3.1.48 \(\int \frac {1}{(-3-5 \cos (c+d x))^3} \, dx\) [48]

Optimal. Leaf size=115 \[ \frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {5 \sin (c+d x)}{32 d (3+5 \cos (c+d x))^2}+\frac {45 \sin (c+d x)}{512 d (3+5 \cos (c+d x))} \]

[Out]

43/2048*ln(2*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c))/d-43/2048*ln(2*cos(1/2*d*x+1/2*c)+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))

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Rubi [A]
time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12, 2738, 213} \begin {gather*} \frac {45 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)}-\frac {5 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}+\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 5*Cos[c + d*x])^(-3),x]

[Out]

(43*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2048*d) - (43*Log[2*Cos[(c + d*x)/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*} \int \frac {1}{(-3-5 \cos (c+d x))^3} \, dx &=-\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}{-8+2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}\\ &=\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\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*}

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Mathematica [A]
time = 0.07, size = 217, normalized size = 1.89 \begin {gather*} \frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {5}{512 d \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{2048 d \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {5}{512 d \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{2048 d \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.07, size = 94, normalized size = 0.82

method	result	size
norman	\(\frac {\frac {35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d}-\frac {85 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right )^{2}}+\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{2048 d}-\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{2048 d}\)	\(83\)
derivativedivides	\(\frac {\frac {25}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {85}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{2048}-\frac {25}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {85}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{2048}}{d}\)	\(94\)
default	\(\frac {\frac {25}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {85}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{2048}-\frac {25}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {85}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{2048}}{d}\)	\(94\)
risch	\(\frac {i \left (215 \,{\mathrm e}^{3 i \left (d x +c \right )}+387 \,{\mathrm e}^{2 i \left (d x +c \right )}+325 \,{\mathrm e}^{i \left (d x +c \right )}+225\right )}{256 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{2}}+\frac {43 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{2048 d}-\frac {43 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{2048 d}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(25/512/(tan(1/2*d*x+1/2*c)+2)^2-85/1024/(tan(1/2*d*x+1/2*c)+2)-43/2048*ln(tan(1/2*d*x+1/2*c)+2)-25/512/(t
an(1/2*d*x+1/2*c)-2)^2-85/1024/(tan(1/2*d*x+1/2*c)-2)+43/2048*ln(tan(1/2*d*x+1/2*c)-2))

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Maxima [A]
time = 0.27, size = 135, normalized size = 1.17 \begin {gather*} -\frac {\frac {20 \, {\left (\frac {28 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {17 \, \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 {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 16} + 43 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 43 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{2048 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 129, normalized size = 1.12 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*co
s(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)*s
in(d*x + c))/(25*d*cos(d*x + c)^2 + 30*d*cos(d*x + c) + 9*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (102) = 204\).
time = 1.27, size = 478, normalized size = 4.16 \begin {gather*} \begin {cases} \frac {x}{\left (-3 - 5 \cos {\left (2 \operatorname {atan}{\left (2 \right )} \right )}\right )^{3}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (2 \right )} \vee c = - d x + 2 \operatorname {atan}{\left (2 \right )} \\\frac {x}{\left (- 5 \cos {\left (c \right )} - 3\right )^{3}} & \text {for}\: d = 0 \\\frac {43 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 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 )} + 32768 d} - \frac {344 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 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 )} + 32768 d} + \frac {688 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{2048 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 )} + 32768 d} - \frac {43 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 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 )} + 32768 d} + \frac {344 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 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 )} + 32768 d} - \frac {688 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{2048 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 )} + 32768 d} - \frac {340 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 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 )} + 32768 d} + \frac {560 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 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 )} + 32768 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(-3 - 5*cos(2*atan(2)))**3, Eq(c, -d*x - 2*atan(2)) | Eq(c, -d*x + 2*atan(2))), (x/(-5*cos(c) - 3
)**3, Eq(d, 0)), (43*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**4/(2048*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c
/2 + d*x/2)**2 + 32768*d) - 344*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**2/(2048*d*tan(c/2 + d*x/2)**4 - 16
384*d*tan(c/2 + d*x/2)**2 + 32768*d) + 688*log(tan(c/2 + d*x/2) - 2)/(2048*d*tan(c/2 + d*x/2)**4 - 16384*d*tan
(c/2 + d*x/2)**2 + 32768*d) - 43*log(tan(c/2 + d*x/2) + 2)*tan(c/2 + d*x/2)**4/(2048*d*tan(c/2 + d*x/2)**4 - 1
6384*d*tan(c/2 + d*x/2)**2 + 32768*d) + 344*log(tan(c/2 + d*x/2) + 2)*tan(c/2 + d*x/2)**2/(2048*d*tan(c/2 + d*
x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 32768*d) - 688*log(tan(c/2 + d*x/2) + 2)/(2048*d*tan(c/2 + d*x/2)**4 -
16384*d*tan(c/2 + d*x/2)**2 + 32768*d) - 340*tan(c/2 + d*x/2)**3/(2048*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c/
2 + d*x/2)**2 + 32768*d) + 560*tan(c/2 + d*x/2)/(2048*d*tan(c/2 + d*x/2)**4 - 16384*d*tan(c/2 + d*x/2)**2 + 32
768*d), True))

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Giac [A]
time = 0.45, size = 78, normalized size = 0.68 \begin {gather*} -\frac {\frac {20 \, {\left (17 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{2}} + 43 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - 43 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{2048 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.67, size = 75, normalized size = 0.65 \begin {gather*} \frac {\frac {35\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}-\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{512}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+16\right )}-\frac {43\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{1024\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(5*cos(c + d*x) + 3)^3,x)

[Out]

((35*tan(c/2 + (d*x)/2))/128 - (85*tan(c/2 + (d*x)/2)^3)/512)/(d*(tan(c/2 + (d*x)/2)^4 - 8*tan(c/2 + (d*x)/2)^
2 + 16)) - (43*atanh(tan(c/2 + (d*x)/2)/2))/(1024*d)

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