3.530 \(\int \frac{1}{(5+3 \sec (c+d x))^4} \, dx\)

Optimal. Leaf size=145 \[ \frac{38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac{519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac{3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac{278151 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{20480000 d}-\frac{278151 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{20480000 d}+\frac{x}{625} \]

[Out]

x/625 + (278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(20480000*d) - (278151*Log[2*Cos[(c + d*x)/2] + Si
n[(c + d*x)/2]])/(20480000*d) + (3*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x])^3) + (519*Tan[c + d*x])/(12800*d*(
5 + 3*Sec[c + d*x])^2) + (38733*Tan[c + d*x])/(1024000*d*(5 + 3*Sec[c + d*x]))

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Rubi [A]  time = 0.180382, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3785, 4060, 3919, 3831, 2659, 206} \[ \frac{38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac{519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac{3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac{278151 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{20480000 d}-\frac{278151 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{20480000 d}+\frac{x}{625} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*Sec[c + d*x])^(-4),x]

[Out]

x/625 + (278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(20480000*d) - (278151*Log[2*Cos[(c + d*x)/2] + Si
n[(c + d*x)/2]])/(20480000*d) + (3*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x])^3) + (519*Tan[c + d*x])/(12800*d*(
5 + 3*Sec[c + d*x])^2) + (38733*Tan[c + d*x])/(1024000*d*(5 + 3*Sec[c + d*x]))

Rule 3785

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] + Dist[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] && IntegerQ[2*n]

Rule 4060

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] + Dist[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]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(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 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(5+3 \sec (c+d x))^4} \, dx &=\frac{3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}-\frac{1}{240} \int \frac{-48+45 \sec (c+d x)-18 \sec ^2(c+d x)}{(5+3 \sec (c+d x))^3} \, dx\\ &=\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{\int \frac{1536-4230 \sec (c+d x)+1557 \sec ^2(c+d x)}{(5+3 \sec (c+d x))^2} \, dx}{38400}\\ &=\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))}-\frac{\int \frac{-24576+152145 \sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{3072000}\\ &=\frac{x}{625}+\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))}-\frac{278151 \int \frac{\sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{5120000}\\ &=\frac{x}{625}+\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))}-\frac{92717 \int \frac{1}{1+\frac{5}{3} \cos (c+d x)} \, dx}{5120000}\\ &=\frac{x}{625}+\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))}-\frac{92717 \operatorname{Subst}\left (\int \frac{1}{\frac{8}{3}-\frac{2 x^2}{3}} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{2560000 d}\\ &=\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))}\\ \end{align*}

Mathematica [B]  time = 0.520187, size = 344, normalized size = 2.37 \[ \frac{52174260 \sin (c+d x)[phone] \sin (2 (c+d x))[phone] \sin (3 (c+d x))+4096000 c \cos (3 (c+d x))+4096000 d x \cos (3 (c+d x))[phone] \cos (3 (c+d x)) \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )[phone] \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+915 \cos (c+d x) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-278151 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+450 \cos (2 (c+d x)) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-278151 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-34768875 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )-155208258 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )[phone] c+18284544 d x}{81920000 d (5 \cos (c+d x)+3)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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]] + 34768875*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 915*C
os[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*Cos[(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)

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Maple [A]  time = 0.053, size = 159, normalized size = 1.1 \begin{align*}{\frac{2}{625\,d}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{27}{10240\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-3}}+{\frac{1431}{102400\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-2}}-{\frac{69093}{2048000\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-1}}-{\frac{278151}{20480000\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }-{\frac{27}{10240\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-3}}-{\frac{1431}{102400\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-2}}-{\frac{69093}{2048000\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-1}}+{\frac{278151}{20480000\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/625/d*arctan(tan(1/2*d*x+1/2*c))-27/10240/d/(tan(1/2*d*x+1/2*c)+2)^3+1431/102400/d/(tan(1/2*d*x+1/2*c)+2)^2-
69093/2048000/d/(tan(1/2*d*x+1/2*c)+2)-278151/20480000/d*ln(tan(1/2*d*x+1/2*c)+2)-27/10240/d/(tan(1/2*d*x+1/2*
c)-2)^3-1431/102400/d/(tan(1/2*d*x+1/2*c)-2)^2-69093/2048000/d/(tan(1/2*d*x+1/2*c)-2)+278151/20480000/d*ln(tan
(1/2*d*x+1/2*c)-2)

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Maxima [A]  time = 1.6035, size = 262, normalized size = 1.81 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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) - 278151*log(sin(d*x + c)/(cos(d*x + c) + 1) - 2))/d

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Fricas [A]  time = 1.75805, size = 672, normalized size = 4.63 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40960000*(8192000*d*x*cos(d*x + c)^3 + 14745600*d*x*cos(d*x + c)^2 + 8847360*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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 \sec{\left (c + d x \right )} + 5\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2832, size = 132, normalized size = 0.91 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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(
tan(1/2*d*x + 1/2*c) - 2)))/d