\(\int \frac {1}{(3-5 \cos (c+d x))^3} \, dx\) [40]

Optimal result

Integrand size = 12, antiderivative size = 113 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))} \]

[Out]

43/2048*ln(cos(1/2*d*x+1/2*c)-2*sin(1/2*d*x+1/2*c))/d-43/2048*ln(cos(1/2*d*x+1/2*c)+2*sin(1/2*d*x+1/2*c))/d-5/
32*sin(d*x+c)/d/(3-5*cos(d*x+c))^2+45/512*sin(d*x+c)/d/(3-5*cos(d*x+c))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12, 2738, 213} \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}-\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (2 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]

[In]

Int[(3 - 5*Cos[c + d*x])^(-3),x]

[Out]

(43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) - (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(20
48*d) - (5*Sin[c + d*x])/(32*d*(3 - 5*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(3 - 5*Cos[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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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*} \text {integral}& = -\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {1}{32} \int \frac {-6-5 \cos (c+d x)}{(3-5 \cos (c+d x))^2} \, dx \\ & = -\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac {1}{512} \int \frac {43}{3-5 \cos (c+d x)} \, dx \\ & = -\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac {43}{512} \int \frac {1}{3-5 \cos (c+d x)} \, dx \\ & = -\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac {43 \text {Subst}\left (\int \frac {1}{-2+8 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{256 d} \\ & = \frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {5}{512 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{1024 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {5}{512 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{1024 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

Integrate[(3 - 5*Cos[c + d*x])^(-3),x]

[Out]

(43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) - (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(20
48*d) - 5/(512*d*(Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(1024*d*(Cos[(c + d*x)/2]
- 2*Sin[(c + d*x)/2])) + 5/(512*d*(Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(1024*d*(
Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]))

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79

method	result	size
norman	\(\frac {-\frac {85 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d}+\frac {35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}^{2}}+\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2048 d}-\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048 d}\)	\(89\)
derivativedivides	\(\frac {-\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2048}+\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048}}{d}\)	\(106\)
default	\(\frac {-\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2048}+\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048}}{d}\)	\(106\)
risch	\(\frac {i \left (215 \,{\mathrm e}^{3 i \left (d x +c \right )}-387 \,{\mathrm e}^{2 i \left (d x +c \right )}+325 \,{\mathrm e}^{i \left (d x +c \right )}-225\right )}{256 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{2}}+\frac {43 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}-\frac {4 i}{5}\right )}{2048 d}-\frac {43 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}+\frac {4 i}{5}\right )}{2048 d}\)	\(107\)
parallelrisch	\(\frac {\left (2580 \cos \left (d x +c \right )-1075 \cos \left (2 d x +2 c \right )-1849\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )+\left (-2580 \cos \left (d x +c \right )+1075 \cos \left (2 d x +2 c \right )+1849\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{2}\right )-440 \sin \left (d x +c \right )+900 \sin \left (2 d x +2 c \right )}{2048 d \left (-25 \cos \left (2 d x +2 c \right )-43+60 \cos \left (d x +c \right )\right )}\)	\(117\)

[In]

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

[Out]

(-85/512/d*tan(1/2*d*x+1/2*c)+35/128/d*tan(1/2*d*x+1/2*c)^3)/(4*tan(1/2*d*x+1/2*c)^2-1)^2+43/2048/d*ln(2*tan(1
/2*d*x+1/2*c)-1)-43/2048/d*ln(2*tan(1/2*d*x+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=-\frac {43 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 43 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 40 \, {\left (45 \, \cos \left (d x + c\right ) - 11\right )} \sin \left (d x + c\right )}{4096 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 30 \, d \cos \left (d x + c\right ) + 9 \, d\right )}} \]

[In]

integrate(1/(3-5*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/4096*(43*(25*cos(d*x + c)^2 - 30*cos(d*x + c) + 9)*log(-3/2*cos(d*x + c) + 2*sin(d*x + c) + 5/2) - 43*(25*c
os(d*x + c)^2 - 30*cos(d*x + c) + 9)*log(-3/2*cos(d*x + c) - 2*sin(d*x + c) + 5/2) + 40*(45*cos(d*x + c) - 11)
*sin(d*x + c))/(25*d*cos(d*x + c)^2 - 30*d*cos(d*x + c) + 9*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (102) = 204\).

Time = 1.19 (sec) , antiderivative size = 490, normalized size of antiderivative = 4.34 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=\begin {cases} \frac {x}{\left (3 - 5 \cos {\left (2 \operatorname {atan}{\left (\frac {1}{2} \right )} \right )}\right )^{3}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (\frac {1}{2} \right )} \vee c = - d x + 2 \operatorname {atan}{\left (\frac {1}{2} \right )} \\\frac {x}{\left (3 - 5 \cos {\left (c \right )}\right )^{3}} & \text {for}\: d = 0 \\\frac {688 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} - \frac {344 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} + \frac {43 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} - \frac {688 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} + \frac {344 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} - \frac {43 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} + \frac {560 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} - \frac {340 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2048 d} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(3-5*cos(d*x+c))**3,x)

[Out]

Piecewise((x/(3 - 5*cos(2*atan(1/2)))**3, Eq(c, -d*x - 2*atan(1/2)) | Eq(c, -d*x + 2*atan(1/2))), (x/(3 - 5*co
s(c))**3, Eq(d, 0)), (688*log(2*tan(c/2 + d*x/2) - 1)*tan(c/2 + d*x/2)**4/(32768*d*tan(c/2 + d*x/2)**4 - 16384
*d*tan(c/2 + d*x/2)**2 + 2048*d) - 344*log(2*tan(c/2 + d*x/2) - 1)*tan(c/2 + d*x/2)**2/(32768*d*tan(c/2 + d*x/
2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) + 43*log(2*tan(c/2 + d*x/2) - 1)/(32768*d*tan(c/2 + d*x/2)**4 -
16384*d*tan(c/2 + d*x/2)**2 + 2048*d) - 688*log(2*tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)**4/(32768*d*tan(c/2 +
d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) + 344*log(2*tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)**2/(3276
8*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) - 43*log(2*tan(c/2 + d*x/2) + 1)/(32768*d*tan(
c/2 + d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) + 560*tan(c/2 + d*x/2)**3/(32768*d*tan(c/2 + d*x/2)**4
- 16384*d*tan(c/2 + d*x/2)**2 + 2048*d) - 340*tan(c/2 + d*x/2)/(32768*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c/2
+ d*x/2)**2 + 2048*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=\frac {\frac {20 \, {\left (\frac {17 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {28 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {16 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1} - 43 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 43 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{2048 \, d} \]

[In]

integrate(1/(3-5*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2048*(20*(17*sin(d*x + c)/(cos(d*x + c) + 1) - 28*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(8*sin(d*x + c)^2/(co
s(d*x + c) + 1)^2 - 16*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 1) - 43*log(2*sin(d*x + c)/(cos(d*x + c) + 1) + 1
) + 43*log(2*sin(d*x + c)/(cos(d*x + c) + 1) - 1))/d

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=\frac {\frac {20 \, {\left (28 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 17 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - 43 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 43 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{2048 \, d} \]

[In]

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

[Out]

1/2048*(20*(28*tan(1/2*d*x + 1/2*c)^3 - 17*tan(1/2*d*x + 1/2*c))/(4*tan(1/2*d*x + 1/2*c)^2 - 1)^2 - 43*log(abs
(2*tan(1/2*d*x + 1/2*c) + 1)) + 43*log(abs(2*tan(1/2*d*x + 1/2*c) - 1)))/d

Mupad [B] (verification not implemented)

Time = 14.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(3-5 \cos (c+d x))^3} \, dx=-\frac {43\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}-\frac {\frac {85\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8192}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2048}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {1}{16}\right )} \]

[In]

int(-1/(5*cos(c + d*x) - 3)^3,x)

[Out]

- (43*atanh(2*tan(c/2 + (d*x)/2)))/(1024*d) - ((85*tan(c/2 + (d*x)/2))/8192 - (35*tan(c/2 + (d*x)/2)^3)/2048)/
(d*(tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)^2/2 + 1/16))