\(\int \frac {1}{(3+5 \sin (c+d x))^4} \, dx\) [37]

Optimal result

Integrand size = 12, antiderivative size = 138 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {279 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))} \]

[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))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2743, 2833, 12, 2739, 630, 31} \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {995 \cos (c+d x)}{24576 d (5 \sin (c+d x)+3)}+\frac {25 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)^2}-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}+\frac {279 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]

[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 630

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 2739

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 2743

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] && Integ
erQ[2*n]

Rule 2833

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*} \text {integral}& = -\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 \text {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 \text {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {837 \text {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*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {2511 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2511 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2320}{\left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {720}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+20 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {80}{\left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {240}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {597}{\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )}\right )}{294912 d} \]

[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)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {125}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{3}}+\frac {75}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{2}}-\frac {345}{8192 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768}-\frac {125}{20736 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {275}{27648 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3505}{221184 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\)	\(132\)
default	\(\frac {-\frac {125}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{3}}+\frac {75}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{2}}-\frac {345}{8192 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768}-\frac {125}{20736 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {275}{27648 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3505}{221184 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\)	\(132\)
risch	\(-\frac {-111042 \,{\mathrm e}^{3 i \left (d x +c \right )}+62775 i {\mathrm e}^{4 i \left (d x +c \right )}-119310 i {\mathrm e}^{2 i \left (d x +c \right )}+20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}+24875 i}{12288 \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-5+6 i {\mathrm e}^{i \left (d x +c \right )}\right )^{3} d}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {4}{5}+\frac {3 i}{5}\right )}{32768 d}-\frac {279 \ln \left (-\frac {4}{5}+\frac {3 i}{5}+{\mathrm e}^{i \left (d x +c \right )}\right )}{32768 d}\)	\(132\)
norman	\(\frac {-\frac {7915}{12288 d}-\frac {63425 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12288 d}-\frac {3047275 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{165888 d}-\frac {15725 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288 d}-\frac {296245 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{18432 d}-\frac {270245 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{36864 d}}{{\left (3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}^{3}}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768 d}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768 d}\)	\(151\)
parallelrisch	\(\frac {\left (-10169550 \cos \left (2 d x +2 c \right )[phone] \sin \left (d x +c \right )-2824875 \sin \left (3 d x +3 c \right )[phone]\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{3}\right )+\left (10169550 \cos \left (2 d x +2 c \right )-20678085 \sin \left (d x +c \right )+2824875 \sin \left (3 d x +3 c \right )-12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )+6105780 \cos \left (d x +c \right )-14247000 \cos \left (2 d x +2 c \right )-2686500 \cos \left (3 d x +3 c \right )[phone] \sin \left (d x +c \right )+5151600 \sin \left (2 d x +2 c \right )-3957500 \sin \left (3 d x +3 c \right )[phone]}{2654208 d \left (-558+125 \sin \left (3 d x +3 c \right )-915 \sin \left (d x +c \right )+450 \cos \left (2 d x +2 c \right )\right )}\)	\(192\)

[In]

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

[Out]

1/d*(-125/768/(tan(1/2*d*x+1/2*c)+3)^3+75/1024/(tan(1/2*d*x+1/2*c)+3)^2-345/8192/(tan(1/2*d*x+1/2*c)+3)+279/32
768*ln(tan(1/2*d*x+1/2*c)+3)-125/20736/(3*tan(1/2*d*x+1/2*c)+1)^3+275/27648/(3*tan(1/2*d*x+1/2*c)+1)^2-3505/22
1184/(3*tan(1/2*d*x+1/2*c)+1)-279/32768*ln(3*tan(1/2*d*x+1/2*c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {199000 \, \cos \left (d x + c\right )^{3} - 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) + 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 190800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 262320 \, \cos \left (d x + c\right )}{196608 \, {\left (225 \, d \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 52 \, d\right )} \sin \left (d x + c\right ) - 252 \, d\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2356 vs. \(2 (126) = 252\).

Time = 3.13 (sec) , antiderivative size = 2356, normalized size of antiderivative = 17.07 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\text {Too large to display} \]

[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) + 3)*tan(c/2 + d*x/2)**6/(716636
16*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*t
an(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*l
og(tan(c/2 + d*x/2) + 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 + 71
6636160*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) + 71663616*d) + 34802460*log(t
an(c/2 + d*x/2) + 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 + 2
603778048*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 + 716636
160*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) + 71663616*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 + 260377804
8*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*ta
n(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) - 610173*log(3*tan(c/2 + d*x/2) + 1)*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(3*tan(c/2 + d*x/2) + 1)*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(3*tan(c/2 + d*x/2) + 1)*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 + 4
087480320*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
) - 34802460*log(3*tan(c/2 + d*x/2) + 1)*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(3*tan(c/2 + d*x/2) + 1)*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 + 4
087480320*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(3*tan(c/2 + d*x/2) + 1)*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(3*tan(c/2 + d*x/2) + 1)/(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 + 4
087480320*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 + 260377804
8*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*ta
n(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 + 408748
0320*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) - 1
709640/(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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (124) = 248\).

Time = 0.19 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {\frac {40 \, {\left (\frac {342495 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1066482 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1218910 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {486441 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84915 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 42741\right )}}{\frac {270 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {981 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {981 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {270 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 27} + 22599 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 22599 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{2654208 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {\frac {40 \, {\left (84915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 486441 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1218910 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1066482 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 342495 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42741\right )}}{{\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}^{3}} + 22599 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 22599 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \right |}\right )}{2654208 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 8.68 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {279\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {5}{4}\right )}{16384\,d}-\frac {\frac {15725\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{331776}+\frac {270245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{995328}+\frac {3047275\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4478976}+\frac {296245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{497664}+\frac {63425\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{331776}+\frac {7915}{331776}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {1540\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{27}+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]

[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))