Integrand size = 12, antiderivative size = 145 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {x}{625}+\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}-\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}+\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))} \]
1/625*x+278151/20480000*ln(2*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c))/d-2781 51/20480000*ln(2*cos(1/2*d*x+1/2*c)+sin(1/2*d*x+1/2*c))/d+3/80*tan(d*x+c)/ d/(5+3*sec(d*x+c))^3+519/12800*tan(d*x+c)/d/(5+3*sec(d*x+c))^2+38733/10240 00*tan(d*x+c)/d/(5+3*sec(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(344\) vs. \(2(145)=290\).
Time = 0.63 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.37 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {18284544 c+18284544 d x+4096000 c \cos (3 (c+d x))+4096000 d x \cos (3 (c+d x))+155208258 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+34768875 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+915 \cos (c+d x) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+450 \cos (2 (c+d x)) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-155208258 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-34768875 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+52174260 \sin (c+d x)+51462000 \sin (2 (c+d x))+24286500 \sin (3 (c+d x))}{81920000 d (3+5 \cos (c+d x))^3} \]
(18284544*c + 18284544*d*x + 4096000*c*Cos[3*(c + d*x)] + 4096000*d*x*Cos[ 3*(c + d*x)] + 155208258*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3476 8875*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 915*Cos [c + d*x]*(32768*(c + d*x) + 278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x) /2]] - 278151*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 450*Cos[2*(c + d*x)]*(32768*(c + d*x) + 278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2] ] - 278151*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 155208258*Log[2*C os[(c + d*x)/2] + Sin[(c + d*x)/2]] - 34768875*Cos[3*(c + d*x)]*Log[2*Cos[ (c + d*x)/2] + Sin[(c + d*x)/2]] + 52174260*Sin[c + d*x] + 51462000*Sin[2* (c + d*x)] + 24286500*Sin[3*(c + d*x)])/(81920000*d*(3 + 5*Cos[c + d*x])^3 )
Time = 0.79 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.81, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {3042, 4272, 27, 3042, 4548, 25, 3042, 4548, 25, 3042, 4407, 3042, 4318, 3042, 3138, 219}
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.
\(\displaystyle \int \frac {1}{(3 \sec (c+d x)+5)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (3 \csc \left (c+d x+\frac {\pi }{2}\right )+5\right )^4}dx\) |
\(\Big \downarrow \) 4272 |
\(\displaystyle \frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}-\frac {1}{240} \int -\frac {3 \left (6 \sec ^2(c+d x)-15 \sec (c+d x)+16\right )}{(3 \sec (c+d x)+5)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{80} \int \frac {6 \sec ^2(c+d x)-15 \sec (c+d x)+16}{(3 \sec (c+d x)+5)^3}dx+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{80} \int \frac {6 \csc \left (c+d x+\frac {\pi }{2}\right )^2-15 \csc \left (c+d x+\frac {\pi }{2}\right )+16}{\left (3 \csc \left (c+d x+\frac {\pi }{2}\right )+5\right )^3}dx+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 4548 |
\(\displaystyle \frac {1}{80} \left (\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}-\frac {1}{160} \int -\frac {519 \sec ^2(c+d x)-1410 \sec (c+d x)+512}{(3 \sec (c+d x)+5)^2}dx\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \int \frac {519 \sec ^2(c+d x)-1410 \sec (c+d x)+512}{(3 \sec (c+d x)+5)^2}dx+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \int \frac {519 \csc \left (c+d x+\frac {\pi }{2}\right )^2-1410 \csc \left (c+d x+\frac {\pi }{2}\right )+512}{\left (3 \csc \left (c+d x+\frac {\pi }{2}\right )+5\right )^2}dx+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 4548 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}-\frac {1}{80} \int -\frac {8192-50715 \sec (c+d x)}{3 \sec (c+d x)+5}dx\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \int \frac {8192-50715 \sec (c+d x)}{3 \sec (c+d x)+5}dx+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \int \frac {8192-50715 \csc \left (c+d x+\frac {\pi }{2}\right )}{3 \csc \left (c+d x+\frac {\pi }{2}\right )+5}dx+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 4407 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {278151}{5} \int \frac {\sec (c+d x)}{3 \sec (c+d x)+5}dx\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {278151}{5} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{3 \csc \left (c+d x+\frac {\pi }{2}\right )+5}dx\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 4318 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {92717}{5} \int \frac {1}{\frac {5}{3} \cos (c+d x)+1}dx\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {92717}{5} \int \frac {1}{\frac {5}{3} \sin \left (c+d x+\frac {\pi }{2}\right )+1}dx\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {185434 \int \frac {1}{\frac {8}{3}-\frac {2}{3} \tan ^2\left (\frac {1}{2} (c+d x)\right )}d\tan \left (\frac {1}{2} (c+d x)\right )}{5 d}\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {278151 \text {arctanh}\left (\frac {1}{2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{10 d}\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\) |
(3*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x])^3) + ((519*Tan[c + d*x])/(160* d*(5 + 3*Sec[c + d*x])^2) + (((8192*x)/5 - (278151*ArcTanh[Tan[(c + d*x)/2 ]/2])/(10*d))/80 + (38733*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x])))/160)/ 80
3.6.30.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(n + 1)*(a^2 - b^2)) Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x ], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ erQ[2*n]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^( m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {x}{625}+\frac {27 i \left (166525 \,{\mathrm e}^{5 i \left (d x +c \right )}+581495 \,{\mathrm e}^{4 i \left (d x +c \right )}+1003842 \,{\mathrm e}^{3 i \left (d x +c \right )}+1064590 \,{\mathrm e}^{2 i \left (d x +c \right )}+643025 \,{\mathrm e}^{i \left (d x +c \right )}+224875\right )}{2560000 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}-\frac {278151 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{20480000 d}+\frac {278151 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{20480000 d}\) | \(132\) |
derivativedivides | \(\frac {-\frac {27}{10240 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {1431}{102400 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {69093}{2048000 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{20480000}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{625}-\frac {27}{10240 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {1431}{102400 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {69093}{2048000 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{20480000}}{d}\) | \(136\) |
default | \(\frac {-\frac {27}{10240 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {1431}{102400 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {69093}{2048000 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{20480000}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{625}-\frac {27}{10240 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {1431}{102400 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {69093}{2048000 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{20480000}}{d}\) | \(136\) |
norman | \(\frac {-\frac {64 x}{625}-\frac {44523 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64000 d}+\frac {13527 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{32000 d}-\frac {69093 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024000 d}+\frac {48 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{625}-\frac {12 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{625}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{625}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-4\right )^{3}}+\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{20480000 d}-\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{20480000 d}\) | \(144\) |
parallelrisch | \(\frac {\left (254508165 \cos \left (d x +c \right )+125167950 \cos \left (2 d x +2 c \right )+34768875 \cos \left (3 d x +3 c \right )+155208258\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (-254508165 \cos \left (d x +c \right )-125167950 \cos \left (2 d x +2 c \right )-34768875 \cos \left (3 d x +3 c \right )-155208258\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )+29982720 d x \cos \left (d x +c \right )+14745600 d x \cos \left (2 d x +2 c \right )+4096000 d x \cos \left (3 d x +3 c \right )+18284544 d x +52174260 \sin \left (d x +c \right )+51462000 \sin \left (2 d x +2 c \right )+24286500 \sin \left (3 d x +3 c \right )}{20480000 d \left (558+125 \cos \left (3 d x +3 c \right )+915 \cos \left (d x +c \right )+450 \cos \left (2 d x +2 c \right )\right )}\) | \(201\) |
1/625*x+27/2560000*I*(166525*exp(5*I*(d*x+c))+581495*exp(4*I*(d*x+c))+1003 842*exp(3*I*(d*x+c))+1064590*exp(2*I*(d*x+c))+643025*exp(I*(d*x+c))+224875 )/d/(5*exp(2*I*(d*x+c))+6*exp(I*(d*x+c))+5)^3-278151/20480000/d*ln(exp(I*( d*x+c))+3/5+4/5*I)+278151/20480000/d*ln(exp(I*(d*x+c))+3/5-4/5*I)
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {8192000 \, d x \cos \left (d x + c\right )^{3} + 14745600 \, d x \cos \left (d x + c\right )^{2} + 8847360 \, d x \cos \left (d x + c\right ) + 1769472 \, d x - 278151 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 278151 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 1080 \, {\left (44975 \, \cos \left (d x + c\right )^{2} + 47650 \, \cos \left (d x + c\right ) + 12911\right )} \sin \left (d x + c\right )}{40960000 \, {\left (125 \, d \cos \left (d x + c\right )^{3} + 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) + 27 \, d\right )}} \]
1/40960000*(8192000*d*x*cos(d*x + c)^3 + 14745600*d*x*cos(d*x + c)^2 + 884 7360*d*x*cos(d*x + c) + 1769472*d*x - 278151*(125*cos(d*x + c)^3 + 225*cos (d*x + c)^2 + 135*cos(d*x + c) + 27)*log(3/2*cos(d*x + c) + 2*sin(d*x + c) + 5/2) + 278151*(125*cos(d*x + c)^3 + 225*cos(d*x + c)^2 + 135*cos(d*x + c) + 27)*log(3/2*cos(d*x + c) - 2*sin(d*x + c) + 5/2) + 1080*(44975*cos(d* x + c)^2 + 47650*cos(d*x + c) + 12911)*sin(d*x + c))/(125*d*cos(d*x + c)^3 + 225*d*cos(d*x + c)^2 + 135*d*cos(d*x + c) + 27*d)
\[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{4}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=-\frac {\frac {540 \, {\left (\frac {26384 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16032 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2559 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} - 65536 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 278151 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 278151 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{20480000 \, d} \]
-1/20480000*(540*(26384*sin(d*x + c)/(cos(d*x + c) + 1) - 16032*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2559*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(48* sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 12*sin(d*x + c)^4/(cos(d*x + c) + 1) ^4 + sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 64) - 65536*arctan(sin(d*x + c) /(cos(d*x + c) + 1)) + 278151*log(sin(d*x + c)/(cos(d*x + c) + 1) + 2) - 2 78151*log(sin(d*x + c)/(cos(d*x + c) + 1) - 2))/d
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {32768 \, d x + 32768 \, c - \frac {540 \, {\left (2559 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16032 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 26384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{3}} - 278151 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 278151 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{20480000 \, d} \]
1/20480000*(32768*d*x + 32768*c - 540*(2559*tan(1/2*d*x + 1/2*c)^5 - 16032 *tan(1/2*d*x + 1/2*c)^3 + 26384*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c )^2 - 4)^3 - 278151*log(abs(tan(1/2*d*x + 1/2*c) + 2)) + 278151*log(abs(ta n(1/2*d*x + 1/2*c) - 2)))/d
Time = 14.47 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {x}{625}-\frac {278151\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{10240000\,d}-\frac {\frac {69093\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024000}-\frac {13527\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32000}+\frac {44523\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64000}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \]