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

Optimal. Leaf size=138 \[ -\frac {995 \cos (c+d x)}{24576 d (5 \sin (c+d x)+3)}+\frac {25 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)^2}-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}+\frac {279 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2664, 2754, 12, 2660, 616, 31} \[ -\frac {995 \cos (c+d x)}{24576 d (5 \sin (c+d x)+3)}+\frac {25 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)^2}-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}+\frac {279 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Sin[c + d*x])^(-4),x]

[Out]

(279*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(32768*d) - (279*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]])/
(32768*d) - (5*Cos[c + d*x])/(48*d*(3 + 5*Sin[c + d*x])^3) + (25*Cos[c + d*x])/(512*d*(3 + 5*Sin[c + d*x])^2)
- (995*Cos[c + d*x])/(24576*d*(3 + 5*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 31

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

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 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}{(3+5 \sin (c+d x))^4} \, dx &=-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {1}{48} \int \frac {-9+10 \sin (c+d x)}{(3+5 \sin (c+d x))^3} \, dx\\ &=-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}+\frac {\int \frac {154-75 \sin (c+d x)}{(3+5 \sin (c+d x))^2} \, dx}{1536}\\ &=-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))}+\frac {\int -\frac {837}{3+5 \sin (c+d x)} \, dx}{24576}\\ &=-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))}-\frac {279 \int \frac {1}{3+5 \sin (c+d x)} \, dx}{8192}\\ &=-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))}-\frac {279 \operatorname {Subst}\left (\int \frac {1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d}\\ &=-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))}-\frac {837 \operatorname {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {837 \operatorname {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}\\ &=\frac {279 \log \left (3+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (1+3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 0.89, size = 235, normalized size = 1.70 \[ \frac {20 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {597}{3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+\frac {240}{\left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )}+\frac {80}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )^3}\right )-\frac {2320}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {720}{\left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+2511 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )-2511 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{294912 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2511*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2511*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]] - 2320/(3*C
os[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 720/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^2 + 20*Sin[(c + d*x)/2]*(8
0/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 199/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 240/(Cos[(c + d*x)
/2] + 3*Sin[(c + d*x)/2])^3 + 597/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(294912*d)

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fricas [A]  time = 0.45, size = 181, normalized size = 1.31 \[ -\frac {199000 \, \cos \left (d x + c\right )^{3} - 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) + 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 190800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 262320 \, \cos \left (d x + c\right )}{196608 \, {\left (225 \, d \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 52 \, d\right )} \sin \left (d x + c\right ) - 252 \, d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/196608*(199000*cos(d*x + c)^3 - 837*(225*cos(d*x + c)^2 + 5*(25*cos(d*x + c)^2 - 52)*sin(d*x + c) - 252)*lo
g(4*cos(d*x + c) + 3*sin(d*x + c) + 5) + 837*(225*cos(d*x + c)^2 + 5*(25*cos(d*x + c)^2 - 52)*sin(d*x + c) - 2
52)*log(-4*cos(d*x + c) + 3*sin(d*x + c) + 5) - 190800*cos(d*x + c)*sin(d*x + c) - 262320*cos(d*x + c))/(225*d
*cos(d*x + c)^2 + 5*(25*d*cos(d*x + c)^2 - 52*d)*sin(d*x + c) - 252*d)

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giac [A]  time = 0.42, size = 133, normalized size = 0.96 \[ -\frac {\frac {40 \, {\left (84915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 486441 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1218910 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1066482 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 342495 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42741\right )}}{{\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}^{3}} + 22599 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 22599 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \right |}\right )}{2654208 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2654208*(40*(84915*tan(1/2*d*x + 1/2*c)^5 + 486441*tan(1/2*d*x + 1/2*c)^4 + 1218910*tan(1/2*d*x + 1/2*c)^3
+ 1066482*tan(1/2*d*x + 1/2*c)^2 + 342495*tan(1/2*d*x + 1/2*c) + 42741)/(3*tan(1/2*d*x + 1/2*c)^2 + 10*tan(1/2
*d*x + 1/2*c) + 3)^3 + 22599*log(abs(3*tan(1/2*d*x + 1/2*c) + 1)) - 22599*log(abs(tan(1/2*d*x + 1/2*c) + 3)))/
d

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maple [A]  time = 0.10, size = 152, normalized size = 1.10 \[ -\frac {125}{768 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{3}}+\frac {75}{1024 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{2}}-\frac {345}{8192 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768 d}-\frac {125}{20736 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {275}{27648 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3505}{221184 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/768/d/(tan(1/2*d*x+1/2*c)+3)^3+75/1024/d/(tan(1/2*d*x+1/2*c)+3)^2-345/8192/d/(tan(1/2*d*x+1/2*c)+3)+279/3
2768/d*ln(tan(1/2*d*x+1/2*c)+3)-125/20736/d/(3*tan(1/2*d*x+1/2*c)+1)^3+275/27648/d/(3*tan(1/2*d*x+1/2*c)+1)^2-
3505/221184/d/(3*tan(1/2*d*x+1/2*c)+1)-279/32768/d*ln(3*tan(1/2*d*x+1/2*c)+1)

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maxima [B]  time = 0.36, size = 275, normalized size = 1.99 \[ -\frac {\frac {40 \, {\left (\frac {342495 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1066482 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1218910 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {486441 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84915 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 42741\right )}}{\frac {270 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {981 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {981 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {270 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 27} + 22599 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 22599 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{2654208 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2654208*(40*(342495*sin(d*x + c)/(cos(d*x + c) + 1) + 1066482*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1218910
*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 486441*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 84915*sin(d*x + c)^5/(cos(
d*x + c) + 1)^5 + 42741)/(270*sin(d*x + c)/(cos(d*x + c) + 1) + 981*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1540
*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 981*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 270*sin(d*x + c)^5/(cos(d*x +
c) + 1)^5 + 27*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 27) + 22599*log(3*sin(d*x + c)/(cos(d*x + c) + 1) + 1) -
22599*log(sin(d*x + c)/(cos(d*x + c) + 1) + 3))/d

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mupad [B]  time = 7.04, size = 168, normalized size = 1.22 \[ \frac {279\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {5}{4}\right )}{16384\,d}-\frac {\frac {15725\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{331776}+\frac {270245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{995328}+\frac {3047275\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4478976}+\frac {296245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{497664}+\frac {63425\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{331776}+\frac {7915}{331776}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {1540\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{27}+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*sin(c + d*x) + 3)^4,x)

[Out]

(279*atanh((3*tan(c/2 + (d*x)/2))/4 + 5/4))/(16384*d) - ((63425*tan(c/2 + (d*x)/2))/331776 + (296245*tan(c/2 +
(d*x)/2)^2)/497664 + (3047275*tan(c/2 + (d*x)/2)^3)/4478976 + (270245*tan(c/2 + (d*x)/2)^4)/995328 + (15725*t
an(c/2 + (d*x)/2)^5)/331776 + 7915/331776)/(d*(10*tan(c/2 + (d*x)/2) + (109*tan(c/2 + (d*x)/2)^2)/3 + (1540*ta
n(c/2 + (d*x)/2)^3)/27 + (109*tan(c/2 + (d*x)/2)^4)/3 + 10*tan(c/2 + (d*x)/2)^5 + tan(c/2 + (d*x)/2)^6 + 1))

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sympy [A]  time = 8.00, size = 2356, normalized size = 17.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(3 - 5*sin(2*atan(1/3)))**4, Eq(c, -d*x - 2*atan(1/3))), (x/(3 - 5*sin(2*atan(3)))**4, Eq(c, -d*x
- 2*atan(3))), (x/(5*sin(c) + 3)**4, Eq(d, 0)), (-610173*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**6/(716
63616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*
d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 610173
0*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**5/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/
2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2
+ 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 22169619*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**4/(7166
3616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d
*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 3480246
0*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**3/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/
2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2
+ 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 22169619*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**2/(7166
3616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d
*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 6101730
*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**
5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 7
16636160*d*tan(c/2 + d*x/2) + 71663616*d) - 610173*log(tan(c/2 + d*x/2) + 1/3)/(71663616*d*tan(c/2 + d*x/2)**6
+ 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 260
3778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) + 610173*log(tan(c/2 + d*x/2) + 3)*
tan(c/2 + d*x/2)**6/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 +
d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2
) + 71663616*d) + 6101730*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)**5/(71663616*d*tan(c/2 + d*x/2)**6 + 7166
36160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048
*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) + 22169619*log(tan(c/2 + d*x/2) + 3)*tan(c
/2 + d*x/2)**4/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/
2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 7
1663616*d) + 34802460*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)**3/(71663616*d*tan(c/2 + d*x/2)**6 + 71663616
0*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*t
an(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) + 22169619*log(tan(c/2 + d*x/2) + 3)*tan(c/2 +
d*x/2)**2/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**
4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663
616*d) + 6101730*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(
c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 +
d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) + 610173*log(tan(c/2 + d*x/2) + 3)/(71663616*d*tan(c/2
+ d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x
/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 3396600*tan(c/2 + d*x
/2)**5/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 +
4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*
d) - 19457640*tan(c/2 + d*x/2)**4/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 26037780
48*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*t
an(c/2 + d*x/2) + 71663616*d) - 48756400*tan(c/2 + d*x/2)**3/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan
(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2
+ d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 42659280*tan(c/2 + d*x/2)**2/(71663616*d*tan(c/2 +
d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 4087480320*d*tan(c/2 + d*x/2)
**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) - 13699800*tan(c/2 + d*x/2
)/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 + 40874
80320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616*d) -
1709640/(71663616*d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*tan(c/2 + d*x/2)**4 +
4087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 716636160*d*tan(c/2 + d*x/2) + 71663616
*d), True))

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