3.6.26 \(\int \frac {1}{(3+5 \sec (c+d x))^4} \, dx\) [526]

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

[Out]

21553/2654208*x+11215/1327104*arctan(sin(d*x+c)/(3+cos(d*x+c)))/d-25/144*tan(d*x+c)/d/(3+5*sec(d*x+c))^3-25/46
08*tan(d*x+c)/d/(3+5*sec(d*x+c))^2-16925/221184*tan(d*x+c)/d/(3+5*sec(d*x+c))

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Rubi [A]
time = 0.11, antiderivative size = 106, normalized size of antiderivative = 1.00, 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 = {3870, 4145, 4004, 3916, 2736} \begin {gather*} \frac {11215 \text {ArcTan}\left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{1327104 d}-\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 {21553 x}{2654208} \end {gather*}

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 2736

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

Rule 3870

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 3916

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

Rule 4004

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 4145

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}{(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*}

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Mathematica [A]
time = 0.56, size = 141, normalized size = 1.33 \begin {gather*} \frac {6307840 c+6307840 d x+8036352 (c+d x) \cos (c+d x)+22430 \text {ArcTan}\left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right ) (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 \cos (c+d x))^3} \end {gather*}

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 = 0.09, size = 87, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{81}+\frac {-\frac {25925 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{221184}-\frac {3575 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6912}-\frac {17675 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{13824}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {11215 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1327104}}{d}\)	\(87\)
default	\(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{81}+\frac {-\frac {25925 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{221184}-\frac {3575 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6912}-\frac {17675 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{13824}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {11215 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1327104}}{d}\)	\(87\)
risch	\(\frac {x}{81}-\frac {25 i \left (164835 \,{\mathrm e}^{5 i \left (d x +c \right )}+931257 \,{\mathrm e}^{4 i \left (d x +c \right )}+1995070 \,{\mathrm e}^{3 i \left (d x +c \right )}+1610514 \,{\mathrm e}^{2 i \left (d x +c \right )}+534735 \,{\mathrm e}^{i \left (d x +c \right )}+69957\right )}{995328 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}+\frac {11215 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{2654208 d}-\frac {11215 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{2654208 d}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2/81*arctan(tan(1/2*d*x+1/2*c))+5/648*(-15555/1024*tan(1/2*d*x+1/2*c)^5-2145/32*tan(1/2*d*x+1/2*c)^3-1060
5/64*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2+4)^3-11215/1327104*arctan(1/2*tan(1/2*d*x+1/2*c)))

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Maxima [A]
time = 0.49, size = 171, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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 = 3.72, size = 159, normalized size = 1.50 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (5 \sec {\left (c + d x \right )} + 3\right )^{4}}\, dx \end {gather*}

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 = 0.44, size = 88, normalized size = 0.83 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 1.09, size = 105, normalized size = 0.99 \begin {gather*} \frac {x}{81}-\frac {11215\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{1327104\,d}-\frac {\frac {25925\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{221184}+\frac {3575\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6912}+\frac {17675\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{13824}}{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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5/cos(c + d*x) + 3)^4,x)

[Out]

x/81 - (11215*atan(tan(c/2 + (d*x)/2)/2))/(1327104*d) - ((17675*tan(c/2 + (d*x)/2))/13824 + (3575*tan(c/2 + (d
*x)/2)^3)/6912 + (25925*tan(c/2 + (d*x)/2)^5)/221184)/(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))

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