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*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]))

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Rubi [A]  time = 0.097106, antiderivative size = 98, normalized size of antiderivative = 1., 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 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 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 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])

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*}

Mathematica [A]  time = 0.245403, 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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Maple [A]  time = 0.016, size = 110, normalized size = 1.1 \begin{align*}{\frac{745}{8192\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-3}}+{\frac{265}{768\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-3}}+{\frac{295}{512\,d}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-3}}-{\frac{279}{16384\,d}\arctan \left ({\frac{1}{2}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}

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 = 1.57465, size = 205, normalized size = 2.09 \begin{align*} \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 )}} \end{align*}

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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Fricas [B]  time = 2.48881, size = 2473, normalized size = 25.23 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (5 \cosh{\left (c + d x \right )} + 3\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*cosh(d*x+c))**4,x)

[Out]

Integral((5*cosh(c + d*x) + 3)**(-4), x)

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Giac [A]  time = 1.13231, size = 134, normalized size = 1.37 \begin{align*} -\frac{279 \, \arctan \left (\frac{5}{4} \, e^{\left (d x + c\right )} + \frac{3}{4}\right )}{16384 \, d} - \frac{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}{12288 \, d{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} + 6 \, e^{\left (d x + c\right )} + 5\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="giac")

[Out]

-279/16384*arctan(5/4*e^(d*x + c) + 3/4)/d - 1/12288*(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)/(d*(5*e^(2*d*x + 2*c) + 6*e^(d*x + c) + 5
)^3)