Integrand size = 12, antiderivative size = 140 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))} \]
279/32768*ln(cos(1/2*d*x+1/2*c)-3*sin(1/2*d*x+1/2*c))/d-279/32768*ln(3*cos (1/2*d*x+1/2*c)-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))
Time = 0.01 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=\frac {2511 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )-2511 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {720}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+20 \left (\frac {240}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {597}{\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )}+\frac {80}{\left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right ) \sin \left (\frac {1}{2} (c+d x)\right )+\frac {2320}{\left (-3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}}{294912 d} \]
(2511*Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] - 2511*Log[3*Cos[(c + d*x )/2] - Sin[(c + d*x)/2]] - 720/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^2 + 20*(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]) + 80/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + 19 9/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*Sin[(c + d*x)/2] + 2320/(-3*Cos [(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(294912*d)
Time = 0.51 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3143, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3139, 1081, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(5 \sin (c+d x)-3)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(5 \sin (c+d x)-3)^4}dx\) |
\(\Big \downarrow \) 3143 |
\(\displaystyle \frac {1}{48} \int -\frac {10 \sin (c+d x)+9}{(3-5 \sin (c+d x))^3}dx+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {1}{48} \int \frac {10 \sin (c+d x)+9}{(3-5 \sin (c+d x))^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {1}{48} \int \frac {10 \sin (c+d x)+9}{(3-5 \sin (c+d x))^3}dx\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle \frac {1}{48} \left (-\frac {1}{32} \int -\frac {75 \sin (c+d x)+154}{(3-5 \sin (c+d x))^2}dx-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {75 \sin (c+d x)+154}{(3-5 \sin (c+d x))^2}dx-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {75 \sin (c+d x)+154}{(3-5 \sin (c+d x))^2}dx-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {1}{16} \int -\frac {837}{3-5 \sin (c+d x)}dx+\frac {995 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}\right )-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}-\frac {837}{16} \int \frac {1}{3-5 \sin (c+d x)}dx\right )-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}-\frac {837}{16} \int \frac {1}{3-5 \sin (c+d x)}dx\right )-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}-\frac {837 \int \frac {1}{3 \tan ^2\left (\frac {1}{2} (c+d x)\right )-10 \tan \left (\frac {1}{2} (c+d x)\right )+3}d\tan \left (\frac {1}{2} (c+d x)\right )}{8 d}\right )-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 1081 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}-\frac {2511 \int \left (\frac {1}{8 \left (1-3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {1}{24 \left (3-\tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )d\tan \left (\frac {1}{2} (c+d x)\right )}{8 d}\right )-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}+\frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}-\frac {2511 \left (\frac {1}{24} \log \left (3-\tan \left (\frac {1}{2} (c+d x)\right )\right )-\frac {1}{24} \log \left (1-3 \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}{8 d}\right )-\frac {75 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}\right )\) |
(((-2511*(-1/24*Log[1 - 3*Tan[(c + d*x)/2]] + Log[3 - Tan[(c + d*x)/2]]/24 ))/(8*d) + (995*Cos[c + d*x])/(16*d*(3 - 5*Sin[c + d*x])))/32 - (75*Cos[c + d*x])/(32*d*(3 - 5*Sin[c + d*x])^2))/48 + (5*Cos[c + d*x])/(48*d*(3 - 5* Sin[c + d*x])^3)
3.1.45.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
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] + Simp [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] && IntegerQ[2*n]
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] + Simp[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]
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94
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 | \(\frac {-111042 \,{\mathrm e}^{3 i \left (d x +c \right )}-62775 i {\mathrm e}^{4 i \left (d x +c \right )}+119310 i {\mathrm e}^{2 i \left (d x +c \right )}+20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}-24875 i}{12288 \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-5-6 i {\mathrm e}^{i \left (d x +c \right )}\right )^{3} d}+\frac {279 \ln \left (-\frac {4}{5}-\frac {3 i}{5}+{\mathrm e}^{i \left (d x +c \right )}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {4}{5}-\frac {3 i}{5}\right )}{32768 d}\) | \(132\) |
norman | \(\frac {\frac {7915}{12288 d}-\frac {15725 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288 d}-\frac {3047275 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{165888 d}-\frac {63425 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12288 d}+\frac {296245 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{18432 d}+\frac {270245 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{36864 d}}{{\left (3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}^{3}}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{32768 d}+\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768 d}\) | \(151\) |
parallelrisch | \(\frac {\left (10169550 \cos \left (2 d x +2 c \right )+20678085 \sin \left (d x +c \right )-2824875 \sin \left (3 d x +3 c \right )-12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3}\right )+\left (-10169550 \cos \left (2 d x +2 c \right )-20678085 \sin \left (d x +c \right )+2824875 \sin \left (3 d x +3 c \right )+12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )-6105780 \cos \left (d x +c \right )+14247000 \cos \left (2 d x +2 c \right )+2686500 \cos \left (3 d x +3 c \right )+28968900 \sin \left (d x +c \right )+5151600 \sin \left (2 d x +2 c \right )-3957500 \sin \left (3 d x +3 c \right )-17666280}{2654208 d \left (-125 \sin \left (3 d x +3 c \right )+915 \sin \left (d x +c \right )-558+450 \cos \left (2 d x +2 c \right )\right )}\) | \(192\) |
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)^2-34 5/8192/(tan(1/2*d*x+1/2*c)-3)-279/32768*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)^2-3505/2211 84/(3*tan(1/2*d*x+1/2*c)-1)+279/32768*ln(3*tan(1/2*d*x+1/2*c)-1))
Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=\frac {199000 \, \cos \left (d x + c\right )^{3} - 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} - 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) + 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} - 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (-4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) + 190800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 262320 \, \cos \left (d x + c\right )}{196608 \, {\left (225 \, d \cos \left (d x + c\right )^{2} - 5 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 52 \, d\right )} \sin \left (d x + c\right ) - 252 \, d\right )}} \]
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)*log(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) - 252) *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)
Leaf count of result is larger than twice the leaf count of optimal. 2353 vs. \(2 (126) = 252\).
Time = 3.12 (sec) , antiderivative size = 2353, normalized size of antiderivative = 16.81 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=\text {Too large to display} \]
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/(71663 616*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*lo g(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 - 4 087480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 - 71663 6160*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(tan(c/2 + d*x/2) - 3)*tan(c/2 + d*x/2)**3/(7 1663616*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*tan(c/2 + d*x/2) + 71663616*d) - 221696 19*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...
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (126) = 252\).
Time = 0.21 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=\frac {\frac {40 \, {\left (\frac {342495 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1066482 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1218910 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {486441 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84915 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 42741\right )}}{\frac {270 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {981 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {981 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {270 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 27} + 22599 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right ) - 22599 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 3\right )}{2654208 \, d} \]
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 - 98 1*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
Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=-\frac {\frac {40 \, {\left (84915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 486441 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1218910 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1066482 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 342495 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 42741\right )}}{{\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}^{3}} - 22599 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 22599 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \right |}\right )}{2654208 \, d} \]
-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 + 342 495*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*lo g(abs(tan(1/2*d*x + 1/2*c) - 3)))/d
Time = 0.00 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(-3+5 \sin (c+d x))^4} \, dx=\frac {279\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {5}{4}\right )}{16384\,d}-\frac {\frac {15725\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{331776}-\frac {270245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{995328}+\frac {3047275\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4478976}-\frac {296245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{497664}+\frac {63425\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{331776}-\frac {7915}{331776}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {1540\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{27}+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
(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*tan (c/2 + (d*x)/2)^5)/331776 - 7915/331776)/(d*((109*tan(c/2 + (d*x)/2)^2)/3 - 10*tan(c/2 + (d*x)/2) - (1540*tan(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))