3.100 \(\int \frac{A+B \cosh (x)}{(1-\cosh (x))^4} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))}-\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))^2}-\frac{(3 A-4 B) \sinh (x)}{35 (1-\cosh (x))^3}-\frac{(A+B) \sinh (x)}{7 (1-\cosh (x))^4} \]

[Out]

-((A + B)*Sinh[x])/(7*(1 - Cosh[x])^4) - ((3*A - 4*B)*Sinh[x])/(35*(1 - Cosh[x])^3) - (2*(3*A - 4*B)*Sinh[x])/
(105*(1 - Cosh[x])^2) - (2*(3*A - 4*B)*Sinh[x])/(105*(1 - Cosh[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0664045, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2750, 2650, 2648} \[ -\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))}-\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))^2}-\frac{(3 A-4 B) \sinh (x)}{35 (1-\cosh (x))^3}-\frac{(A+B) \sinh (x)}{7 (1-\cosh (x))^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(1 - Cosh[x])^4,x]

[Out]

-((A + B)*Sinh[x])/(7*(1 - Cosh[x])^4) - ((3*A - 4*B)*Sinh[x])/(35*(1 - Cosh[x])^3) - (2*(3*A - 4*B)*Sinh[x])/
(105*(1 - Cosh[x])^2) - (2*(3*A - 4*B)*Sinh[x])/(105*(1 - Cosh[x]))

Rule 2750

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((b
*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)
), Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 -
b^2, 0] && LtQ[m, -2^(-1)]

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{A+B \cosh (x)}{(1-\cosh (x))^4} \, dx &=-\frac{(A+B) \sinh (x)}{7 (1-\cosh (x))^4}+\frac{1}{7} (3 A-4 B) \int \frac{1}{(1-\cosh (x))^3} \, dx\\ &=-\frac{(A+B) \sinh (x)}{7 (1-\cosh (x))^4}-\frac{(3 A-4 B) \sinh (x)}{35 (1-\cosh (x))^3}+\frac{1}{35} (2 (3 A-4 B)) \int \frac{1}{(1-\cosh (x))^2} \, dx\\ &=-\frac{(A+B) \sinh (x)}{7 (1-\cosh (x))^4}-\frac{(3 A-4 B) \sinh (x)}{35 (1-\cosh (x))^3}-\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))^2}+\frac{1}{105} (2 (3 A-4 B)) \int \frac{1}{1-\cosh (x)} \, dx\\ &=-\frac{(A+B) \sinh (x)}{7 (1-\cosh (x))^4}-\frac{(3 A-4 B) \sinh (x)}{35 (1-\cosh (x))^3}-\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))^2}-\frac{2 (3 A-4 B) \sinh (x)}{105 (1-\cosh (x))}\\ \end{align*}

Mathematica [A]  time = 0.0878756, size = 57, normalized size = 0.7 \[ \frac{\sinh (x) (29 (3 A-4 B) \cosh (x)-8 (3 A-4 B) \cosh (2 x)+3 A \cosh (3 x)-96 A-4 B \cosh (3 x)+58 B)}{210 (\cosh (x)-1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(1 - Cosh[x])^4,x]

[Out]

((-96*A + 58*B + 29*(3*A - 4*B)*Cosh[x] - 8*(3*A - 4*B)*Cosh[2*x] + 3*A*Cosh[3*x] - 4*B*Cosh[3*x])*Sinh[x])/(2
10*(-1 + Cosh[x])^4)

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 56, normalized size = 0.7 \begin{align*} -{\frac{-A+B}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{3\,A-B}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{A+B}{56} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-7}}-{\frac{-3\,A-B}{40} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cosh(x))/(1-cosh(x))^4,x)

[Out]

-1/8*(-A+B)/tanh(1/2*x)-1/24*(3*A-B)/tanh(1/2*x)^3-1/56*(A+B)/tanh(1/2*x)^7-1/40*(-3*A-B)/tanh(1/2*x)^5

________________________________________________________________________________________

Maxima [B]  time = 1.066, size = 609, normalized size = 7.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))^4,x, algorithm="maxima")

[Out]

-8/105*B*(14*e^(-x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x)
- 1) - 42*e^(-2*x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) -
1) + 35*e^(-3*x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) -
1) - 35*e^(-4*x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) - 1
) - 2/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) - 1)) + 4/35*A
*(7*e^(-x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) - 1) - 21
*e^(-2*x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) - 1) + 35*
e^(-3*x)/(7*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) - 1) - 1/(7
*e^(-x) - 21*e^(-2*x) + 35*e^(-3*x) - 35*e^(-4*x) + 21*e^(-5*x) - 7*e^(-6*x) + e^(-7*x) - 1))

________________________________________________________________________________________

Fricas [B]  time = 2.08389, size = 585, normalized size = 7.22 \begin{align*} \frac{4 \,{\left ({\left (3 \, A - 74 \, B\right )} \cosh \left (x\right )^{2} +{\left (3 \, A - 74 \, B\right )} \sinh \left (x\right )^{2} - 14 \,{\left (9 \, A - 7 \, B\right )} \cosh \left (x\right ) - 6 \,{\left ({\left (A + 22 \, B\right )} \cosh \left (x\right ) + 14 \, A - 7 \, B\right )} \sinh \left (x\right ) + 63 \, A - 84 \, B\right )}}{105 \,{\left (\cosh \left (x\right )^{5} +{\left (5 \, \cosh \left (x\right ) - 7\right )} \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} - 7 \, \cosh \left (x\right )^{4} +{\left (10 \, \cosh \left (x\right )^{2} - 28 \, \cosh \left (x\right ) + 21\right )} \sinh \left (x\right )^{3} + 21 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} - 42 \, \cosh \left (x\right )^{2} + 63 \, \cosh \left (x\right ) - 36\right )} \sinh \left (x\right )^{2} - 36 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} - 28 \, \cosh \left (x\right )^{3} + 63 \, \cosh \left (x\right )^{2} - 68 \, \cosh \left (x\right ) + 28\right )} \sinh \left (x\right ) + 42 \, \cosh \left (x\right ) - 21\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))^4,x, algorithm="fricas")

[Out]

4/105*((3*A - 74*B)*cosh(x)^2 + (3*A - 74*B)*sinh(x)^2 - 14*(9*A - 7*B)*cosh(x) - 6*((A + 22*B)*cosh(x) + 14*A
- 7*B)*sinh(x) + 63*A - 84*B)/(cosh(x)^5 + (5*cosh(x) - 7)*sinh(x)^4 + sinh(x)^5 - 7*cosh(x)^4 + (10*cosh(x)^
2 - 28*cosh(x) + 21)*sinh(x)^3 + 21*cosh(x)^3 + (10*cosh(x)^3 - 42*cosh(x)^2 + 63*cosh(x) - 36)*sinh(x)^2 - 36
*cosh(x)^2 + (5*cosh(x)^4 - 28*cosh(x)^3 + 63*cosh(x)^2 - 68*cosh(x) + 28)*sinh(x) + 42*cosh(x) - 21)

________________________________________________________________________________________

Sympy [A]  time = 5.14603, size = 78, normalized size = 0.96 \begin{align*} \frac{A}{8 \tanh{\left (\frac{x}{2} \right )}} - \frac{A}{8 \tanh ^{3}{\left (\frac{x}{2} \right )}} + \frac{3 A}{40 \tanh ^{5}{\left (\frac{x}{2} \right )}} - \frac{A}{56 \tanh ^{7}{\left (\frac{x}{2} \right )}} - \frac{B}{8 \tanh{\left (\frac{x}{2} \right )}} + \frac{B}{24 \tanh ^{3}{\left (\frac{x}{2} \right )}} + \frac{B}{40 \tanh ^{5}{\left (\frac{x}{2} \right )}} - \frac{B}{56 \tanh ^{7}{\left (\frac{x}{2} \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))**4,x)

[Out]

A/(8*tanh(x/2)) - A/(8*tanh(x/2)**3) + 3*A/(40*tanh(x/2)**5) - A/(56*tanh(x/2)**7) - B/(8*tanh(x/2)) + B/(24*t
anh(x/2)**3) + B/(40*tanh(x/2)**5) - B/(56*tanh(x/2)**7)

________________________________________________________________________________________

Giac [A]  time = 1.18975, size = 81, normalized size = 1. \begin{align*} -\frac{4 \,{\left (70 \, B e^{\left (4 \, x\right )} + 105 \, A e^{\left (3 \, x\right )} - 70 \, B e^{\left (3 \, x\right )} - 63 \, A e^{\left (2 \, x\right )} + 84 \, B e^{\left (2 \, x\right )} + 21 \, A e^{x} - 28 \, B e^{x} - 3 \, A + 4 \, B\right )}}{105 \,{\left (e^{x} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(1-cosh(x))^4,x, algorithm="giac")

[Out]

-4/105*(70*B*e^(4*x) + 105*A*e^(3*x) - 70*B*e^(3*x) - 63*A*e^(2*x) + 84*B*e^(2*x) + 21*A*e^x - 28*B*e^x - 3*A
+ 4*B)/(e^x - 1)^7