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

Optimal. Leaf size=120 \[ \frac {963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac {9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {x}{125} \]

[Out]

1/125*x+8361/256000*ln(2*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c))/d-8361/256000*ln(2*cos(1/2*d*x+1/2*c)+sin(1/2*
d*x+1/2*c))/d+9/160*tan(d*x+c)/d/(5+3*sec(d*x+c))^2+963/12800*tan(d*x+c)/d/(5+3*sec(d*x+c))

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Rubi [A]  time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3785, 4060, 3919, 3831, 2659, 206} \[ \frac {963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac {9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {x}{125} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*Sec[c + d*x])^(-3),x]

[Out]

x/125 + (8361*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(256000*d) - (8361*Log[2*Cos[(c + d*x)/2] + Sin[(c +
d*x)/2]])/(256000*d) + (9*Tan[c + d*x])/(160*d*(5 + 3*Sec[c + d*x])^2) + (963*Tan[c + d*x])/(12800*d*(5 + 3*S
ec[c + d*x]))

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])

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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {1}{(5+3 \sec (c+d x))^3} \, dx &=\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}-\frac {1}{160} \int \frac {-32+30 \sec (c+d x)-9 \sec ^2(c+d x)}{(5+3 \sec (c+d x))^2} \, dx\\ &=\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}+\frac {\int \frac {512-1365 \sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{12800}\\ &=\frac {x}{125}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}-\frac {8361 \int \frac {\sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{64000}\\ &=\frac {x}{125}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}-\frac {2787 \int \frac {1}{1+\frac {5}{3} \cos (c+d x)} \, dx}{64000}\\ &=\frac {x}{125}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}-\frac {2787 \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 )}{32000 d}\\ &=\frac {x}{125}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}\\ \end {align*}

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Mathematica [B]  time = 0.32, size = 241, normalized size = 2.01 \[ \frac {115560 \sin (c+d x)+110700 \sin (2 (c+d x))+359523 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \cos (c+d x) \left (2048 (c+d x)+8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+25 \cos (2 (c+d x)) \left (2048 (c+d x)+8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-359523 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )+88064 c+88064 d x}{512000 d (5 \cos (c+d x)+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(88064*c + 88064*d*x + 359523*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 60*Cos[c + d*x]*(2048*(c + d*x) + 8
361*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 8361*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 25*Cos[2*(
c + d*x)]*(2048*(c + d*x) + 8361*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 8361*Log[2*Cos[(c + d*x)/2] + Si
n[(c + d*x)/2]]) - 359523*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 115560*Sin[c + d*x] + 110700*Sin[2*(c +
d*x)])/(512000*d*(3 + 5*Cos[c + d*x])^2)

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fricas [A]  time = 0.49, size = 155, normalized size = 1.29 \[ \frac {102400 \, d x \cos \left (d x + c\right )^{2} + 122880 \, d x \cos \left (d x + c\right ) + 36864 \, d x - 8361 \, {\left (25 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 8361 \, {\left (25 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 1080 \, {\left (205 \, \cos \left (d x + c\right ) + 107\right )} \sin \left (d x + c\right )}{512000 \, {\left (25 \, d \cos \left (d x + c\right )^{2} + 30 \, d \cos \left (d x + c\right ) + 9 \, d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/512000*(102400*d*x*cos(d*x + c)^2 + 122880*d*x*cos(d*x + c) + 36864*d*x - 8361*(25*cos(d*x + c)^2 + 30*cos(d
*x + c) + 9)*log(3/2*cos(d*x + c) + 2*sin(d*x + c) + 5/2) + 8361*(25*cos(d*x + c)^2 + 30*cos(d*x + c) + 9)*log
(3/2*cos(d*x + c) - 2*sin(d*x + c) + 5/2) + 1080*(205*cos(d*x + c) + 107)*sin(d*x + c))/(25*d*cos(d*x + c)^2 +
30*d*cos(d*x + c) + 9*d)

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giac [A]  time = 0.17, size = 85, normalized size = 0.71 \[ \frac {2048 \, d x + 2048 \, c - \frac {540 \, {\left (49 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 156 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{2}} - 8361 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 8361 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{256000 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256000*(2048*d*x + 2048*c - 540*(49*tan(1/2*d*x + 1/2*c)^3 - 156*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)
^2 - 4)^2 - 8361*log(abs(tan(1/2*d*x + 1/2*c) + 2)) + 8361*log(abs(tan(1/2*d*x + 1/2*c) - 2)))/d

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maple [A]  time = 0.43, size = 123, normalized size = 1.02 \[ -\frac {27}{2560 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {1323}{25600 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{256000 d}+\frac {27}{2560 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {1323}{25600 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{256000 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{125 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/2560/d/(tan(1/2*d*x+1/2*c)-2)^2-1323/25600/d/(tan(1/2*d*x+1/2*c)-2)+8361/256000/d*ln(tan(1/2*d*x+1/2*c)-2)
+27/2560/d/(tan(1/2*d*x+1/2*c)+2)^2-1323/25600/d/(tan(1/2*d*x+1/2*c)+2)-8361/256000/d*ln(tan(1/2*d*x+1/2*c)+2)
+2/125/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 0.49, size = 155, normalized size = 1.29 \[ -\frac {\frac {540 \, {\left (\frac {156 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {49 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 16} - 4096 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 8361 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 8361 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{256000 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256000*(540*(156*sin(d*x + c)/(cos(d*x + c) + 1) - 49*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(8*sin(d*x + c)^
2/(cos(d*x + c) + 1)^2 - sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 16) - 4096*arctan(sin(d*x + c)/(cos(d*x + c) +
1)) + 8361*log(sin(d*x + c)/(cos(d*x + c) + 1) + 2) - 8361*log(sin(d*x + c)/(cos(d*x + c) + 1) - 2))/d

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mupad [B]  time = 0.95, size = 78, normalized size = 0.65 \[ \frac {x}{125}-\frac {8361\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{128000\,d}+\frac {\frac {1053\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3200}-\frac {1323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12800}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+16\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3/cos(c + d*x) + 5)^3,x)

[Out]

x/125 - (8361*atanh(tan(c/2 + (d*x)/2)/2))/(128000*d) + ((1053*tan(c/2 + (d*x)/2))/3200 - (1323*tan(c/2 + (d*x
)/2)^3)/12800)/(d*(tan(c/2 + (d*x)/2)^4 - 8*tan(c/2 + (d*x)/2)^2 + 16))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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