3.1.37 \(\int \frac {1}{(3+5 \cos (c+d x))^4} \, dx\) [37]

Optimal. Leaf size=140 \[ \frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))} \]

[Out]

279/32768*ln(2*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c))/d-279/32768*ln(2*cos(1/2*d*x+1/2*c)+sin(1/2*d*x+1/2*c))/
d+5/48*sin(d*x+c)/d/(3+5*cos(d*x+c))^3-25/512*sin(d*x+c)/d/(3+5*cos(d*x+c))^2+995/24576*sin(d*x+c)/d/(3+5*cos(
d*x+c))

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Rubi [A]
time = 0.08, antiderivative size = 140, 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 {995 \sin (c+d x)}{24576 d (5 \cos (c+d x)+3)}-\frac {25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Cos[c + d*x])^(-4),x]

[Out]

(279*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(32768*d) - (279*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/
(32768*d) + (5*Sin[c + d*x])/(48*d*(3 + 5*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x])^2)
+ (995*Sin[c + d*x])/(24576*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 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 \cos (c+d x))^4} \, dx &=\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}+\frac {1}{48} \int \frac {-9+10 \cos (c+d x)}{(3+5 \cos (c+d x))^3} \, dx\\ &=\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {\int \frac {154-75 \cos (c+d x)}{(3+5 \cos (c+d x))^2} \, dx}{1536}\\ &=\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))}+\frac {\int -\frac {837}{3+5 \cos (c+d x)} \, dx}{24576}\\ &=\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))}-\frac {279 \int \frac {1}{3+5 \cos (c+d x)} \, dx}{8192}\\ &=\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))}-\frac {279 \text {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d}\\ &=\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(296\) vs. \(2(140)=280\).
time = 0.20, size = 296, normalized size = 2.11 \begin {gather*} \frac {467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+765855 \cos (c+d x) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+226140 \sin (c+d x)+190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))}{393216 d (3+5 \cos (c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*Cos[c + d*x])^(-4),x]

[Out]

(467046*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c +
d*x)/2]] + 765855*Cos[c + d*x]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + S
in[(c + d*x)/2]]) - 467046*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c +
d*x)/2] + Sin[(c + d*x)/2]] + 226140*Sin[c + d*x] + 190800*Sin[2*(c + d*x)] + 99500*Sin[3*(c + d*x)])/(393216
*d*(3 + 5*Cos[c + d*x])^3)

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Maple [A]
time = 0.10, size = 124, normalized size = 0.89

method	result	size
norman	\(\frac {-\frac {295 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d}+\frac {265 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}-\frac {745 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8192 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right )^{3}}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768 d}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768 d}\)	\(99\)
derivativedivides	\(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\)	\(124\)
default	\(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\)	\(124\)
risch	\(\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}+119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}+24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-125/6144/(tan(1/2*d*x+1/2*c)+2)^3+175/4096/(tan(1/2*d*x+1/2*c)+2)^2-745/16384/(tan(1/2*d*x+1/2*c)+2)-279
/32768*ln(tan(1/2*d*x+1/2*c)+2)-125/6144/(tan(1/2*d*x+1/2*c)-2)^3-175/4096/(tan(1/2*d*x+1/2*c)-2)^2-745/16384/
(tan(1/2*d*x+1/2*c)-2)+279/32768*ln(tan(1/2*d*x+1/2*c)-2))

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Maxima [A]
time = 0.30, size = 174, normalized size = 1.24 \begin {gather*} -\frac {\frac {20 \, {\left (\frac {2832 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {447 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} + 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{98304 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/98304*(20*(2832*sin(d*x + c)/(cos(d*x + c) + 1) - 1696*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 447*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5)/(48*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 12*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 +
sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 64) + 837*log(sin(d*x + c)/(cos(d*x + c) + 1) + 2) - 837*log(sin(d*x + c
)/(cos(d*x + c) + 1) - 2))/d

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Fricas [A]
time = 0.38, size = 170, normalized size = 1.21 \begin {gather*} -\frac {837 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 837 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\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 (4975 \, \cos \left (d x + c\right )^{2} + 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \, {\left (125 \, d \cos \left (d x + c\right )^{3} + 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) + 27 \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/196608*(837*(125*cos(d*x + c)^3 + 225*cos(d*x + c)^2 + 135*cos(d*x + c) + 27)*log(3/2*cos(d*x + c) + 2*sin(
d*x + c) + 5/2) - 837*(125*cos(d*x + c)^3 + 225*cos(d*x + c)^2 + 135*cos(d*x + c) + 27)*log(3/2*cos(d*x + c) -
2*sin(d*x + c) + 5/2) - 40*(4975*cos(d*x + c)^2 + 4770*cos(d*x + c) + 1583)*sin(d*x + c))/(125*d*cos(d*x + c)
^3 + 225*d*cos(d*x + c)^2 + 135*d*cos(d*x + c) + 27*d)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (126) = 252\).
time = 2.76, size = 813, normalized size = 5.81 \begin {gather*} \begin {cases} \frac {x}{\left (5 \cos {\left (2 \operatorname {atan}{\left (2 \right )} \right )} + 3\right )^{4}} & \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 )^{4}} & \text {for}\: d = 0 \\\frac {837 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {10044 \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 )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {40176 \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 )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {53568 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {837 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {10044 \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 )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {40176 \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 )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {53568 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {8940 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {33920 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {56640 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 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))**4,x)

[Out]

Piecewise((x/(5*cos(2*atan(2)) + 3)**4, Eq(c, -d*x - 2*atan(2)) | Eq(c, -d*x + 2*atan(2))), (x/(5*cos(c) + 3)*
*4, Eq(d, 0)), (837*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**6/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan
(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 10044*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/
2)**4/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d
) + 40176*log(tan(c/2 + d*x/2) - 2)*tan(c/2 + d*x/2)**2/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x
/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 53568*log(tan(c/2 + d*x/2) - 2)/(98304*d*tan(c/2 + d*x/2
)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 837*log(tan(c/2 + d*x/2) +
2)*tan(c/2 + d*x/2)**6/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x
/2)**2 - 6291456*d) + 10044*log(tan(c/2 + d*x/2) + 2)*tan(c/2 + d*x/2)**4/(98304*d*tan(c/2 + d*x/2)**6 - 11796
48*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 40176*log(tan(c/2 + d*x/2) + 2)*tan(c/
2 + d*x/2)**2/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6
291456*d) + 53568*log(tan(c/2 + d*x/2) + 2)/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 471
8592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 8940*tan(c/2 + d*x/2)**5/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*ta
n(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) + 33920*tan(c/2 + d*x/2)**3/(98304*d*tan(c/2 +
d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d) - 56640*tan(c/2 + d*x/2
)/(98304*d*tan(c/2 + d*x/2)**6 - 1179648*d*tan(c/2 + d*x/2)**4 + 4718592*d*tan(c/2 + d*x/2)**2 - 6291456*d), T
rue))

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Giac [A]
time = 0.50, size = 91, normalized size = 0.65 \begin {gather*} -\frac {\frac {20 \, {\left (447 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2832 \, \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 )}^{3}} + 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{98304 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/98304*(20*(447*tan(1/2*d*x + 1/2*c)^5 - 1696*tan(1/2*d*x + 1/2*c)^3 + 2832*tan(1/2*d*x + 1/2*c))/(tan(1/2*d
*x + 1/2*c)^2 - 4)^3 + 837*log(abs(tan(1/2*d*x + 1/2*c) + 2)) - 837*log(abs(tan(1/2*d*x + 1/2*c) - 2)))/d

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Mupad [B]
time = 1.97, size = 102, normalized size = 0.73 \begin {gather*} -\frac {279\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{16384\,d}-\frac {\frac {745\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{768}+\frac {295\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*cos(c + d*x) + 3)^4,x)

[Out]

- (279*atanh(tan(c/2 + (d*x)/2)/2))/(16384*d) - ((295*tan(c/2 + (d*x)/2))/512 - (265*tan(c/2 + (d*x)/2)^3)/768
+ (745*tan(c/2 + (d*x)/2)^5)/8192)/(d*(48*tan(c/2 + (d*x)/2)^2 - 12*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)
^6 - 64))

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