Integrand size = 12, antiderivative size = 138 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {279 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \cos (c+d x)}{48 d (3+5 \sin (c+d x))^3}+\frac {25 \cos (c+d x)}{512 d (3+5 \sin (c+d x))^2}-\frac {995 \cos (c+d x)}{24576 d (3+5 \sin (c+d x))} \] Output:
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))
Time = 1.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {2511 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2511 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2320}{\left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {720}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+20 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {80}{\left (3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {240}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {597}{\cos \left (\frac {1}{2} (c+d x)\right )+3 \sin \left (\frac {1}{2} (c+d x)\right )}\right )}{294912 d} \] Input:
Integrate[(3 + 5*Sin[c + d*x])^(-4),x]
Output:
(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*Cos[(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]*(80/ (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)
Time = 0.56 (sec) , antiderivative size = 130, 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 |
\(\displaystyle \frac {1}{48} \int -\frac {9-10 \sin (c+d x)}{(5 \sin (c+d x)+3)^3}dx-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{48} \int \frac {9-10 \sin (c+d x)}{(5 \sin (c+d x)+3)^3}dx-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{48} \int \frac {9-10 \sin (c+d x)}{(5 \sin (c+d x)+3)^3}dx-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle \frac {1}{48} \left (\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}-\frac {1}{32} \int -\frac {154-75 \sin (c+d x)}{(5 \sin (c+d x)+3)^2}dx\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 \sin (c+d x)}{(5 \sin (c+d x)+3)^2}dx+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 \sin (c+d x)}{(5 \sin (c+d x)+3)^2}dx+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {1}{16} \int -\frac {837}{5 \sin (c+d x)+3}dx-\frac {995 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}\right )+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {837}{16} \int \frac {1}{5 \sin (c+d x)+3}dx-\frac {995 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}\right )+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {837}{16} \int \frac {1}{5 \sin (c+d x)+3}dx-\frac {995 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}\right )+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {837 \int \frac {1}{3 \tan ^2\left (\frac {1}{2} (c+d x)\right )+10 \tan \left (\frac {1}{2} (c+d x)\right )+3}d\tan \left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {995 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}\right )+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 1081 |
\(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (-\frac {2511 \int \left (\frac {1}{8 \left (3 \tan \left (\frac {1}{2} (c+d x)\right )+1\right )}-\frac {1}{24 \left (\tan \left (\frac {1}{2} (c+d x)\right )+3\right )}\right )d\tan \left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {995 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}\right )+\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{48} \left (\frac {75 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}+\frac {1}{32} \left (-\frac {2511 \left (\frac {1}{24} \log \left (3 \tan \left (\frac {1}{2} (c+d x)\right )+1\right )-\frac {1}{24} \log \left (\tan \left (\frac {1}{2} (c+d x)\right )+3\right )\right )}{8 d}-\frac {995 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}\right )\right )-\frac {5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}\) |
Input:
Int[(3 + 5*Sin[c + d*x])^(-4),x]
Output:
(-5*Cos[c + d*x])/(48*d*(3 + 5*Sin[c + d*x])^3) + ((75*Cos[c + d*x])/(32*d *(3 + 5*Sin[c + d*x])^2) + ((-2511*(-1/24*Log[3 + Tan[(c + d*x)/2]] + Log[ 1 + 3*Tan[(c + d*x)/2]]/24))/(8*d) - (995*Cos[c + d*x])/(16*d*(3 + 5*Sin[c + d*x])))/32)/48
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 = 0.44 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {125}{20736 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {275}{27648 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3505}{221184 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}-\frac {125}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{3}}+\frac {75}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{2}}-\frac {345}{8192 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768}}{d}\) | \(132\) |
default | \(\frac {-\frac {125}{20736 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {275}{27648 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3505}{221184 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}-\frac {125}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{3}}+\frac {75}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{2}}-\frac {345}{8192 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768}}{d}\) | \(132\) |
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 | \(\frac {-\frac {7915}{12288 d}-\frac {63425 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12288 d}-\frac {3047275 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{165888 d}-\frac {15725 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12288 d}-\frac {296245 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{18432 d}-\frac {270245 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{36864 d}}{\left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{3}}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{32768 d}-\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768 d}\) | \(151\) |
parallelrisch | \(\frac {\left (-10169550 \cos \left (2 d x +2 c \right )+20678085 \sin \left (d x +c \right )-2824875 \sin \left (3 d x +3 c \right )+12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1}{3}\right )+\left (10169550 \cos \left (2 d x +2 c \right )-20678085 \sin \left (d x +c \right )+2824875 \sin \left (3 d x +3 c \right )-12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )+6105780 \cos \left (d x +c \right )-14247000 \cos \left (2 d x +2 c \right )-2686500 \cos \left (3 d x +3 c \right )+28968900 \sin \left (d x +c \right )+5151600 \sin \left (2 d x +2 c \right )-3957500 \sin \left (3 d x +3 c \right )+17666280}{2654208 d \left (-558+125 \sin \left (3 d x +3 c \right )-915 \sin \left (d x +c \right )+450 \cos \left (2 d x +2 c \right )\right )}\) | \(192\) |
Input:
int(1/(3+5*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(-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)^2-3505/221184/(3*tan(1/2*d*x+1/2*c)+1)-279/32768*ln(3*tan(1/2*d*x+1/2* c)+1)-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))
Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {199000 \, \cos \left (d x + c\right )^{3} - 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) + 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 190800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 262320 \, \cos \left (d x + c\right )}{196608 \, {\left (225 \, d \cos \left (d x + c\right )^{2} + 5 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 52 \, d\right )} \sin \left (d x + c\right ) - 252 \, d\right )}} \] Input:
integrate(1/(3+5*sin(d*x+c))^4,x, algorithm="fricas")
Output:
-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. 2356 vs. \(2 (126) = 252\).
Time = 3.22 (sec) , antiderivative size = 2356, normalized size of antiderivative = 17.07 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(3+5*sin(d*x+c))**4,x)
Output:
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/(71663616 *d*tan(c/2 + d*x/2)**6 + 716636160*d*tan(c/2 + d*x/2)**5 + 2603778048*d*ta n(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(t an(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 + 4087 480320*d*tan(c/2 + d*x/2)**3 + 2603778048*d*tan(c/2 + d*x/2)**2 + 71663616 0*d*tan(c/2 + d*x/2) + 71663616*d) + 22169619*log(tan(c/2 + d*x/2) + 3)*ta n(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/(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*ta n(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 + ...
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (124) = 248\).
Time = 0.04 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {\frac {40 \, {\left (\frac {342495 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1066482 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1218910 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {486441 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84915 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 42741\right )}}{\frac {270 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {981 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {981 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {270 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 27} + 22599 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 22599 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{2654208 \, d} \] Input:
integrate(1/(3+5*sin(d*x+c))^4,x, algorithm="maxima")
Output:
-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 + 9 81*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 = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=-\frac {\frac {40 \, {\left (84915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 486441 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1218910 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1066482 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 342495 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42741\right )}}{{\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}^{3}} + 22599 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 22599 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \right |}\right )}{2654208 \, d} \] Input:
integrate(1/(3+5*sin(d*x+c))^4,x, algorithm="giac")
Output:
-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 = 28.67 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {279\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {5}{4}\right )}{16384\,d}-\frac {\frac {15725\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{331776}+\frac {270245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{995328}+\frac {3047275\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4478976}+\frac {296245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{497664}+\frac {63425\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{331776}+\frac {7915}{331776}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {1540\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{27}+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \] Input:
int(1/(5*sin(c + d*x) + 3)^4,x)
Output:
(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*(10*tan(c/2 + (d*x)/2) + (109* tan(c/2 + (d*x)/2)^2)/3 + (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))
Time = 0.17 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.95 \[ \int \frac {1}{(3+5 \sin (c+d x))^4} \, dx=\frac {-298500 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-286200 \cos \left (d x +c \right ) \sin \left (d x +c \right )-94980 \cos \left (d x +c \right )+313875 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right ) \sin \left (d x +c \right )^{3}+564975 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right ) \sin \left (d x +c \right )^{2}+338985 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right ) \sin \left (d x +c \right )+67797 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )-313875 \,\mathrm {log}\left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{3}-564975 \,\mathrm {log}\left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}-338985 \,\mathrm {log}\left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )-67797 \,\mathrm {log}\left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-265000 \sin \left (d x +c \right )^{3}-477000 \sin \left (d x +c \right )^{2}-286200 \sin \left (d x +c \right )-57240}{294912 d \left (125 \sin \left (d x +c \right )^{3}+225 \sin \left (d x +c \right )^{2}+135 \sin \left (d x +c \right )+27\right )} \] Input:
int(1/(3+5*sin(d*x+c))^4,x)
Output:
( - 298500*cos(c + d*x)*sin(c + d*x)**2 - 286200*cos(c + d*x)*sin(c + d*x) - 94980*cos(c + d*x) + 313875*log(tan((c + d*x)/2) + 3)*sin(c + d*x)**3 + 564975*log(tan((c + d*x)/2) + 3)*sin(c + d*x)**2 + 338985*log(tan((c + d* x)/2) + 3)*sin(c + d*x) + 67797*log(tan((c + d*x)/2) + 3) - 313875*log(3*t an((c + d*x)/2) + 1)*sin(c + d*x)**3 - 564975*log(3*tan((c + d*x)/2) + 1)* sin(c + d*x)**2 - 338985*log(3*tan((c + d*x)/2) + 1)*sin(c + d*x) - 67797* log(3*tan((c + d*x)/2) + 1) - 265000*sin(c + d*x)**3 - 477000*sin(c + d*x) **2 - 286200*sin(c + d*x) - 57240)/(294912*d*(125*sin(c + d*x)**3 + 225*si n(c + d*x)**2 + 135*sin(c + d*x) + 27))