3.78 \(\int \frac{1}{(5+3 \cosh (c+d x))^4} \, dx\)
Optimal. Leaf size=106 \[ -\frac{311 \sinh (c+d x)}{8192 d (3 \cosh (c+d x)+5)}-\frac{25 \sinh (c+d x)}{512 d (3 \cosh (c+d x)+5)^2}-\frac{\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}-\frac{385 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{16384 d}+\frac{385 x}{32768} \]
[Out]
(385*x)/32768 - (385*ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])])/(16384*d) - Sinh[c + d*x]/(16*d*(5 + 3*Cosh[c
+ d*x])^3) - (25*Sinh[c + d*x])/(512*d*(5 + 3*Cosh[c + d*x])^2) - (311*Sinh[c + d*x])/(8192*d*(5 + 3*Cosh[c +
d*x]))
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Rubi [A] time = 0.0965579, antiderivative size = 106, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{2664, 2754, 12, 2657} \[ -\frac{311 \sinh (c+d x)}{8192 d (3 \cosh (c+d x)+5)}-\frac{25 \sinh (c+d x)}{512 d (3 \cosh (c+d x)+5)^2}-\frac{\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}-\frac{385 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{16384 d}+\frac{385 x}{32768} \]
Antiderivative was successfully verified.
[In]
Int[(5 + 3*Cosh[c + d*x])^(-4),x]
[Out]
(385*x)/32768 - (385*ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])])/(16384*d) - Sinh[c + d*x]/(16*d*(5 + 3*Cosh[c
+ d*x])^3) - (25*Sinh[c + d*x])/(512*d*(5 + 3*Cosh[c + d*x])^2) - (311*Sinh[c + d*x])/(8192*d*(5 + 3*Cosh[c +
d*x]))
Rule 2664
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]
Rule 2754
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 + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 2657
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]
Rubi steps
\begin{align*} \int \frac{1}{(5+3 \cosh (c+d x))^4} \, dx &=-\frac{\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac{1}{48} \int \frac{-15+6 \cosh (c+d x)}{(5+3 \cosh (c+d x))^3} \, dx\\ &=-\frac{\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac{25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}+\frac{\int \frac{186-75 \cosh (c+d x)}{(5+3 \cosh (c+d x))^2} \, dx}{1536}\\ &=-\frac{\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac{25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac{311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}-\frac{\int -\frac{1155}{5+3 \cosh (c+d x)} \, dx}{24576}\\ &=-\frac{\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac{25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac{311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}+\frac{385 \int \frac{1}{5+3 \cosh (c+d x)} \, dx}{8192}\\ &=\frac{385 x}{32768}-\frac{385 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{16384 d}-\frac{\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac{25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac{311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.247145, size = 68, normalized size = 0.64 \[ \frac{770 \tanh ^{-1}\left (\frac{1}{2} \tanh \left (\frac{1}{2} (c+d x)\right )\right )-\frac{9 (4883 \sinh (c+d x)+2340 \sinh (2 (c+d x))+311 \sinh (3 (c+d x)))}{(3 \cosh (c+d x)+5)^3}}{32768 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(5 + 3*Cosh[c + d*x])^(-4),x]
[Out]
(770*ArcTanh[Tanh[(c + d*x)/2]/2] - (9*(4883*Sinh[c + d*x] + 2340*Sinh[2*(c + d*x)] + 311*Sinh[3*(c + d*x)]))/
(5 + 3*Cosh[c + d*x])^3)/(32768*d)
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Maple [A] time = 0.019, size = 144, normalized size = 1.4 \begin{align*}{\frac{9}{2048\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-3}}-{\frac{81}{4096\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-2}}+{\frac{639}{16384\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-1}}+{\frac{385}{32768\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }+{\frac{9}{2048\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-3}}+{\frac{81}{4096\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-2}}+{\frac{639}{16384\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-1}}-{\frac{385}{32768\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(5+3*cosh(d*x+c))^4,x)
[Out]
9/2048/d/(tanh(1/2*d*x+1/2*c)+2)^3-81/4096/d/(tanh(1/2*d*x+1/2*c)+2)^2+639/16384/d/(tanh(1/2*d*x+1/2*c)+2)+385
/32768/d*ln(tanh(1/2*d*x+1/2*c)+2)+9/2048/d/(tanh(1/2*d*x+1/2*c)-2)^3+81/4096/d/(tanh(1/2*d*x+1/2*c)-2)^2+639/
16384/d/(tanh(1/2*d*x+1/2*c)-2)-385/32768/d*ln(tanh(1/2*d*x+1/2*c)-2)
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Maxima [A] time = 1.0722, size = 228, normalized size = 2.15 \begin{align*} -\frac{385 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{32768 \, d} + \frac{385 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{32768 \, d} - \frac{73575 \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} + 239470 \, e^{\left (-3 \, d x - 3 \, c\right )} + 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 \, e^{\left (-5 \, d x - 5 \, c\right )} + 8397}{12288 \, d{\left (270 \, e^{\left (-d x - c\right )} + 981 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1540 \, e^{\left (-3 \, d x - 3 \, c\right )} + 981 \, e^{\left (-4 \, d x - 4 \, c\right )} + 270 \, e^{\left (-5 \, d x - 5 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 27\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="maxima")
[Out]
-385/32768*log(3*e^(-d*x - c) + 1)/d + 385/32768*log(e^(-d*x - c) + 3)/d - 1/12288*(73575*e^(-d*x - c) + 21846
6*e^(-2*d*x - 2*c) + 239470*e^(-3*d*x - 3*c) + 86625*e^(-4*d*x - 4*c) + 10395*e^(-5*d*x - 5*c) + 8397)/(d*(270
*e^(-d*x - c) + 981*e^(-2*d*x - 2*c) + 1540*e^(-3*d*x - 3*c) + 981*e^(-4*d*x - 4*c) + 270*e^(-5*d*x - 5*c) + 2
7*e^(-6*d*x - 6*c) + 27))
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Fricas [B] time = 2.24239, size = 3320, normalized size = 31.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="fricas")
[Out]
1/98304*(83160*cosh(d*x + c)^5 + 138600*(3*cosh(d*x + c) + 5)*sinh(d*x + c)^4 + 83160*sinh(d*x + c)^5 + 693000
*cosh(d*x + c)^4 + 6160*(135*cosh(d*x + c)^2 + 450*cosh(d*x + c) + 311)*sinh(d*x + c)^3 + 1915760*cosh(d*x + c
)^3 + 48*(17325*cosh(d*x + c)^3 + 86625*cosh(d*x + c)^2 + 119735*cosh(d*x + c) + 36411)*sinh(d*x + c)^2 + 1747
728*cosh(d*x + c)^2 + 1155*(27*cosh(d*x + c)^6 + 54*(3*cosh(d*x + c) + 5)*sinh(d*x + c)^5 + 27*sinh(d*x + c)^6
+ 270*cosh(d*x + c)^5 + 9*(45*cosh(d*x + c)^2 + 150*cosh(d*x + c) + 109)*sinh(d*x + c)^4 + 981*cosh(d*x + c)^
4 + 4*(135*cosh(d*x + c)^3 + 675*cosh(d*x + c)^2 + 981*cosh(d*x + c) + 385)*sinh(d*x + c)^3 + 1540*cosh(d*x +
c)^3 + 3*(135*cosh(d*x + c)^4 + 900*cosh(d*x + c)^3 + 1962*cosh(d*x + c)^2 + 1540*cosh(d*x + c) + 327)*sinh(d*
x + c)^2 + 981*cosh(d*x + c)^2 + 6*(27*cosh(d*x + c)^5 + 225*cosh(d*x + c)^4 + 654*cosh(d*x + c)^3 + 770*cosh(
d*x + c)^2 + 327*cosh(d*x + c) + 45)*sinh(d*x + c) + 270*cosh(d*x + c) + 27)*log(3*cosh(d*x + c) + 3*sinh(d*x
+ c) + 1) - 1155*(27*cosh(d*x + c)^6 + 54*(3*cosh(d*x + c) + 5)*sinh(d*x + c)^5 + 27*sinh(d*x + c)^6 + 270*cos
h(d*x + c)^5 + 9*(45*cosh(d*x + c)^2 + 150*cosh(d*x + c) + 109)*sinh(d*x + c)^4 + 981*cosh(d*x + c)^4 + 4*(135
*cosh(d*x + c)^3 + 675*cosh(d*x + c)^2 + 981*cosh(d*x + c) + 385)*sinh(d*x + c)^3 + 1540*cosh(d*x + c)^3 + 3*(
135*cosh(d*x + c)^4 + 900*cosh(d*x + c)^3 + 1962*cosh(d*x + c)^2 + 1540*cosh(d*x + c) + 327)*sinh(d*x + c)^2 +
981*cosh(d*x + c)^2 + 6*(27*cosh(d*x + c)^5 + 225*cosh(d*x + c)^4 + 654*cosh(d*x + c)^3 + 770*cosh(d*x + c)^2
+ 327*cosh(d*x + c) + 45)*sinh(d*x + c) + 270*cosh(d*x + c) + 27)*log(cosh(d*x + c) + sinh(d*x + c) + 3) + 24
*(17325*cosh(d*x + c)^4 + 115500*cosh(d*x + c)^3 + 239470*cosh(d*x + c)^2 + 145644*cosh(d*x + c) + 24525)*sinh
(d*x + c) + 588600*cosh(d*x + c) + 67176)/(27*d*cosh(d*x + c)^6 + 27*d*sinh(d*x + c)^6 + 270*d*cosh(d*x + c)^5
+ 54*(3*d*cosh(d*x + c) + 5*d)*sinh(d*x + c)^5 + 981*d*cosh(d*x + c)^4 + 9*(45*d*cosh(d*x + c)^2 + 150*d*cosh
(d*x + c) + 109*d)*sinh(d*x + c)^4 + 1540*d*cosh(d*x + c)^3 + 4*(135*d*cosh(d*x + c)^3 + 675*d*cosh(d*x + c)^2
+ 981*d*cosh(d*x + c) + 385*d)*sinh(d*x + c)^3 + 981*d*cosh(d*x + c)^2 + 3*(135*d*cosh(d*x + c)^4 + 900*d*cos
h(d*x + c)^3 + 1962*d*cosh(d*x + c)^2 + 1540*d*cosh(d*x + c) + 327*d)*sinh(d*x + c)^2 + 270*d*cosh(d*x + c) +
6*(27*d*cosh(d*x + c)^5 + 225*d*cosh(d*x + c)^4 + 654*d*cosh(d*x + c)^3 + 770*d*cosh(d*x + c)^2 + 327*d*cosh(d
*x + c) + 45*d)*sinh(d*x + c) + 27*d)
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Sympy [A] time = 12.9858, size = 784, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5+3*cosh(d*x+c))**4,x)
[Out]
Piecewise((-385*log(tanh(c/2 + d*x/2) - 2)*tanh(c/2 + d*x/2)**6/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(
c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 4620*log(tanh(c/2 + d*x/2) - 2)*tanh(c/2 + d*x
/2)**4/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 209715
2*d) - 18480*log(tanh(c/2 + d*x/2) - 2)*tanh(c/2 + d*x/2)**2/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2
+ d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 24640*log(tanh(c/2 + d*x/2) - 2)/(32768*d*tanh(c/
2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 385*log(tanh(c/2
+ d*x/2) + 2)*tanh(c/2 + d*x/2)**6/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*
tanh(c/2 + d*x/2)**2 - 2097152*d) - 4620*log(tanh(c/2 + d*x/2) + 2)*tanh(c/2 + d*x/2)**4/(32768*d*tanh(c/2 + d
*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 18480*log(tanh(c/2 +
d*x/2) + 2)*tanh(c/2 + d*x/2)**2/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tan
h(c/2 + d*x/2)**2 - 2097152*d) - 24640*log(tanh(c/2 + d*x/2) + 2)/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tan
h(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 2556*tanh(c/2 + d*x/2)**5/(32768*d*tanh(c/2
+ d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) - 14976*tanh(c/2 + d
*x/2)**3/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097
152*d) + 23616*tanh(c/2 + d*x/2)/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tan
h(c/2 + d*x/2)**2 - 2097152*d), Ne(d, 0)), (x/(3*cosh(c) + 5)**4, True))
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Giac [A] time = 1.13681, size = 153, normalized size = 1.44 \begin{align*} \frac{385 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right )}{32768 \, d} - \frac{385 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{32768 \, d} + \frac{10395 \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} + 239470 \, e^{\left (3 \, d x + 3 \, c\right )} + 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 \, e^{\left (d x + c\right )} + 8397}{12288 \, d{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="giac")
[Out]
385/32768*log(3*e^(d*x + c) + 1)/d - 385/32768*log(e^(d*x + c) + 3)/d + 1/12288*(10395*e^(5*d*x + 5*c) + 86625
*e^(4*d*x + 4*c) + 239470*e^(3*d*x + 3*c) + 218466*e^(2*d*x + 2*c) + 73575*e^(d*x + c) + 8397)/(d*(3*e^(2*d*x
+ 2*c) + 10*e^(d*x + c) + 3)^3)