3.21 \(\int \frac {1}{(5+3 \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=106 \[ \frac {311 \cos (c+d x)}{8192 d (3 \sin (c+d x)+5)}+\frac {25 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)^2}+\frac {\cos (c+d x)}{16 d (3 \sin (c+d x)+5)^3}+\frac {385 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{16384 d}+\frac {385 x}{32768} \]

[Out]

385/32768*x+385/16384*arctan(cos(d*x+c)/(3+sin(d*x+c)))/d+1/16*cos(d*x+c)/d/(5+3*sin(d*x+c))^3+25/512*cos(d*x+
c)/d/(5+3*sin(d*x+c))^2+311/8192*cos(d*x+c)/d/(5+3*sin(d*x+c))

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Rubi [A]  time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2664, 2754, 12, 2657} \[ \frac {311 \cos (c+d x)}{8192 d (3 \sin (c+d x)+5)}+\frac {25 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)^2}+\frac {\cos (c+d x)}{16 d (3 \sin (c+d x)+5)^3}+\frac {385 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{16384 d}+\frac {385 x}{32768} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*Sin[c + d*x])^(-4),x]

[Out]

(385*x)/32768 + (385*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(16384*d) + Cos[c + d*x]/(16*d*(5 + 3*Sin[c + d*
x])^3) + (25*Cos[c + d*x])/(512*d*(5 + 3*Sin[c + d*x])^2) + (311*Cos[c + d*x])/(8192*d*(5 + 3*Sin[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 2657

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rule 2664

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] && Integer
Q[2*n]

Rule 2754

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}{(5+3 \sin (c+d x))^4} \, dx &=\frac {\cos (c+d x)}{16 d (5+3 \sin (c+d x))^3}-\frac {1}{48} \int \frac {-15+6 \sin (c+d x)}{(5+3 \sin (c+d x))^3} \, dx\\ &=\frac {\cos (c+d x)}{16 d (5+3 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (5+3 \sin (c+d x))^2}+\frac {\int \frac {186-75 \sin (c+d x)}{(5+3 \sin (c+d x))^2} \, dx}{1536}\\ &=\frac {\cos (c+d x)}{16 d (5+3 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (5+3 \sin (c+d x))^2}+\frac {311 \cos (c+d x)}{8192 d (5+3 \sin (c+d x))}-\frac {\int -\frac {1155}{5+3 \sin (c+d x)} \, dx}{24576}\\ &=\frac {\cos (c+d x)}{16 d (5+3 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (5+3 \sin (c+d x))^2}+\frac {311 \cos (c+d x)}{8192 d (5+3 \sin (c+d x))}+\frac {385 \int \frac {1}{5+3 \sin (c+d x)} \, dx}{8192}\\ &=\frac {385 x}{32768}+\frac {385 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{16384 d}+\frac {\cos (c+d x)}{16 d (5+3 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (5+3 \sin (c+d x))^2}+\frac {311 \cos (c+d x)}{8192 d (5+3 \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 0.46, size = 133, normalized size = 1.25 \[ \frac {\frac {-305091 \sin (c+d x)+105300 \sin (2 (c+d x))+8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)+5)^3}+1925 \tan ^{-1}\left (\frac {2 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{81920 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1925*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (-239470 + 219
735*Cos[c + d*x] + 83970*Cos[2*(c + d*x)] - 13995*Cos[3*(c + d*x)] - 305091*Sin[c + d*x] + 105300*Sin[2*(c + d
*x)] + 8397*Sin[3*(c + d*x)])/(2*(5 + 3*Sin[c + d*x])^3))/(81920*d)

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fricas [A]  time = 0.43, size = 130, normalized size = 1.23 \[ \frac {11196 \, \cos \left (d x + c\right )^{3} + 385 \, {\left (135 \, \cos \left (d x + c\right )^{2} + 9 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 28\right )} \sin \left (d x + c\right ) - 260\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right ) - 42120 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 52344 \, \cos \left (d x + c\right )}{32768 \, {\left (135 \, d \cos \left (d x + c\right )^{2} + 9 \, {\left (3 \, d \cos \left (d x + c\right )^{2} - 28 \, d\right )} \sin \left (d x + c\right ) - 260 \, d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32768*(11196*cos(d*x + c)^3 + 385*(135*cos(d*x + c)^2 + 9*(3*cos(d*x + c)^2 - 28)*sin(d*x + c) - 260)*arctan
(1/4*(5*sin(d*x + c) + 3)/cos(d*x + c)) - 42120*cos(d*x + c)*sin(d*x + c) - 52344*cos(d*x + c))/(135*d*cos(d*x
+ c)^2 + 9*(3*d*cos(d*x + c)^2 - 28*d)*sin(d*x + c) - 260*d)

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giac [A]  time = 0.54, size = 147, normalized size = 1.39 \[ \frac {48125 \, d x + 48125 \, c + \frac {72 \, {\left (110925 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 373735 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 637794 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 672110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 403425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 142875\right )}}{{\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}^{3}} + 96250 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{4096000 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4096000*(48125*d*x + 48125*c + 72*(110925*tan(1/2*d*x + 1/2*c)^5 + 373735*tan(1/2*d*x + 1/2*c)^4 + 637794*ta
n(1/2*d*x + 1/2*c)^3 + 672110*tan(1/2*d*x + 1/2*c)^2 + 403425*tan(1/2*d*x + 1/2*c) + 142875)/(5*tan(1/2*d*x +
1/2*c)^2 + 6*tan(1/2*d*x + 1/2*c) + 5)^3 + 96250*arctan(-(3*cos(d*x + c) + sin(d*x + c) + 3)/(cos(d*x + c) - 3
*sin(d*x + c) - 9)))/d

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maple [B]  time = 0.09, size = 272, normalized size = 2.57 \[ \frac {39933 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {672723 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{102400 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {2870073 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256000 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {604899 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{51200 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {145233 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{20480 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {10287}{4096 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {385 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{16384 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*sin(d*x+c))^4,x)

[Out]

39933/20480/d/(5*tan(1/2*d*x+1/2*c)^2+6*tan(1/2*d*x+1/2*c)+5)^3*tan(1/2*d*x+1/2*c)^5+672723/102400/d/(5*tan(1/
2*d*x+1/2*c)^2+6*tan(1/2*d*x+1/2*c)+5)^3*tan(1/2*d*x+1/2*c)^4+2870073/256000/d/(5*tan(1/2*d*x+1/2*c)^2+6*tan(1
/2*d*x+1/2*c)+5)^3*tan(1/2*d*x+1/2*c)^3+604899/51200/d/(5*tan(1/2*d*x+1/2*c)^2+6*tan(1/2*d*x+1/2*c)+5)^3*tan(1
/2*d*x+1/2*c)^2+145233/20480/d/(5*tan(1/2*d*x+1/2*c)^2+6*tan(1/2*d*x+1/2*c)+5)^3*tan(1/2*d*x+1/2*c)+10287/4096
/d/(5*tan(1/2*d*x+1/2*c)^2+6*tan(1/2*d*x+1/2*c)+5)^3+385/16384/d*arctan(5/4*tan(1/2*d*x+1/2*c)+3/4)

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maxima [B]  time = 0.73, size = 253, normalized size = 2.39 \[ \frac {\frac {36 \, {\left (\frac {403425 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {672110 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {637794 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {373735 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {110925 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 142875\right )}}{\frac {450 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {915 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1116 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {915 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {450 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {125 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 125} + 48125 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right )}{2048000 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2048000*(36*(403425*sin(d*x + c)/(cos(d*x + c) + 1) + 672110*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 637794*si
n(d*x + c)^3/(cos(d*x + c) + 1)^3 + 373735*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 110925*sin(d*x + c)^5/(cos(d*
x + c) + 1)^5 + 142875)/(450*sin(d*x + c)/(cos(d*x + c) + 1) + 915*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1116*
sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 915*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 450*sin(d*x + c)^5/(cos(d*x +
c) + 1)^5 + 125*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 125) + 48125*arctan(5/4*sin(d*x + c)/(cos(d*x + c) + 1)
+ 3/4))/d

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mupad [B]  time = 5.06, size = 187, normalized size = 1.76 \[ \frac {385\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3}{4}\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}+\frac {\frac {39933\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2560000}+\frac {672723\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{12800000}+\frac {2870073\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32000000}+\frac {604899\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6400000}+\frac {145233\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2560000}+\frac {10287}{512000}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {183\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{25}+\frac {1116\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{125}+\frac {183\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{25}+\frac {18\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*sin(c + d*x) + 5)^4,x)

[Out]

(385*atan((5*tan(c/2 + (d*x)/2))/4 + 3/4))/(16384*d) - (385*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(16384*d) +
((145233*tan(c/2 + (d*x)/2))/2560000 + (604899*tan(c/2 + (d*x)/2)^2)/6400000 + (2870073*tan(c/2 + (d*x)/2)^3)/
32000000 + (672723*tan(c/2 + (d*x)/2)^4)/12800000 + (39933*tan(c/2 + (d*x)/2)^5)/2560000 + 10287/512000)/(d*((
18*tan(c/2 + (d*x)/2))/5 + (183*tan(c/2 + (d*x)/2)^2)/25 + (1116*tan(c/2 + (d*x)/2)^3)/125 + (183*tan(c/2 + (d
*x)/2)^4)/25 + (18*tan(c/2 + (d*x)/2)^5)/5 + tan(c/2 + (d*x)/2)^6 + 1))

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sympy [A]  time = 10.83, size = 1693, normalized size = 15.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(5 - 3*sin(2*atan(3/5 - 4*I/5)))**4, Eq(c, -d*x - 2*atan(3/5 - 4*I/5))), (x/(5 - 3*sin(2*atan(3/5
+ 4*I/5)))**4, Eq(c, -d*x - 2*atan(3/5 + 4*I/5))), (x/(3*sin(c) + 5)**4, Eq(d, 0)), (6015625*(atan(5*tan(c/2
+ d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**6/(256000000*d*tan(c/2 + d*x/2)**6 +
921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 187392
0000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 21656250*(atan(5*tan(c/2 + d*x/2)/4
+ 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**5/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*
d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan
(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 44034375*(atan(5*tan(c/2 + d*x/2)/4 + 3/4) +
pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**4/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2
+ d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x
/2)**2 + 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 53707500*(atan(5*tan(c/2 + d*x/2)/4 + 3/4) + pi*floor((
c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**3/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**
5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 9
21600000*d*tan(c/2 + d*x/2) + 256000000*d) + 44034375*(atan(5*tan(c/2 + d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/
2 - pi/2)/pi))*tan(c/2 + d*x/2)**2/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 187392
0000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d
*tan(c/2 + d*x/2) + 256000000*d) + 21656250*(atan(5*tan(c/2 + d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/
pi))*tan(c/2 + d*x/2)/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/
2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*
x/2) + 256000000*d) + 6015625*(atan(5*tan(c/2 + d*x/2)/4 + 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(25600000
0*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*ta
n(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 3993300*t
an(c/2 + d*x/2)**5/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 +
d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*x/2
) + 256000000*d) + 13454460*tan(c/2 + d*x/2)**4/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2
)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2
+ 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 22960584*tan(c/2 + d*x/2)**3/(256000000*d*tan(c/2 + d*x/2)**6
+ 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873
920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 24195960*tan(c/2 + d*x/2)**2/(256
000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 + 2285568000
*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 1452
3300*tan(c/2 + d*x/2)/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/
2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(c/2 + d*
x/2) + 256000000*d) + 5143500/(256000000*d*tan(c/2 + d*x/2)**6 + 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*
d*tan(c/2 + d*x/2)**4 + 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 + 921600000*d*tan(
c/2 + d*x/2) + 256000000*d), True))

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