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

Optimal. Leaf size=106 \[ -\frac{16925 \tan (c+d x)}{221184 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)^2}-\frac{25 \tan (c+d x)}{144 d (5 \sec (c+d x)+3)^3}+\frac{11215 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{1327104 d}+\frac{21553 x}{2654208} \]

[Out]

(21553*x)/2654208 + (11215*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1327104*d) - (25*Tan[c + d*x])/(144*d*(3
+ 5*Sec[c + d*x])^3) - (25*Tan[c + d*x])/(4608*d*(3 + 5*Sec[c + d*x])^2) - (16925*Tan[c + d*x])/(221184*d*(3 +
5*Sec[c + d*x]))

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Rubi [A]  time = 0.157904, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3785, 4060, 3919, 3831, 2657} \[ -\frac{16925 \tan (c+d x)}{221184 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)^2}-\frac{25 \tan (c+d x)}{144 d (5 \sec (c+d x)+3)^3}+\frac{11215 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{1327104 d}+\frac{21553 x}{2654208} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Sec[c + d*x])^(-4),x]

[Out]

(21553*x)/2654208 + (11215*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1327104*d) - (25*Tan[c + d*x])/(144*d*(3
+ 5*Sec[c + d*x])^3) - (25*Tan[c + d*x])/(4608*d*(3 + 5*Sec[c + d*x])^2) - (16925*Tan[c + d*x])/(221184*d*(3 +
5*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 2657

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{1}{(3+5 \sec (c+d x))^4} \, dx &=-\frac{25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}+\frac{1}{144} \int \frac{48+45 \sec (c+d x)-50 \sec ^2(c+d x)}{(3+5 \sec (c+d x))^3} \, dx\\ &=-\frac{25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac{25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}+\frac{\int \frac{1536-870 \sec (c+d x)-75 \sec ^2(c+d x)}{(3+5 \sec (c+d x))^2} \, dx}{13824}\\ &=-\frac{25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac{25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac{16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}+\frac{\int \frac{24576+29745 \sec (c+d x)}{3+5 \sec (c+d x)} \, dx}{663552}\\ &=\frac{x}{81}-\frac{25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac{25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac{16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}-\frac{11215 \int \frac{\sec (c+d x)}{3+5 \sec (c+d x)} \, dx}{663552}\\ &=\frac{x}{81}-\frac{25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac{25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac{16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}-\frac{2243 \int \frac{1}{1+\frac{3}{5} \cos (c+d x)} \, dx}{663552}\\ &=\frac{21553 x}{2654208}+\frac{11215 \tan ^{-1}\left (\frac{\sin (c+d x)}{3+\cos (c+d x)}\right )}{1327104 d}-\frac{25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac{25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac{16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.529908, size = 141, normalized size = 1.33 \[ \frac{-5660475 \sin (c+d x)-3082500 \sin (2 (c+d x))-582975 \sin (3 (c+d x))+8036352 (c+d x) \cos (c+d x)+2211840 c \cos (2 (c+d x))+2211840 d x \cos (2 (c+d x))+221184 c \cos (3 (c+d x))+221184 d x \cos (3 (c+d x))+22430 (3 \cos (c+d x)+5)^3 \tan ^{-1}\left (2 \cot \left (\frac{1}{2} (c+d x)\right )\right )+6307840 c+6307840 d x}{2654208 d (3 \cos (c+d x)+5)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6307840*c + 6307840*d*x + 8036352*(c + d*x)*Cos[c + d*x] + 22430*ArcTan[2*Cot[(c + d*x)/2]]*(5 + 3*Cos[c + d*
x])^3 + 2211840*c*Cos[2*(c + d*x)] + 2211840*d*x*Cos[2*(c + d*x)] + 221184*c*Cos[3*(c + d*x)] + 221184*d*x*Cos
[3*(c + d*x)] - 5660475*Sin[c + d*x] - 3082500*Sin[2*(c + d*x)] - 582975*Sin[3*(c + d*x)])/(2654208*d*(5 + 3*C
os[c + d*x])^3)

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Maple [A]  time = 0.05, size = 125, normalized size = 1.2 \begin{align*}{\frac{2}{81\,d}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{25925}{221184\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-3}}-{\frac{3575}{6912\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-3}}-{\frac{17675}{13824\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+4 \right ) ^{-3}}-{\frac{11215}{1327104\,d}\arctan \left ({\frac{1}{2}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/81/d*arctan(tan(1/2*d*x+1/2*c))-25925/221184/d/(tan(1/2*d*x+1/2*c)^2+4)^3*tan(1/2*d*x+1/2*c)^5-3575/6912/d/(
tan(1/2*d*x+1/2*c)^2+4)^3*tan(1/2*d*x+1/2*c)^3-17675/13824/d/(tan(1/2*d*x+1/2*c)^2+4)^3*tan(1/2*d*x+1/2*c)-112
15/1327104/d*arctan(1/2*tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.95531, size = 231, normalized size = 2.18 \begin{align*} -\frac{\frac{150 \,{\left (\frac{11312 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{4576 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1037 \, \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} - 32768 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 11215 \, \arctan \left (\frac{\sin \left (d x + c\right )}{2 \,{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{1327104 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1327104*(150*(11312*sin(d*x + c)/(cos(d*x + c) + 1) + 4576*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1037*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) - 32768*arctan(sin(d*x + c)/(cos(d*x + c) + 1)) + 11215*arctan(
1/2*sin(d*x + c)/(cos(d*x + c) + 1)))/d

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Fricas [A]  time = 1.73109, size = 504, normalized size = 4.75 \begin{align*} \frac{884736 \, d x \cos \left (d x + c\right )^{3} + 4423680 \, d x \cos \left (d x + c\right )^{2} + 7372800 \, d x \cos \left (d x + c\right ) + 4096000 \, d x + 11215 \,{\left (27 \, \cos \left (d x + c\right )^{3} + 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) + 125\right )} \arctan \left (\frac{5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) - 300 \,{\left (7773 \, \cos \left (d x + c\right )^{2} + 20550 \, \cos \left (d x + c\right ) + 16925\right )} \sin \left (d x + c\right )}{2654208 \,{\left (27 \, d \cos \left (d x + c\right )^{3} + 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) + 125 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2654208*(884736*d*x*cos(d*x + c)^3 + 4423680*d*x*cos(d*x + c)^2 + 7372800*d*x*cos(d*x + c) + 4096000*d*x + 1
1215*(27*cos(d*x + c)^3 + 135*cos(d*x + c)^2 + 225*cos(d*x + c) + 125)*arctan(1/4*(5*cos(d*x + c) + 3)/sin(d*x
+ c)) - 300*(7773*cos(d*x + c)^2 + 20550*cos(d*x + c) + 16925)*sin(d*x + c))/(27*d*cos(d*x + c)^3 + 135*d*cos
(d*x + c)^2 + 225*d*cos(d*x + c) + 125*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21062, size = 119, normalized size = 1.12 \begin{align*} \frac{21553 \, d x + 21553 \, c - \frac{300 \,{\left (1037 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4576 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 11312 \, \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}} + 22430 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{2654208 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2654208*(21553*d*x + 21553*c - 300*(1037*tan(1/2*d*x + 1/2*c)^5 + 4576*tan(1/2*d*x + 1/2*c)^3 + 11312*tan(1/
2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 4)^3 + 22430*arctan(sin(d*x + c)/(cos(d*x + c) + 3)))/d