-2*b*x - 2*a) + 1) + 5*a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15*2^m
)*b) + 1/8*m*e^((b*x + a)*m + 5*b*x + m*log(e^(-2*b*x - 2*a) + 1) + 5*a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15
*2^m)*b) - 1/32*(3*m^2 + 28*m + 25)*e^((b*x + a)*m + 3*b*x + m*log(e^(-2*b*x - 2*a) + 1) + 3*a)/((2^m*m^3 + 9*
2^m*m^2 + 23*2^m*m + 15*2^m)*b) + 1/16*(m^2 + 12*m + 75)*e^((b*x + a)*m + b*x + m*log(e^(-2*b*x - 2*a) + 1) +
a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15*2^m)*b) + 1/16*(m^2 + 12*m + 75)*e^((b*x + a)*m - b*x + m*log(e^(-2*b
*x - 2*a) + 1) - a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15*2^m)*b) - 1/32*(3*m^2 + 28*m + 25)*e^((b*x + a)*m -
3*b*x + m*log(e^(-2*b*x - 2*a) + 1) - 3*a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15*2^m)*b) + 1/32*(m^2 + 4*m + 3
)*e^((b*x + a)*m - 5*b*x + m*log(e^(-2*b*x - 2*a) + 1) - 5*a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15*2^m)*b) +
3/32*e^((b*x + a)*m + 5*b*x + m*log(e^(-2*b*x - 2*a) + 1) + 5*a)/((2^m*m^3 + 9*2^m*m^2 + 23*2^m*m + 15*2^m)*b)
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mupad [B] time = 1.75, size = 254, normalized size = 4.31 \[ {\mathrm {e}}^{-5\,a-5\,b\,x}\,{\left (\frac {{\mathrm {e}}^{a+b\,x}}{2}+\frac {{\mathrm {e}}^{-a-b\,x}}{2}\right )}^m\,\left (\frac {m^2+4\,m+3}{32\,b\,\left (m^3+9\,m^2+23\,m+15\right )}+\frac {{\mathrm {e}}^{10\,a+10\,b\,x}\,\left (m^2+4\,m+3\right )}{32\,b\,\left (m^3+9\,m^2+23\,m+15\right )}-\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (3\,m^2+28\,m+25\right )}{32\,b\,\left (m^3+9\,m^2+23\,m+15\right )}-\frac {{\mathrm {e}}^{8\,a+8\,b\,x}\,\left (3\,m^2+28\,m+25\right )}{32\,b\,\left (m^3+9\,m^2+23\,m+15\right )}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}\,\left (2\,m^2+24\,m+150\right )}{32\,b\,\left (m^3+9\,m^2+23\,m+15\right )}+\frac {{\mathrm {e}}^{6\,a+6\,b\,x}\,\left (2\,m^2+24\,m+150\right )}{32\,b\,\left (m^3+9\,m^2+23\,m+15\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(a + b*x)^m*sinh(a + b*x)^5,x)
[Out]
exp(- 5*a - 5*b*x)*(exp(a + b*x)/2 + exp(- a - b*x)/2)^m*((4*m + m^2 + 3)/(32*b*(23*m + 9*m^2 + m^3 + 15)) + (
exp(10*a + 10*b*x)*(4*m + m^2 + 3))/(32*b*(23*m + 9*m^2 + m^3 + 15)) - (exp(2*a + 2*b*x)*(28*m + 3*m^2 + 25))/
(32*b*(23*m + 9*m^2 + m^3 + 15)) - (exp(8*a + 8*b*x)*(28*m + 3*m^2 + 25))/(32*b*(23*m + 9*m^2 + m^3 + 15)) + (
exp(4*a + 4*b*x)*(24*m + 2*m^2 + 150))/(32*b*(23*m + 9*m^2 + m^3 + 15)) + (exp(6*a + 6*b*x)*(24*m + 2*m^2 + 15
0))/(32*b*(23*m + 9*m^2 + m^3 + 15)))
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sympy [A] time = 42.81, size = 2351, normalized size = 39.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)**m*sinh(b*x+a)**5,x)
[Out]
Piecewise((x*sinh(a)**5*cosh(a)**m, Eq(b, 0)), (log(cosh(a + b*x))/b - sinh(a + b*x)**4/(4*b*cosh(a + b*x)**4)
- sinh(a + b*x)**2/(2*b*cosh(a + b*x)**2), Eq(m, -5)), (-2*b*x*tanh(a/2 + b*x/2)**8/(b*tanh(a/2 + b*x/2)**8 -
2*b*tanh(a/2 + b*x/2)**4 + b) + 4*b*x*tanh(a/2 + b*x/2)**4/(b*tanh(a/2 + b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4
+ b) - 2*b*x/(b*tanh(a/2 + b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) + 4*log(tanh(a/2 + b*x/2) + 1)*tanh(a/2
+ b*x/2)**8/(b*tanh(a/2 + b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) - 8*log(tanh(a/2 + b*x/2) + 1)*tanh(a/2 +
b*x/2)**4/(b*tanh(a/2 + b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) + 4*log(tanh(a/2 + b*x/2) + 1)/(b*tanh(a/2 +
b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) - 2*log(tanh(a/2 + b*x/2)**2 + 1)*tanh(a/2 + b*x/2)**8/(b*tanh(a/2
+ b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) + 4*log(tanh(a/2 + b*x/2)**2 + 1)*tanh(a/2 + b*x/2)**4/(b*tanh(a/2
+ b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) - 2*log(tanh(a/2 + b*x/2)**2 + 1)/(b*tanh(a/2 + b*x/2)**8 - 2*b*t
anh(a/2 + b*x/2)**4 + b) + 4*tanh(a/2 + b*x/2)**6/(b*tanh(a/2 + b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b) + 4*
tanh(a/2 + b*x/2)**2/(b*tanh(a/2 + b*x/2)**8 - 2*b*tanh(a/2 + b*x/2)**4 + b), Eq(m, -3)), (b*x*tanh(a/2 + b*x/
2)**8/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2
+ b) - 4*b*x*tanh(a/2 + b*x/2)**6/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)*
*4 - 4*b*tanh(a/2 + b*x/2)**2 + b) + 6*b*x*tanh(a/2 + b*x/2)**4/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2
)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 4*b*x*tanh(a/2 + b*x/2)**2/(b*tanh(a/2 + b*x
/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) + b*x/(b*tanh(a/2
+ b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 2*log(tan
h(a/2 + b*x/2) + 1)*tanh(a/2 + b*x/2)**8/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b
*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) + 8*log(tanh(a/2 + b*x/2) + 1)*tanh(a/2 + b*x/2)**6/(b*tanh(a/2 + b*x
/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 12*log(tanh(a/2
+ b*x/2) + 1)*tanh(a/2 + b*x/2)**4/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)
**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) + 8*log(tanh(a/2 + b*x/2) + 1)*tanh(a/2 + b*x/2)**2/(b*tanh(a/2 + b*x/2)**
8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 2*log(tanh(a/2 + b*x
/2) + 1)/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)
**2 + b) + log(tanh(a/2 + b*x/2)**2 + 1)*tanh(a/2 + b*x/2)**8/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)*
*6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 4*log(tanh(a/2 + b*x/2)**2 + 1)*tanh(a/2 + b*x
/2)**6/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**
2 + b) + 6*log(tanh(a/2 + b*x/2)**2 + 1)*tanh(a/2 + b*x/2)**4/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)*
*6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 4*log(tanh(a/2 + b*x/2)**2 + 1)*tanh(a/2 + b*x
/2)**2/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**
2 + b) + log(tanh(a/2 + b*x/2)**2 + 1)/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x
/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 2*tanh(a/2 + b*x/2)**6/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2
)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) + 8*tanh(a/2 + b*x/2)**4/(b*tanh(a/2 + b*x/2)*
*8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 + b) - 2*tanh(a/2 + b*x/2)
**2/(b*tanh(a/2 + b*x/2)**8 - 4*b*tanh(a/2 + b*x/2)**6 + 6*b*tanh(a/2 + b*x/2)**4 - 4*b*tanh(a/2 + b*x/2)**2 +
b), Eq(m, -1)), (m**2*sinh(a + b*x)**4*cosh(a + b*x)*cosh(a + b*x)**m/(b*m**3 + 9*b*m**2 + 23*b*m + 15*b) + 8
*m*sinh(a + b*x)**4*cosh(a + b*x)*cosh(a + b*x)**m/(b*m**3 + 9*b*m**2 + 23*b*m + 15*b) - 4*m*sinh(a + b*x)**2*
cosh(a + b*x)**3*cosh(a + b*x)**m/(b*m**3 + 9*b*m**2 + 23*b*m + 15*b) + 15*sinh(a + b*x)**4*cosh(a + b*x)*cosh
(a + b*x)**m/(b*m**3 + 9*b*m**2 + 23*b*m + 15*b) - 20*sinh(a + b*x)**2*cosh(a + b*x)**3*cosh(a + b*x)**m/(b*m*
*3 + 9*b*m**2 + 23*b*m + 15*b) + 8*cosh(a + b*x)**5*cosh(a + b*x)**m/(b*m**3 + 9*b*m**2 + 23*b*m + 15*b), True
))
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