3.6.30 \(\int \frac {1}{(5+3 \sec (c+d x))^4} \, dx\) [530]

3.6.30.1 Optimal result

Integrand size = 12, antiderivative size = 145 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {x}{625}+\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}-\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}+\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))} \]

output

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))
 

3.6.30.2 Mathematica [B] (verified)

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 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {18284544 c+18284544 d x+4096000 c \cos (3 (c+d x))+4096000 d x \cos (3 (c+d x))[phone] \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )[phone] \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+915 \cos (c+d x) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+450 \cos (2 (c+d x)) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-155208258 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-34768875 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )[phone] \sin (c+d x)[phone] \sin (2 (c+d x))[phone] \sin (3 (c+d x))}{81920000 d (3+5 \cos (c+d x))^3} \]

input

Integrate[(5 + 3*Sec[c + d*x])^(-4),x]
 

output

(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 
)
 

3.6.30.3 Rubi [A] (verified)

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

\(\displaystyle \frac {1}{80} \left (\frac {1}{160} \left (\frac {1}{80} \left (\frac {8192 x}{5}-\frac {278151 \text {arctanh}\left (\frac {1}{2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{10 d}\right )+\frac {38733 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}\right )+\frac {519 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}\right )+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}\)

input

Int[(5 + 3*Sec[c + d*x])^(-4),x]
 

output

(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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

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]
 

rule 4272

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]
 

rule 4318

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]
 

rule 4407

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]
 

rule 4548

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]
 

3.6.30.4 Maple [C] (verified)

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	\(\frac {\left (254508165 \cos \left (d x +c \right )[phone] \cos \left (2 d x +2 c \right )[phone] \cos \left (3 d x +3 c \right )[phone]\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (-254508165 \cos \left (d x +c \right )-125167950 \cos \left (2 d x +2 c \right )-34768875 \cos \left (3 d x +3 c \right )-155208258\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )[phone] d x \cos \left (d x +c \right )[phone] d x \cos \left (2 d x +2 c \right )+4096000 d x \cos \left (3 d x +3 c \right )[phone] d x [phone] \sin \left (d x +c \right )[phone] \sin \left (2 d x +2 c \right )[phone] \sin \left (3 d x +3 c \right )}{20480000 d \left (558+125 \cos \left (3 d x +3 c \right )+915 \cos \left (d x +c \right )+450 \cos \left (2 d x +2 c \right )\right )}\)	\(201\)

input

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

output

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)
 

3.6.30.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {8192000 \, d x \cos \left (d x + c\right )^{3} + 14745600 \, d x \cos \left (d x + c\right )^{2} + 8847360 \, d x \cos \left (d x + c\right ) + 1769472 \, d x - 278151 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 278151 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 1080 \, {\left (44975 \, \cos \left (d x + c\right )^{2} + 47650 \, \cos \left (d x + c\right ) + 12911\right )} \sin \left (d x + c\right )}{40960000 \, {\left (125 \, d \cos \left (d x + c\right )^{3} + 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) + 27 \, d\right )}} \]

input

integrate(1/(5+3*sec(d*x+c))^4,x, algorithm="fricas")
 

output

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)
 

3.6.30.6 Sympy [F]

\[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{4}}\, dx \]

input

integrate(1/(5+3*sec(d*x+c))**4,x)
 

output

Integral((3*sec(c + d*x) + 5)**(-4), x)
 

3.6.30.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=-\frac {\frac {540 \, {\left (\frac {26384 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16032 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2559 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} - 65536 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 278151 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 278151 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{20480000 \, d} \]

input

integrate(1/(5+3*sec(d*x+c))^4,x, algorithm="maxima")
 

output

-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
 

3.6.30.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {32768 \, d x + 32768 \, c - \frac {540 \, {\left (2559 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16032 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 26384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{3}} - 278151 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 278151 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{20480000 \, d} \]

input

integrate(1/(5+3*sec(d*x+c))^4,x, algorithm="giac")
 

output

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
 

3.6.30.9 Mupad [B] (verification not implemented)

Time = 14.47 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx=\frac {x}{625}-\frac {278151\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{10240000\,d}-\frac {\frac {69093\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024000}-\frac {13527\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32000}+\frac {44523\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64000}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \]

input

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

output

x/625 - (278151*atanh(tan(c/2 + (d*x)/2)/2))/(10240000*d) - ((44523*tan(c/ 
2 + (d*x)/2))/64000 - (13527*tan(c/2 + (d*x)/2)^3)/32000 + (69093*tan(c/2 
+ (d*x)/2)^5)/1024000)/(d*(48*tan(c/2 + (d*x)/2)^2 - 12*tan(c/2 + (d*x)/2) 
^4 + tan(c/2 + (d*x)/2)^6 - 64))