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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]

-1/7*(A+B)*sinh(x)/(1-cosh(x))^4-1/35*(3*A-4*B)*sinh(x)/(1-cosh(x))^3-2/105*(3*A-4*B)*sinh(x)/(1-cosh(x))^2-2/
105*(3*A-4*B)*sinh(x)/(1-cosh(x))

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Rubi [A]  time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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 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)]

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*}

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Mathematica [A]  time = 0.10, size = 57, normalized size = 0.70 \[ \frac {\sinh (x) (29 (3 A-4 B) \cosh (x)-8 (3 A-4 B) \cosh (2 x)+3 A \cosh (3 x)-96 A-4 B \cosh (3 x)+58 B)}{210 (\cosh (x)-1)^4} \]

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)

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fricas [B]  time = 0.84, size = 175, normalized size = 2.16 \[ \frac {4 \, {\left ({\left (3 \, A - 74 \, B\right )} \cosh \relax (x)^{2} + {\left (3 \, A - 74 \, B\right )} \sinh \relax (x)^{2} - 14 \, {\left (9 \, A - 7 \, B\right )} \cosh \relax (x) - 6 \, {\left ({\left (A + 22 \, B\right )} \cosh \relax (x) + 14 \, A - 7 \, B\right )} \sinh \relax (x) + 63 \, A - 84 \, B\right )}}{105 \, {\left (\cosh \relax (x)^{5} + {\left (5 \, \cosh \relax (x) - 7\right )} \sinh \relax (x)^{4} + \sinh \relax (x)^{5} - 7 \, \cosh \relax (x)^{4} + {\left (10 \, \cosh \relax (x)^{2} - 28 \, \cosh \relax (x) + 21\right )} \sinh \relax (x)^{3} + 21 \, \cosh \relax (x)^{3} + {\left (10 \, \cosh \relax (x)^{3} - 42 \, \cosh \relax (x)^{2} + 63 \, \cosh \relax (x) - 36\right )} \sinh \relax (x)^{2} - 36 \, \cosh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{4} - 28 \, \cosh \relax (x)^{3} + 63 \, \cosh \relax (x)^{2} - 68 \, \cosh \relax (x) + 28\right )} \sinh \relax (x) + 42 \, \cosh \relax (x) - 21\right )}} \]

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)

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giac [A]  time = 0.12, size = 60, normalized size = 0.74 \[ -\frac {4 \, {\left (70 \, B e^{\left (4 \, x\right )} + 105 \, A e^{\left (3 \, x\right )} - 70 \, B e^{\left (3 \, x\right )} - 63 \, A e^{\left (2 \, x\right )} + 84 \, B e^{\left (2 \, x\right )} + 21 \, A e^{x} - 28 \, B e^{x} - 3 \, A + 4 \, B\right )}}{105 \, {\left (e^{x} - 1\right )}^{7}} \]

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

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maple [A]  time = 0.06, size = 56, normalized size = 0.69 \[ -\frac {A +B}{56 \tanh \left (\frac {x}{2}\right )^{7}}-\frac {3 A -B}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-A +B}{8 \tanh \left (\frac {x}{2}\right )}-\frac {-3 A -B}{40 \tanh \left (\frac {x}{2}\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.34, size = 451, normalized size = 5.57 \[ -\frac {8}{105} \, B {\left (\frac {14 \, e^{\left (-x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} - \frac {42 \, e^{\left (-2 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} + \frac {35 \, e^{\left (-3 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} - \frac {35 \, e^{\left (-4 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} - \frac {2}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1}\right )} + \frac {4}{35} \, A {\left (\frac {7 \, e^{\left (-x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} - \frac {21 \, e^{\left (-2 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} + \frac {35 \, e^{\left (-3 \, x\right )}}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1} - \frac {1}{7 \, e^{\left (-x\right )} - 21 \, e^{\left (-2 \, x\right )} + 35 \, e^{\left (-3 \, x\right )} - 35 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} - 7 \, e^{\left (-6 \, x\right )} + e^{\left (-7 \, x\right )} - 1}\right )} \]

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))

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mupad [B]  time = 0.92, size = 233, normalized size = 2.88 \[ \frac {\frac {8\,B}{105}+\frac {16\,A\,{\mathrm {e}}^x}{35}+\frac {16\,B\,{\mathrm {e}}^{2\,x}}{35}}{10\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{5\,x}-5\,{\mathrm {e}}^x+1}-\frac {\frac {4\,A}{35}+\frac {8\,B\,{\mathrm {e}}^x}{35}}{6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1}-\frac {\frac {8\,B\,{\mathrm {e}}^x}{21}+\frac {8\,A\,{\mathrm {e}}^{2\,x}}{7}+\frac {16\,B\,{\mathrm {e}}^{3\,x}}{21}}{15\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{3\,x}+15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{5\,x}+{\mathrm {e}}^{6\,x}-6\,{\mathrm {e}}^x+1}+\frac {8\,B}{105\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}+\frac {\frac {16\,A\,{\mathrm {e}}^{3\,x}}{7}+\frac {8\,B\,{\mathrm {e}}^{2\,x}}{7}+\frac {8\,B\,{\mathrm {e}}^{4\,x}}{7}}{21\,{\mathrm {e}}^{2\,x}-35\,{\mathrm {e}}^{3\,x}+35\,{\mathrm {e}}^{4\,x}-21\,{\mathrm {e}}^{5\,x}+7\,{\mathrm {e}}^{6\,x}-{\mathrm {e}}^{7\,x}-7\,{\mathrm {e}}^x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((8*B)/105 + (16*A*exp(x))/35 + (16*B*exp(2*x))/35)/(10*exp(2*x) - 10*exp(3*x) + 5*exp(4*x) - exp(5*x) - 5*exp
(x) + 1) - ((4*A)/35 + (8*B*exp(x))/35)/(6*exp(2*x) - 4*exp(3*x) + exp(4*x) - 4*exp(x) + 1) - ((8*B*exp(x))/21
 + (8*A*exp(2*x))/7 + (16*B*exp(3*x))/21)/(15*exp(2*x) - 20*exp(3*x) + 15*exp(4*x) - 6*exp(5*x) + exp(6*x) - 6
*exp(x) + 1) + (8*B)/(105*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) + ((16*A*exp(3*x))/7 + (8*B*exp(2*x))/7 + (8
*B*exp(4*x))/7)/(21*exp(2*x) - 35*exp(3*x) + 35*exp(4*x) - 21*exp(5*x) + 7*exp(6*x) - exp(7*x) - 7*exp(x) + 1)

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sympy [A]  time = 2.72, size = 78, normalized size = 0.96 \[ \frac {A}{8 \tanh {\left (\frac {x}{2} \right )}} - \frac {A}{8 \tanh ^{3}{\left (\frac {x}{2} \right )}} + \frac {3 A}{40 \tanh ^{5}{\left (\frac {x}{2} \right )}} - \frac {A}{56 \tanh ^{7}{\left (\frac {x}{2} \right )}} - \frac {B}{8 \tanh {\left (\frac {x}{2} \right )}} + \frac {B}{24 \tanh ^{3}{\left (\frac {x}{2} \right )}} + \frac {B}{40 \tanh ^{5}{\left (\frac {x}{2} \right )}} - \frac {B}{56 \tanh ^{7}{\left (\frac {x}{2} \right )}} \]

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)

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