3.369 \(\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx\)

Optimal. Leaf size=207 \[ -\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]

[Out]

-(a*(5*a^2 + 3*c^2)*Log[a + c*Tan[(d + e*x)/2]])/(32*c^7*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(48*c^2*e*(a +
a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(96*c^4*e*(a + a*Cos[d + e*x]
+ c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] - a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a + a*C
os[d + e*x] + c*Sin[d + e*x]))

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Rubi [A]  time = 0.248415, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3129, 3156, 3153, 3124, 31} \[ -\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]

[Out]

-(a*(5*a^2 + 3*c^2)*Log[a + c*Tan[(d + e*x)/2]])/(32*c^7*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(48*c^2*e*(a +
a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(96*c^4*e*(a + a*Cos[d + e*x]
+ c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] - a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a + a*C
os[d + e*x] + c*Sin[d + e*x]))

Rule 3129

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[((-(c*Cos[d
+ e*x]) + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] +
Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*C
os[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n
, -1] && NeQ[n, -3/2]

Rule 3156

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> -Simp[((c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/
((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C)
+ (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A,
B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]

Rule 3153

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)
*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B - c*C)/(a^2 -
b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[
a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx &=-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac{\int \frac{-6 a+4 a \cos (d+e x)+4 c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx}{12 c^2}\\ &=-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}+\frac{\int \frac{8 \left (5 a^2+2 c^2\right )-20 a^2 \cos (d+e x)-20 a c \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx}{96 c^4}\\ &=-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}-\frac{\left (a \left (5 a^2+3 c^2\right )\right ) \int \frac{1}{2 a+2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{16 c^6}\\ &=-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}-\frac{\left (a \left (5 a^2+3 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a+4 c x} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{8 c^6 e}\\ &=-\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a+a \cos (d+e x)+c \sin (d+e x))^3}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a+a \cos (d+e x)+c \sin (d+e x))^2}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a+a \cos (d+e x)+c \sin (d+e x))}\\ \end{align*}

Mathematica [B]  time = 1.6272, size = 492, normalized size = 2.38 \[ \frac{\cos \left (\frac{1}{2} (d+e x)\right ) \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right ) \left (\frac{c \left (255 a^4 c^2 \sin (d+e x)+72 a^4 c^2 \sin (2 (d+e x))-37 a^4 c^2 \sin (3 (d+e x))+129 a^2 c^4 \sin (d+e x)+36 a^2 c^4 \sin (2 (d+e x))-27 a^2 c^4 \sin (3 (d+e x))-35 a^3 c^3 \cos (3 (d+e x))+3 a c \left (25 a^2 c^2+25 a^4-4 c^4\right ) \cos (d+e x)-6 \left (15 a^3 c^3+25 a^5 c+4 a c^5\right ) \cos (2 (d+e x))+130 a^3 c^3-75 a^5 c \cos (3 (d+e x))+150 a^5 c+150 a^6 \sin (d+e x)+120 a^6 \sin (2 (d+e x))+30 a^6 \sin (3 (d+e x))-4 a c^5 \cos (3 (d+e x))+24 a c^5+12 c^6 \sin (d+e x)-4 c^6 \sin (3 (d+e x))\right )}{a}+192 \left (5 a^3+3 a c^2\right ) \cos ^3\left (\frac{1}{2} (d+e x)\right ) \log \left (\cos \left (\frac{1}{2} (d+e x)\right )\right ) \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )^3-192 \left (5 a^3+3 a c^2\right ) \cos ^3\left (\frac{1}{2} (d+e x)\right ) \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )^3 \log \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )\right )}{384 c^7 e (a \cos (d+e x)+a+c \sin (d+e x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]

[Out]

(Cos[(d + e*x)/2]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])*(192*(5*a^3 + 3*a*c^2)*Cos[(d + e*x)/2]^3*Log[Cos[
(d + e*x)/2]]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^3 - 192*(5*a^3 + 3*a*c^2)*Cos[(d + e*x)/2]^3*Log[a*Cos
[(d + e*x)/2] + c*Sin[(d + e*x)/2]]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^3 + (c*(150*a^5*c + 130*a^3*c^3
+ 24*a*c^5 + 3*a*c*(25*a^4 + 25*a^2*c^2 - 4*c^4)*Cos[d + e*x] - 6*(25*a^5*c + 15*a^3*c^3 + 4*a*c^5)*Cos[2*(d +
e*x)] - 75*a^5*c*Cos[3*(d + e*x)] - 35*a^3*c^3*Cos[3*(d + e*x)] - 4*a*c^5*Cos[3*(d + e*x)] + 150*a^6*Sin[d +
e*x] + 255*a^4*c^2*Sin[d + e*x] + 129*a^2*c^4*Sin[d + e*x] + 12*c^6*Sin[d + e*x] + 120*a^6*Sin[2*(d + e*x)] +
72*a^4*c^2*Sin[2*(d + e*x)] + 36*a^2*c^4*Sin[2*(d + e*x)] + 30*a^6*Sin[3*(d + e*x)] - 37*a^4*c^2*Sin[3*(d + e*
x)] - 27*a^2*c^4*Sin[3*(d + e*x)] - 4*c^6*Sin[3*(d + e*x)]))/a))/(384*c^7*e*(a + a*Cos[d + e*x] + c*Sin[d + e*
x])^4)

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Maple [A]  time = 0.222, size = 378, normalized size = 1.8 \begin{align*}{\frac{1}{384\,{c}^{4}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{3}}-{\frac{a}{64\,{c}^{5}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{2}}+{\frac{5\,{a}^{2}}{64\,e{c}^{6}}\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) }+{\frac{3}{128\,{c}^{4}e}\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) }+{\frac{3\,{a}^{5}}{128\,e{c}^{7}} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}+{\frac{3\,{a}^{3}}{64\,{c}^{5}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}+{\frac{3\,a}{128\,{c}^{3}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}-{\frac{{a}^{6}}{384\,e{c}^{7}} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{{a}^{4}}{128\,{c}^{5}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{{a}^{2}}{128\,{c}^{3}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{1}{384\,ce} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{5\,{a}^{3}}{32\,e{c}^{7}}\ln \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{3\,a}{32\,{c}^{5}e}\ln \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{15\,{a}^{4}}{128\,e{c}^{7}} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{9\,{a}^{2}}{64\,{c}^{5}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{3}{128\,{c}^{3}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x)

[Out]

1/384/e/c^4*tan(1/2*d+1/2*e*x)^3-1/64/e/c^5*tan(1/2*d+1/2*e*x)^2*a+5/64/e/c^6*a^2*tan(1/2*d+1/2*e*x)+3/128/e/c
^4*tan(1/2*d+1/2*e*x)+3/128/e*a^5/c^7/(a+c*tan(1/2*d+1/2*e*x))^2+3/64/e*a^3/c^5/(a+c*tan(1/2*d+1/2*e*x))^2+3/1
28/e*a/c^3/(a+c*tan(1/2*d+1/2*e*x))^2-1/384/e/c^7/(a+c*tan(1/2*d+1/2*e*x))^3*a^6-1/128/e/c^5/(a+c*tan(1/2*d+1/
2*e*x))^3*a^4-1/128/e/c^3/(a+c*tan(1/2*d+1/2*e*x))^3*a^2-1/384/e/c/(a+c*tan(1/2*d+1/2*e*x))^3-5/32/e*a^3/c^7*l
n(a+c*tan(1/2*d+1/2*e*x))-3/32/e*a/c^5*ln(a+c*tan(1/2*d+1/2*e*x))-15/128/e/c^7/(a+c*tan(1/2*d+1/2*e*x))*a^4-9/
64/e/c^5/(a+c*tan(1/2*d+1/2*e*x))*a^2-3/128/e/c^3/(a+c*tan(1/2*d+1/2*e*x))

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Maxima [A]  time = 1.14598, size = 414, normalized size = 2. \begin{align*} -\frac{\frac{37 \, a^{6} + 39 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6} + \frac{9 \,{\left (9 \, a^{5} c + 10 \, a^{3} c^{3} + a c^{5}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac{9 \,{\left (5 \, a^{4} c^{2} + 6 \, a^{2} c^{4} + c^{6}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}}{a^{3} c^{7} + \frac{3 \, a^{2} c^{8} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac{3 \, a c^{9} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac{c^{10} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}} + \frac{\frac{6 \, a c \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac{c^{2} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} - \frac{3 \,{\left (10 \, a^{2} + 3 \, c^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{c^{6}} + \frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (a + \frac{c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="maxima")

[Out]

-1/384*((37*a^6 + 39*a^4*c^2 + 3*a^2*c^4 + c^6 + 9*(9*a^5*c + 10*a^3*c^3 + a*c^5)*sin(e*x + d)/(cos(e*x + d) +
1) + 9*(5*a^4*c^2 + 6*a^2*c^4 + c^6)*sin(e*x + d)^2/(cos(e*x + d) + 1)^2)/(a^3*c^7 + 3*a^2*c^8*sin(e*x + d)/(
cos(e*x + d) + 1) + 3*a*c^9*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 + c^10*sin(e*x + d)^3/(cos(e*x + d) + 1)^3) +
(6*a*c*sin(e*x + d)^2/(cos(e*x + d) + 1)^2 - c^2*sin(e*x + d)^3/(cos(e*x + d) + 1)^3 - 3*(10*a^2 + 3*c^2)*sin(
e*x + d)/(cos(e*x + d) + 1))/c^6 + 12*(5*a^3 + 3*a*c^2)*log(a + c*sin(e*x + d)/(cos(e*x + d) + 1))/c^7)/e

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Fricas [B]  time = 2.70019, size = 1782, normalized size = 8.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="fricas")

[Out]

1/192*(60*a^4*c^2 + 6*a^2*c^4 - 2*(45*a^4*c^2 - 3*a^2*c^4 - 4*c^6)*cos(e*x + d)^3 - 12*(10*a^4*c^2 + a^2*c^4)*
cos(e*x + d)^2 + 6*(5*a^4*c^2 - 2*a^2*c^4 - 2*c^6)*cos(e*x + d) - 3*(5*a^6 + 18*a^4*c^2 + 9*a^2*c^4 + (5*a^6 -
12*a^4*c^2 - 9*a^2*c^4)*cos(e*x + d)^3 + 3*(5*a^6 - 2*a^4*c^2 - 3*a^2*c^4)*cos(e*x + d)^2 + 3*(5*a^6 + 8*a^4*
c^2 + 3*a^2*c^4)*cos(e*x + d) + (15*a^5*c + 14*a^3*c^3 + 3*a*c^5 + (15*a^5*c + 4*a^3*c^3 - 3*a*c^5)*cos(e*x +
d)^2 + 6*(5*a^5*c + 3*a^3*c^3)*cos(e*x + d))*sin(e*x + d))*log(a*c*sin(e*x + d) + 1/2*a^2 + 1/2*c^2 + 1/2*(a^2
- c^2)*cos(e*x + d)) + 3*(5*a^6 + 18*a^4*c^2 + 9*a^2*c^4 + (5*a^6 - 12*a^4*c^2 - 9*a^2*c^4)*cos(e*x + d)^3 +
3*(5*a^6 - 2*a^4*c^2 - 3*a^2*c^4)*cos(e*x + d)^2 + 3*(5*a^6 + 8*a^4*c^2 + 3*a^2*c^4)*cos(e*x + d) + (15*a^5*c
+ 14*a^3*c^3 + 3*a*c^5 + (15*a^5*c + 4*a^3*c^3 - 3*a*c^5)*cos(e*x + d)^2 + 6*(5*a^5*c + 3*a^3*c^3)*cos(e*x + d
))*sin(e*x + d))*log(1/2*cos(e*x + d) + 1/2) + 2*(15*a^5*c + 14*a^3*c^3 + 6*a*c^5 + (15*a^5*c - 41*a^3*c^3 - 1
2*a*c^5)*cos(e*x + d)^2 + 3*(10*a^5*c - 9*a^3*c^3 - a*c^5)*cos(e*x + d))*sin(e*x + d))/((a^3*c^7 - 3*a*c^9)*e*
cos(e*x + d)^3 + 3*(a^3*c^7 - a*c^9)*e*cos(e*x + d)^2 + 3*(a^3*c^7 + a*c^9)*e*cos(e*x + d) + (a^3*c^7 + 3*a*c^
9)*e + (6*a^2*c^8*e*cos(e*x + d) + (3*a^2*c^8 - c^10)*e*cos(e*x + d)^2 + (3*a^2*c^8 + c^10)*e)*sin(e*x + d))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.14161, size = 410, normalized size = 1.98 \begin{align*} -\frac{1}{384} \,{\left (\frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a \right |}\right )}{c^{7}} - \frac{110 \, a^{3} c^{3} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 66 \, a c^{5} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 285 \, a^{4} c^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 144 \, a^{2} c^{4} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} - 9 \, c^{6} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 249 \, a^{5} c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 108 \, a^{3} c^{3} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - 9 \, a c^{5} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 73 \, a^{6} + 27 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - c^{6}}{{\left (c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a\right )}^{3} c^{7}} - \frac{c^{8} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} - 6 \, a c^{7} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 30 \, a^{2} c^{6} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 9 \, c^{8} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )}{c^{12}}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*a+2*a*cos(e*x+d)+2*c*sin(e*x+d))^4,x, algorithm="giac")

[Out]

-1/384*(12*(5*a^3 + 3*a*c^2)*log(abs(c*tan(1/2*x*e + 1/2*d) + a))/c^7 - (110*a^3*c^3*tan(1/2*x*e + 1/2*d)^3 +
66*a*c^5*tan(1/2*x*e + 1/2*d)^3 + 285*a^4*c^2*tan(1/2*x*e + 1/2*d)^2 + 144*a^2*c^4*tan(1/2*x*e + 1/2*d)^2 - 9*
c^6*tan(1/2*x*e + 1/2*d)^2 + 249*a^5*c*tan(1/2*x*e + 1/2*d) + 108*a^3*c^3*tan(1/2*x*e + 1/2*d) - 9*a*c^5*tan(1
/2*x*e + 1/2*d) + 73*a^6 + 27*a^4*c^2 - 3*a^2*c^4 - c^6)/((c*tan(1/2*x*e + 1/2*d) + a)^3*c^7) - (c^8*tan(1/2*x
*e + 1/2*d)^3 - 6*a*c^7*tan(1/2*x*e + 1/2*d)^2 + 30*a^2*c^6*tan(1/2*x*e + 1/2*d) + 9*c^8*tan(1/2*x*e + 1/2*d))
/c^12)*e^(-1)