### 3.49 $$\int \frac{1}{(-3-5 \sin (c+d x))^4} \, dx$$

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*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]))

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Rubi [A]  time = 0.112822, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, 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 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 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 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 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}{(-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}{-9-3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\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{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*}

Mathematica [A]  time = 0.823022, size = 235, normalized size = 1.7 $\frac{20 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\frac{597}{3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{240}{\left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{199}{\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )}+\frac{80}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )^3}\right )-\frac{2320}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{720}{\left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+2511 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )-2511 \log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{294912 d}$

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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Maple [A]  time = 0.04, size = 152, normalized size = 1.1 \begin{align*} -{\frac{125}{20736\,d} \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}}+{\frac{275}{27648\,d} \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}}-{\frac{3505}{221184\,d} \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1}}-{\frac{279}{32768\,d}\ln \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }-{\frac{125}{768\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) ^{-3}}+{\frac{75}{1024\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) ^{-2}}-{\frac{345}{8192\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) ^{-1}}+{\frac{279}{32768\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)-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)+279/32768/d*ln(tan(1/2*d*x+1/2*c)+3)

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Maxima [B]  time = 1.01203, size = 371, normalized size = 2.69 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 2.02264, size = 543, normalized size = 3.93 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 32.9948, size = 2358, normalized size = 17.09 \begin{align*} \text{result too large to display} \end{align*}

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)), (-22599*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**6/(2
654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*t
an(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) - 225990*log(ta
n(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**5/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96
436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*t
an(c/2 + d*x/2) + 2654208*d) - 821097*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**4/(2654208*d*tan(c/2 + d*x
/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 9
6436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) - 1288980*log(tan(c/2 + d*x/2) + 1/3)
*tan(c/2 + d*x/2)**3/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*
x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 265
4208*d) - 821097*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)**2/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*t
an(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d
*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) - 225990*log(tan(c/2 + d*x/2) + 1/3)*tan(c/2 + d*x/2)/(265
4208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan
(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) - 22599*log(tan(c
/2 + d*x/2) + 1/3)/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/
2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 26542
08*d) + 22599*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)**6/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/
2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)
**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 225990*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)**5/(2654208
*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2
+ d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 821097*log(tan(c/2
+ d*x/2) + 3)*tan(c/2 + d*x/2)**4/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d
*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 +
d*x/2) + 2654208*d) + 1288980*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)**3/(2654208*d*tan(c/2 + d*x/2)**6 +
26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d
*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 821097*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d
*x/2)**2/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 15
1388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 22
5990*log(tan(c/2 + d*x/2) + 3)*tan(c/2 + d*x/2)/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**
5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 265420
80*d*tan(c/2 + d*x/2) + 2654208*d) + 22599*log(tan(c/2 + d*x/2) + 3)/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080
*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2
+ d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 31660*tan(c/2 + d*x/2)**6/(2654208*d*tan(c/2 + d*x/2
)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 964
36224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 190800*tan(c/2 + d*x/2)**5/(2654208*d
*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 +
d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) + 429660*tan(c/2 + d*x/
2)**4/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d*x/2)**4 + 15138
8160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 2654208*d) - 42966
0*tan(c/2 + d*x/2)**2/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan(c/2 + d
*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x/2) + 26
54208*d) - 190800*tan(c/2 + d*x/2)/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*
d*tan(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2
+ d*x/2) + 2654208*d) - 31660/(2654208*d*tan(c/2 + d*x/2)**6 + 26542080*d*tan(c/2 + d*x/2)**5 + 96436224*d*tan
(c/2 + d*x/2)**4 + 151388160*d*tan(c/2 + d*x/2)**3 + 96436224*d*tan(c/2 + d*x/2)**2 + 26542080*d*tan(c/2 + d*x
/2) + 2654208*d), True))

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Giac [A]  time = 1.18768, size = 180, normalized size = 1.3 \begin{align*} -\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} \end{align*}

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