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]
385/32768*x-385/16384*arctanh(sinh(d*x+c)/(3+cosh(d*x+c)))/d-1/16*sinh(d*x+c)/d/(5+3*cosh(d*x+c))^3-25/512*sin
h(d*x+c)/d/(5+3*cosh(d*x+c))^2-311/8192*sinh(d*x+c)/d/(5+3*cosh(d*x+c))
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Rubi [A] time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00,
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 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]
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]
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*}
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Mathematica [A] time = 0.27, 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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fricas [B] time = 0.66, size = 1078, normalized size = 10.17 \[ \text {result too large to display} \]
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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giac [A] time = 0.12, size = 109, normalized size = 1.03 \[ \frac {\frac {8 \, {\left (10395 \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} + 239470 \, e^{\left (3 \, d x + 3 \, c\right )} + 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{3}} + 1155 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 1155 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{98304 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="giac")
[Out]
1/98304*(8*(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)/(3*e^(2*d*x + 2*c) + 10*e^(d*x + c) + 3)^3 + 1155*log(3*e^(d*x + c) + 1) - 1155*log(
e^(d*x + c) + 3))/d
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maple [A] time = 0.08, size = 144, normalized size = 1.36 \[ \frac {9}{2048 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768 d}+\frac {9}{2048 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768 d} \]
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 = 0.34, size = 169, normalized size = 1.59 \[ -\frac {385 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{32768 \, d} + \frac {385 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{32768 \, d} - \frac {73575 \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} + 239470 \, e^{\left (-3 \, d x - 3 \, c\right )} + 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 \, e^{\left (-5 \, d x - 5 \, c\right )} + 8397}{12288 \, d {\left (270 \, e^{\left (-d x - c\right )} + 981 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1540 \, e^{\left (-3 \, d x - 3 \, c\right )} + 981 \, e^{\left (-4 \, d x - 4 \, c\right )} + 270 \, e^{\left (-5 \, d x - 5 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 27\right )}} \]
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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mupad [B] time = 0.11, size = 226, normalized size = 2.13 \[ \frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {1925}{12288\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {385\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{16384\,\sqrt {-d^2}}-\frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{1152\,d}+\frac {3461}{3456\,d}}{60\,{\mathrm {e}}^{c+d\,x}+118\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+9\,{\mathrm {e}}^{4\,c+4\,d\,x}+9}+\frac {\frac {365\,{\mathrm {e}}^{c+d\,x}}{54\,d}+\frac {41}{18\,d}}{270\,{\mathrm {e}}^{c+d\,x}+981\,{\mathrm {e}}^{2\,c+2\,d\,x}+1540\,{\mathrm {e}}^{3\,c+3\,d\,x}+981\,{\mathrm {e}}^{4\,c+4\,d\,x}+270\,{\mathrm {e}}^{5\,c+5\,d\,x}+27\,{\mathrm {e}}^{6\,c+6\,d\,x}+27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3*cosh(c + d*x) + 5)^4,x)
[Out]
((385*exp(c + d*x))/(4096*d) + 1925/(12288*d))/(10*exp(c + d*x) + 3*exp(2*c + 2*d*x) + 3) - (385*atan((5/(4*d)
+ (3*exp(d*x)*exp(c))/(4*d))*(-d^2)^(1/2)))/(16384*(-d^2)^(1/2)) - ((385*exp(c + d*x))/(1152*d) + 3461/(3456*
d))/(60*exp(c + d*x) + 118*exp(2*c + 2*d*x) + 60*exp(3*c + 3*d*x) + 9*exp(4*c + 4*d*x) + 9) + ((365*exp(c + d*
x))/(54*d) + 41/(18*d))/(270*exp(c + d*x) + 981*exp(2*c + 2*d*x) + 1540*exp(3*c + 3*d*x) + 981*exp(4*c + 4*d*x
) + 270*exp(5*c + 5*d*x) + 27*exp(6*c + 6*d*x) + 27)
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sympy [A] time = 7.95, size = 784, normalized size = 7.40 \[ \text {result too large to display} \]
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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