3.1.74 \(\int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx\) [74]

Optimal. Leaf size=98 \[ -\frac {279 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))} \]

[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 = 0.07, 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 = {2743, 2833, 12, 2738, 212} \begin {gather*} -\frac {279 \text {ArcTan}\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} \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2738

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 2743

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] && Integ
erQ[2*n]

Rule 2833

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) \text {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 = 0.19, size = 65, normalized size = 0.66 \begin {gather*} \frac {-837 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )+\frac {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} \end {gather*}

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 = 0.86, size = 75, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {745 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {265 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\)	\(75\)
default	\(\frac {-\frac {-\frac {745 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {265 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\)	\(75\)
risch	\(-\frac {20925 \,{\mathrm e}^{5 d x +5 c}+62775 \,{\mathrm e}^{4 d x +4 c}+111042 \,{\mathrm e}^{3 d x +3 c}+119310 \,{\mathrm e}^{2 d x +2 c}+68625 \,{\mathrm e}^{d x +c}+24875}{12288 d \left (5 \,{\mathrm e}^{2 d x +2 c}+6 \,{\mathrm e}^{d x +c}+5\right )^{3}}+\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}-\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+5*cosh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/8*(-745/1024*tanh(1/2*d*x+1/2*c)^5-265/96*tanh(1/2*d*x+1/2*c)^3-295/64*tanh(1/2*d*x+1/2*c))/(tanh(1/2*
d*x+1/2*c)^2+4)^3-279/16384*arctan(1/2*tanh(1/2*d*x+1/2*c)))

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Maxima [A]
time = 0.48, size = 152, normalized size = 1.55 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (87) = 174\).
time = 0.37, size = 793, normalized size = 8.09 \begin {gather*} -\frac {83700 \, \cosh \left (d x + c\right )^{5} + 83700 \, {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{4} + 83700 \, \sinh \left (d x + c\right )^{5} + 251100 \, \cosh \left (d x + c\right )^{4} + 2232 \, {\left (375 \, \cosh \left (d x + c\right )^{2} + 450 \, \cosh \left (d x + c\right ) + 199\right )} \sinh \left (d x + c\right )^{3} + 444168 \, \cosh \left (d x + c\right )^{3} + 24 \, {\left (34875 \, \cosh \left (d x + c\right )^{3} + 62775 \, \cosh \left (d x + c\right )^{2} + 55521 \, \cosh \left (d x + c\right ) + 19885\right )} \sinh \left (d x + c\right )^{2} + 837 \, {\left (125 \, \cosh \left (d x + c\right )^{6} + 150 \, {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{5} + 125 \, \sinh \left (d x + c\right )^{6} + 450 \, \cosh \left (d x + c\right )^{5} + 15 \, {\left (125 \, \cosh \left (d x + c\right )^{2} + 150 \, \cosh \left (d x + c\right ) + 61\right )} \sinh \left (d x + c\right )^{4} + 915 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (625 \, \cosh \left (d x + c\right )^{3} + 1125 \, \cosh \left (d x + c\right )^{2} + 915 \, \cosh \left (d x + c\right ) + 279\right )} \sinh \left (d x + c\right )^{3} + 1116 \, \cosh \left (d x + c\right )^{3} + 3 \, {\left (625 \, \cosh \left (d x + c\right )^{4} + 1500 \, \cosh \left (d x + c\right )^{3} + 1830 \, \cosh \left (d x + c\right )^{2} + 1116 \, \cosh \left (d x + c\right ) + 305\right )} \sinh \left (d x + c\right )^{2} + 915 \, \cosh \left (d x + c\right )^{2} + 6 \, {\left (125 \, \cosh \left (d x + c\right )^{5} + 375 \, \cosh \left (d x + c\right )^{4} + 610 \, \cosh \left (d x + c\right )^{3} + 558 \, \cosh \left (d x + c\right )^{2} + 305 \, \cosh \left (d x + c\right ) + 75\right )} \sinh \left (d x + c\right ) + 450 \, \cosh \left (d x + c\right ) + 125\right )} \arctan \left (\frac {5}{4} \, \cosh \left (d x + c\right ) + \frac {5}{4} \, \sinh \left (d x + c\right ) + \frac {3}{4}\right ) + 477240 \, \cosh \left (d x + c\right )^{2} + 12 \, {\left (34875 \, \cosh \left (d x + c\right )^{4} + 83700 \, \cosh \left (d x + c\right )^{3} + 111042 \, \cosh \left (d x + c\right )^{2} + 79540 \, \cosh \left (d x + c\right ) + 22875\right )} \sinh \left (d x + c\right ) + 274500 \, \cosh \left (d x + c\right ) + 99500}{49152 \, {\left (125 \, d \cosh \left (d x + c\right )^{6} + 125 \, d \sinh \left (d x + c\right )^{6} + 450 \, d \cosh \left (d x + c\right )^{5} + 150 \, {\left (5 \, d \cosh \left (d x + c\right ) + 3 \, d\right )} \sinh \left (d x + c\right )^{5} + 915 \, d \cosh \left (d x + c\right )^{4} + 15 \, {\left (125 \, d \cosh \left (d x + c\right )^{2} + 150 \, d \cosh \left (d x + c\right ) + 61 \, d\right )} \sinh \left (d x + c\right )^{4} + 1116 \, d \cosh \left (d x + c\right )^{3} + 4 \, {\left (625 \, d \cosh \left (d x + c\right )^{3} + 1125 \, d \cosh \left (d x + c\right )^{2} + 915 \, d \cosh \left (d x + c\right ) + 279 \, d\right )} \sinh \left (d x + c\right )^{3} + 915 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (625 \, d \cosh \left (d x + c\right )^{4} + 1500 \, d \cosh \left (d x + c\right )^{3} + 1830 \, d \cosh \left (d x + c\right )^{2} + 1116 \, d \cosh \left (d x + c\right ) + 305 \, d\right )} \sinh \left (d x + c\right )^{2} + 450 \, d \cosh \left (d x + c\right ) + 6 \, {\left (125 \, d \cosh \left (d x + c\right )^{5} + 375 \, d \cosh \left (d x + c\right )^{4} + 610 \, d \cosh \left (d x + c\right )^{3} + 558 \, d \cosh \left (d x + c\right )^{2} + 305 \, d \cosh \left (d x + c\right ) + 75 \, d\right )} \sinh \left (d x + c\right ) + 125 \, d\right )}} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 4.95, size = 908, normalized size = 9.27 \begin {gather*} \begin {cases} \frac {x}{625 \cosh ^{4}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1500 \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1350 \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 540 \cosh {\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 81} & \text {for}\: c = \log {\left (\left (- \frac {3}{5} - \frac {4 i}{5}\right ) e^{- d x} \right )} \\- \frac {\log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )}}{625 d \cosh ^{4}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1500 d \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1350 d \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 540 d \cosh {\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 81 d} + \frac {\log {\left (5 \right )}}{625 d \cosh ^{4}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1500 d \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1350 d \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 540 d \cosh {\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 81 d} & \text {for}\: c = \log {\left (\left (- \frac {3}{5} + \frac {4 i}{5}\right ) e^{- d x} \right )} \\\frac {x}{\left (5 \cosh {\left (c \right )} + 3\right )^{4}} & \text {for}\: d = 0 \\- \frac {837 \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} + \frac {4470 \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} - \frac {10044 \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} + \frac {16960 \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} - \frac {40176 \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} + \frac {28320 \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} - \frac {53568 \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(625*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5))**4 + 1500*cosh(d*x + log(-3*exp(-d*x)
- 4*I*exp(-d*x)) - log(5))**3 + 1350*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5))**2 + 540*cosh(d*x
+ log(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5)) + 81), Eq(c, log((-3/5 - 4*I/5)*exp(-d*x)))), (-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) + log(5)/(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*c
osh(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((-3/5 + 4*I/5)*exp(-d*x)))), (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*t
anh(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)/(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
) + 16960*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)/(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) - 53
568*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), True))

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Giac [A]
time = 0.41, size = 98, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}

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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Mupad [B]
time = 0.95, size = 223, normalized size = 2.28 \begin {gather*} \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} \end {gather*}

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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