### 3.25 $$\int \frac{1}{(5-3 \sin (c+d x))^4} \, dx$$

Optimal. Leaf size=108 $-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{385 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}+\frac{385 x}{32768}$

[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]))

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Rubi [A]  time = 0.0960948, antiderivative size = 108, normalized size of antiderivative = 1., 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} $-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{385 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}+\frac{385 x}{32768}$

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 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]

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]

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*}

Mathematica [A]  time = 0.435363, size = 133, normalized size = 1.23 $\frac{\frac{305091 \sin (c+d x)-105300 \sin (2 (c+d x))-8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)-5)^3}-1925 \tan ^{-1}\left (\frac{2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}\right )}{81920 d}$

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 + 21
9735*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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Maple [B]  time = 0.037, size = 272, normalized size = 2.5 \begin{align*}{\frac{39933}{20480\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}-{\frac{672723}{102400\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}+{\frac{2870073}{256000\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}-{\frac{604899}{51200\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}+{\frac{145233}{20480\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}-{\frac{10287}{4096\,d} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}+{\frac{385}{16384\,d}\arctan \left ({\frac{5}{4}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{3}{4}} \right ) } \end{align*}

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 = 1.46694, size = 342, normalized size = 3.17 \begin{align*} -\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} \end{align*}

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*s
in(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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Fricas [A]  time = 2.01296, size = 383, normalized size = 3.55 \begin{align*} -\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 )}} \end{align*}

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)*arcta
n(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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Sympy [A]  time = 73.7717, size = 1690, normalized size = 15.65 \begin{align*} \text{result too large to display} \end{align*}

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/(5 - 3*sin(c))**4, Eq(d, 0)), (48125*(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/(2048000*d*tan(c/2 + d*x/2)**6 - 7372
800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c
/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) + 2048000*d) - 173250*(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/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5
+ 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d
*tan(c/2 + d*x/2) + 2048000*d) + 352275*(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/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2 + d*x
/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) + 204800
0*d) - 429660*(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/(2048
000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/
2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) + 2048000*d) + 352275*(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/(2048000*d*tan(c/2 + d*x/2)**6
- 7372800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d
*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) + 2048000*d) - 173250*(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)/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)*
*5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 737280
0*d*tan(c/2 + d*x/2) + 2048000*d) + 48125*(atan(5*tan(c/2 + d*x/2)/4 - 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi
))/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*
d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) + 2048000*d) + 20574*tan(c
/2 + d*x/2)**6/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2 + d*x/2)**4
- 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) + 2048000*d) -
42120*tan(c/2 + d*x/2)**5/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5 + 14991360*d*tan(c/2
+ d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2 + d*x/2) +
2048000*d) + 42966*tan(c/2 + d*x/2)**4/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)**5 + 149913
60*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372800*d*tan(c/2
+ d*x/2) + 2048000*d) - 42966*tan(c/2 + d*x/2)**2/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2)
**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 73728
00*d*tan(c/2 + d*x/2) + 2048000*d) + 42120*tan(c/2 + d*x/2)/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2
+ d*x/2)**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**
2 - 7372800*d*tan(c/2 + d*x/2) + 2048000*d) - 20574/(2048000*d*tan(c/2 + d*x/2)**6 - 7372800*d*tan(c/2 + d*x/2
)**5 + 14991360*d*tan(c/2 + d*x/2)**4 - 18284544*d*tan(c/2 + d*x/2)**3 + 14991360*d*tan(c/2 + d*x/2)**2 - 7372
800*d*tan(c/2 + d*x/2) + 2048000*d), True))

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Giac [A]  time = 1.17046, size = 200, normalized size = 1.85 \begin{align*} \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} \end{align*}

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