3.74 \(\int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx\)
Optimal. Leaf size=98 \[ -\frac {279 \tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {995 \sinh (c+d x)}{24576 d (5 \cosh (c+d x)+3)}-\frac {25 \sinh (c+d x)}{512 d (5 \cosh (c+d x)+3)^2}+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3} \]
[Out]
-279/16384*arctan(1/2*tanh(1/2*d*x+1/2*c))/d+5/48*sinh(d*x+c)/d/(3+5*cosh(d*x+c))^3-25/512*sinh(d*x+c)/d/(3+5*
cosh(d*x+c))^2+995/24576*sinh(d*x+c)/d/(3+5*cosh(d*x+c))
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Rubi [A] time = 0.10, antiderivative size = 98, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used =
{2664, 2754, 12, 2659, 206} \[ -\frac {279 \tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {995 \sinh (c+d x)}{24576 d (5 \cosh (c+d x)+3)}-\frac {25 \sinh (c+d x)}{512 d (5 \cosh (c+d x)+3)^2}+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3} \]
Antiderivative was successfully verified.
[In]
Int[(3 + 5*Cosh[c + d*x])^(-4),x]
[Out]
(-279*ArcTan[Tanh[(c + d*x)/2]/2])/(16384*d) + (5*Sinh[c + d*x])/(48*d*(3 + 5*Cosh[c + d*x])^3) - (25*Sinh[c +
d*x])/(512*d*(3 + 5*Cosh[c + d*x])^2) + (995*Sinh[c + d*x])/(24576*d*(3 + 5*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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 2659
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]
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}{(3+5 \cosh (c+d x))^4} \, dx &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}+\frac {1}{48} \int \frac {-9+10 \cosh (c+d x)}{(3+5 \cosh (c+d x))^3} \, dx\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {\int \frac {154-75 \cosh (c+d x)}{(3+5 \cosh (c+d x))^2} \, dx}{1536}\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}+\frac {\int -\frac {837}{3+5 \cosh (c+d x)} \, dx}{24576}\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}-\frac {279 \int \frac {1}{3+5 \cosh (c+d x)} \, dx}{8192}\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}+\frac {(279 i) \operatorname {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{4096 d}\\ &=-\frac {279 \tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 65, normalized size = 0.66 \[ \frac {\frac {5 \sinh (c+d x) (9540 \cosh (c+d x)+4975 \cosh (2 (c+d x))+8141)}{(5 \cosh (c+d x)+3)^3}-837 \tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{49152 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(3 + 5*Cosh[c + d*x])^(-4),x]
[Out]
(-837*ArcTan[Tanh[(c + d*x)/2]/2] + (5*(8141 + 9540*Cosh[c + d*x] + 4975*Cosh[2*(c + d*x)])*Sinh[c + d*x])/(3
+ 5*Cosh[c + d*x])^3)/(49152*d)
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fricas [B] time = 0.67, size = 793, normalized size = 8.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="fricas")
[Out]
-1/49152*(83700*cosh(d*x + c)^5 + 83700*(5*cosh(d*x + c) + 3)*sinh(d*x + c)^4 + 83700*sinh(d*x + c)^5 + 251100
*cosh(d*x + c)^4 + 2232*(375*cosh(d*x + c)^2 + 450*cosh(d*x + c) + 199)*sinh(d*x + c)^3 + 444168*cosh(d*x + c)
^3 + 24*(34875*cosh(d*x + c)^3 + 62775*cosh(d*x + c)^2 + 55521*cosh(d*x + c) + 19885)*sinh(d*x + c)^2 + 837*(1
25*cosh(d*x + c)^6 + 150*(5*cosh(d*x + c) + 3)*sinh(d*x + c)^5 + 125*sinh(d*x + c)^6 + 450*cosh(d*x + c)^5 + 1
5*(125*cosh(d*x + c)^2 + 150*cosh(d*x + c) + 61)*sinh(d*x + c)^4 + 915*cosh(d*x + c)^4 + 4*(625*cosh(d*x + c)^
3 + 1125*cosh(d*x + c)^2 + 915*cosh(d*x + c) + 279)*sinh(d*x + c)^3 + 1116*cosh(d*x + c)^3 + 3*(625*cosh(d*x +
c)^4 + 1500*cosh(d*x + c)^3 + 1830*cosh(d*x + c)^2 + 1116*cosh(d*x + c) + 305)*sinh(d*x + c)^2 + 915*cosh(d*x
+ c)^2 + 6*(125*cosh(d*x + c)^5 + 375*cosh(d*x + c)^4 + 610*cosh(d*x + c)^3 + 558*cosh(d*x + c)^2 + 305*cosh(
d*x + c) + 75)*sinh(d*x + c) + 450*cosh(d*x + c) + 125)*arctan(5/4*cosh(d*x + c) + 5/4*sinh(d*x + c) + 3/4) +
477240*cosh(d*x + c)^2 + 12*(34875*cosh(d*x + c)^4 + 83700*cosh(d*x + c)^3 + 111042*cosh(d*x + c)^2 + 79540*co
sh(d*x + c) + 22875)*sinh(d*x + c) + 274500*cosh(d*x + c) + 99500)/(125*d*cosh(d*x + c)^6 + 125*d*sinh(d*x + c
)^6 + 450*d*cosh(d*x + c)^5 + 150*(5*d*cosh(d*x + c) + 3*d)*sinh(d*x + c)^5 + 915*d*cosh(d*x + c)^4 + 15*(125*
d*cosh(d*x + c)^2 + 150*d*cosh(d*x + c) + 61*d)*sinh(d*x + c)^4 + 1116*d*cosh(d*x + c)^3 + 4*(625*d*cosh(d*x +
c)^3 + 1125*d*cosh(d*x + c)^2 + 915*d*cosh(d*x + c) + 279*d)*sinh(d*x + c)^3 + 915*d*cosh(d*x + c)^2 + 3*(625
*d*cosh(d*x + c)^4 + 1500*d*cosh(d*x + c)^3 + 1830*d*cosh(d*x + c)^2 + 1116*d*cosh(d*x + c) + 305*d)*sinh(d*x
+ c)^2 + 450*d*cosh(d*x + c) + 6*(125*d*cosh(d*x + c)^5 + 375*d*cosh(d*x + c)^4 + 610*d*cosh(d*x + c)^3 + 558*
d*cosh(d*x + c)^2 + 305*d*cosh(d*x + c) + 75*d)*sinh(d*x + c) + 125*d)
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giac [A] time = 0.15, size = 98, normalized size = 1.00 \[ -\frac {\frac {4 \, {\left (20925 \, e^{\left (5 \, d x + 5 \, c\right )} + 62775 \, e^{\left (4 \, d x + 4 \, c\right )} + 111042 \, e^{\left (3 \, d x + 3 \, c\right )} + 119310 \, e^{\left (2 \, d x + 2 \, c\right )} + 68625 \, e^{\left (d x + c\right )} + 24875\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} + 6 \, e^{\left (d x + c\right )} + 5\right )}^{3}} + 837 \, \arctan \left (\frac {5}{4} \, e^{\left (d x + c\right )} + \frac {3}{4}\right )}{49152 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="giac")
[Out]
-1/49152*(4*(20925*e^(5*d*x + 5*c) + 62775*e^(4*d*x + 4*c) + 111042*e^(3*d*x + 3*c) + 119310*e^(2*d*x + 2*c) +
68625*e^(d*x + c) + 24875)/(5*e^(2*d*x + 2*c) + 6*e^(d*x + c) + 5)^3 + 837*arctan(5/4*e^(d*x + c) + 3/4))/d
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maple [A] time = 0.07, size = 110, normalized size = 1.12 \[ \frac {745 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8192 d \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}+\frac {265 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}+\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3+5*cosh(d*x+c))^4,x)
[Out]
745/8192/d/(tanh(1/2*d*x+1/2*c)^2+4)^3*tanh(1/2*d*x+1/2*c)^5+265/768/d/(tanh(1/2*d*x+1/2*c)^2+4)^3*tanh(1/2*d*
x+1/2*c)^3+295/512/d/(tanh(1/2*d*x+1/2*c)^2+4)^3*tanh(1/2*d*x+1/2*c)-279/16384*arctan(1/2*tanh(1/2*d*x+1/2*c))
/d
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maxima [A] time = 0.41, size = 152, normalized size = 1.55 \[ \frac {279 \, \arctan \left (\frac {5}{4} \, e^{\left (-d x - c\right )} + \frac {3}{4}\right )}{16384 \, d} + \frac {68625 \, e^{\left (-d x - c\right )} + 119310 \, e^{\left (-2 \, d x - 2 \, c\right )} + 111042 \, e^{\left (-3 \, d x - 3 \, c\right )} + 62775 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20925 \, e^{\left (-5 \, d x - 5 \, c\right )} + 24875}{12288 \, d {\left (450 \, e^{\left (-d x - c\right )} + 915 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1116 \, e^{\left (-3 \, d x - 3 \, c\right )} + 915 \, e^{\left (-4 \, d x - 4 \, c\right )} + 450 \, e^{\left (-5 \, d x - 5 \, c\right )} + 125 \, e^{\left (-6 \, d x - 6 \, c\right )} + 125\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="maxima")
[Out]
279/16384*arctan(5/4*e^(-d*x - c) + 3/4)/d + 1/12288*(68625*e^(-d*x - c) + 119310*e^(-2*d*x - 2*c) + 111042*e^
(-3*d*x - 3*c) + 62775*e^(-4*d*x - 4*c) + 20925*e^(-5*d*x - 5*c) + 24875)/(d*(450*e^(-d*x - c) + 915*e^(-2*d*x
- 2*c) + 1116*e^(-3*d*x - 3*c) + 915*e^(-4*d*x - 4*c) + 450*e^(-5*d*x - 5*c) + 125*e^(-6*d*x - 6*c) + 125))
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mupad [B] time = 0.95, size = 223, normalized size = 2.28 \[ \frac {\frac {39\,{\mathrm {e}}^{c+d\,x}}{50\,d}+\frac {7}{30\,d}}{450\,{\mathrm {e}}^{c+d\,x}+915\,{\mathrm {e}}^{2\,c+2\,d\,x}+1116\,{\mathrm {e}}^{3\,c+3\,d\,x}+915\,{\mathrm {e}}^{4\,c+4\,d\,x}+450\,{\mathrm {e}}^{5\,c+5\,d\,x}+125\,{\mathrm {e}}^{6\,c+6\,d\,x}+125}-\frac {\frac {93\,{\mathrm {e}}^{c+d\,x}}{640\,d}+\frac {791}{3200\,d}}{60\,{\mathrm {e}}^{c+d\,x}+86\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+25\,{\mathrm {e}}^{4\,c+4\,d\,x}+25}-\frac {279\,\mathrm {atan}\left (\left (\frac {3}{4\,d}+\frac {5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {d^2}\right )}{16384\,\sqrt {d^2}}-\frac {\frac {279\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {837}{20480\,d}}{6\,{\mathrm {e}}^{c+d\,x}+5\,{\mathrm {e}}^{2\,c+2\,d\,x}+5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(5*cosh(c + d*x) + 3)^4,x)
[Out]
((39*exp(c + d*x))/(50*d) + 7/(30*d))/(450*exp(c + d*x) + 915*exp(2*c + 2*d*x) + 1116*exp(3*c + 3*d*x) + 915*e
xp(4*c + 4*d*x) + 450*exp(5*c + 5*d*x) + 125*exp(6*c + 6*d*x) + 125) - ((93*exp(c + d*x))/(640*d) + 791/(3200*
d))/(60*exp(c + d*x) + 86*exp(2*c + 2*d*x) + 60*exp(3*c + 3*d*x) + 25*exp(4*c + 4*d*x) + 25) - (279*atan((3/(4
*d) + (5*exp(d*x)*exp(c))/(4*d))*(d^2)^(1/2)))/(16384*(d^2)^(1/2)) - ((279*exp(c + d*x))/(4096*d) + 837/(20480
*d))/(6*exp(c + d*x) + 5*exp(2*c + 2*d*x) + 5)
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sympy [A] time = 14.13, size = 809, normalized size = 8.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3+5*cosh(d*x+c))**4,x)
[Out]
Piecewise((-log(-3*exp(-d*x) + 4*I*exp(-d*x))/(625*d*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5))**4
+ 1500*d*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5))**3 + 1350*d*cosh(d*x + log(-3*exp(-d*x) + 4*I
*exp(-d*x)) - log(5))**2 + 540*d*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5)) + 81*d), Eq(c, log((-3
+ 4*I)*exp(-d*x)) - log(5))), (-log(-3*exp(-d*x) - 4*I*exp(-d*x))/(625*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*ex
p(-d*x)) - log(5))**4 + 1500*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5))**3 + 1350*d*cosh(d*x + l
og(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5))**2 + 540*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5)) +
81*d), Eq(c, log(-(3 + 4*I)*exp(-d*x)) - log(5))), (x/(5*cosh(c) + 3)**4, Eq(d, 0)), (-837*tanh(c/2 + d*x/2)*
*6*atan(tanh(c/2 + d*x/2)/2)/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/
2 + d*x/2)**2 + 3145728*d) + 4470*tanh(c/2 + d*x/2)**5/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x
/2)**4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) - 10044*tanh(c/2 + d*x/2)**4*atan(tanh(c/2 + d*x/2)/2)/(4
9152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) + 16
960*tanh(c/2 + d*x/2)**3/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/2 +
d*x/2)**2 + 3145728*d) - 40176*tanh(c/2 + d*x/2)**2*atan(tanh(c/2 + d*x/2)/2)/(49152*d*tanh(c/2 + d*x/2)**6 +
589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) + 28320*tanh(c/2 + d*x/2)/(49152*d
*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) - 53568*at
an(tanh(c/2 + d*x/2)/2)/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/2 + d
*x/2)**2 + 3145728*d), True))
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