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

Optimal. Leaf size=88 \[ -\frac{5}{19652 d (5 \tan (c+d x)+3)}-\frac{15}{1156 d (5 \tan (c+d x)+3)^2}-\frac{5}{102 d (5 \tan (c+d x)+3)^3}-\frac{60 \log (5 \sin (c+d x)+3 \cos (c+d x))}{83521 d}-\frac{161 x}{334084} \]

[Out]

(-161*x)/334084 - (60*Log[3*Cos[c + d*x] + 5*Sin[c + d*x]])/(83521*d) - 5/(102*d*(3 + 5*Tan[c + d*x])^3) - 15/
(1156*d*(3 + 5*Tan[c + d*x])^2) - 5/(19652*d*(3 + 5*Tan[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.120047, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3483, 3529, 3531, 3530} \[ -\frac{5}{19652 d (5 \tan (c+d x)+3)}-\frac{15}{1156 d (5 \tan (c+d x)+3)^2}-\frac{5}{102 d (5 \tan (c+d x)+3)^3}-\frac{60 \log (5 \sin (c+d x)+3 \cos (c+d x))}{83521 d}-\frac{161 x}{334084} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Tan[c + d*x])^(-4),x]

[Out]

(-161*x)/334084 - (60*Log[3*Cos[c + d*x] + 5*Sin[c + d*x]])/(83521*d) - 5/(102*d*(3 + 5*Tan[c + d*x])^3) - 15/
(1156*d*(3 + 5*Tan[c + d*x])^2) - 5/(19652*d*(3 + 5*Tan[c + d*x]))

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(3+5 \tan (c+d x))^4} \, dx &=-\frac{5}{102 d (3+5 \tan (c+d x))^3}+\frac{1}{34} \int \frac{3-5 \tan (c+d x)}{(3+5 \tan (c+d x))^3} \, dx\\ &=-\frac{5}{102 d (3+5 \tan (c+d x))^3}-\frac{15}{1156 d (3+5 \tan (c+d x))^2}+\frac{\int \frac{-16-30 \tan (c+d x)}{(3+5 \tan (c+d x))^2} \, dx}{1156}\\ &=-\frac{5}{102 d (3+5 \tan (c+d x))^3}-\frac{15}{1156 d (3+5 \tan (c+d x))^2}-\frac{5}{19652 d (3+5 \tan (c+d x))}+\frac{\int \frac{-198-10 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx}{39304}\\ &=-\frac{161 x}{334084}-\frac{5}{102 d (3+5 \tan (c+d x))^3}-\frac{15}{1156 d (3+5 \tan (c+d x))^2}-\frac{5}{19652 d (3+5 \tan (c+d x))}-\frac{60 \int \frac{5-3 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx}{83521}\\ &=-\frac{161 x}{334084}-\frac{60 \log (3 \cos (c+d x)+5 \sin (c+d x))}{83521 d}-\frac{5}{102 d (3+5 \tan (c+d x))^3}-\frac{15}{1156 d (3+5 \tan (c+d x))^2}-\frac{5}{19652 d (3+5 \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 1.13588, size = 87, normalized size = 0.99 \[ \frac{-\frac{170 \left (75 \tan ^2(c+d x)+855 \tan (c+d x)+1064\right )}{(5 \tan (c+d x)+3)^3}+(720+483 i) \log (-\tan (c+d x)+i)+(720-483 i) \log (\tan (c+d x)+i)-1440 \log (5 \tan (c+d x)+3)}{2004504 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((720 + 483*I)*Log[I - Tan[c + d*x]] + (720 - 483*I)*Log[I + Tan[c + d*x]] - 1440*Log[3 + 5*Tan[c + d*x]] - (1
70*(1064 + 855*Tan[c + d*x] + 75*Tan[c + d*x]^2))/(3 + 5*Tan[c + d*x])^3)/(2004504*d)

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 97, normalized size = 1.1 \begin{align*}{\frac{30\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{83521\,d}}-{\frac{161\,\arctan \left ( \tan \left ( dx+c \right ) \right ) }{334084\,d}}-{\frac{5}{102\,d \left ( 3+5\,\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{15}{1156\,d \left ( 3+5\,\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5}{19652\,d \left ( 3+5\,\tan \left ( dx+c \right ) \right ) }}-{\frac{60\,\ln \left ( 3+5\,\tan \left ( dx+c \right ) \right ) }{83521\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30/83521/d*ln(1+tan(d*x+c)^2)-161/334084/d*arctan(tan(d*x+c))-5/102/d/(3+5*tan(d*x+c))^3-15/1156/d/(3+5*tan(d*
x+c))^2-5/19652/d/(3+5*tan(d*x+c))-60/83521/d*ln(3+5*tan(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.65634, size = 126, normalized size = 1.43 \begin{align*} -\frac{483 \, d x + 483 \, c + \frac{85 \,{\left (75 \, \tan \left (d x + c\right )^{2} + 855 \, \tan \left (d x + c\right ) + 1064\right )}}{125 \, \tan \left (d x + c\right )^{3} + 225 \, \tan \left (d x + c\right )^{2} + 135 \, \tan \left (d x + c\right ) + 27} - 360 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 720 \, \log \left (5 \, \tan \left (d x + c\right ) + 3\right )}{1002252 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1002252*(483*d*x + 483*c + 85*(75*tan(d*x + c)^2 + 855*tan(d*x + c) + 1064)/(125*tan(d*x + c)^3 + 225*tan(d
*x + c)^2 + 135*tan(d*x + c) + 27) - 360*log(tan(d*x + c)^2 + 1) + 720*log(5*tan(d*x + c) + 3))/d

________________________________________________________________________________________

Fricas [B]  time = 1.62391, size = 483, normalized size = 5.49 \begin{align*} -\frac{375 \,{\left (161 \, d x + 135\right )} \tan \left (d x + c\right )^{3} + 75 \,{\left (1449 \, d x + 1300\right )} \tan \left (d x + c\right )^{2} + 13041 \, d x + 360 \,{\left (125 \, \tan \left (d x + c\right )^{3} + 225 \, \tan \left (d x + c\right )^{2} + 135 \, \tan \left (d x + c\right ) + 27\right )} \log \left (\frac{25 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 9}{\tan \left (d x + c\right )^{2} + 1}\right ) + 45 \,{\left (1449 \, d x + 2830\right )} \tan \left (d x + c\right ) + 101375}{1002252 \,{\left (125 \, d \tan \left (d x + c\right )^{3} + 225 \, d \tan \left (d x + c\right )^{2} + 135 \, d \tan \left (d x + c\right ) + 27 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1002252*(375*(161*d*x + 135)*tan(d*x + c)^3 + 75*(1449*d*x + 1300)*tan(d*x + c)^2 + 13041*d*x + 360*(125*ta
n(d*x + c)^3 + 225*tan(d*x + c)^2 + 135*tan(d*x + c) + 27)*log((25*tan(d*x + c)^2 + 30*tan(d*x + c) + 9)/(tan(
d*x + c)^2 + 1)) + 45*(1449*d*x + 2830)*tan(d*x + c) + 101375)/(125*d*tan(d*x + c)^3 + 225*d*tan(d*x + c)^2 +
135*d*tan(d*x + c) + 27*d)

________________________________________________________________________________________

Sympy [A]  time = 1.56405, size = 790, normalized size = 8.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-181125*d*x*tan(c + d*x)**3/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*
d*tan(c + d*x) + 81182412*d) - 326025*d*x*tan(c + d*x)**2/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d
*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) - 195615*d*x*tan(c + d*x)/(375844500*d*tan(c + d*x)**3 + 67652
0100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) - 39123*d*x/(375844500*d*tan(c + d*x)**3 + 676
520100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) - 270000*log(tan(c + d*x) + 3/5)*tan(c + d*x
)**3/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) - 486
000*log(tan(c + d*x) + 3/5)*tan(c + d*x)**2/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 40591
2060*d*tan(c + d*x) + 81182412*d) - 291600*log(tan(c + d*x) + 3/5)*tan(c + d*x)/(375844500*d*tan(c + d*x)**3 +
676520100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) - 58320*log(tan(c + d*x) + 3/5)/(3758445
00*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) + 135000*log(tan(c
+ d*x)**2 + 1)*tan(c + d*x)**3/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*d*tan(c
+ d*x) + 81182412*d) + 243000*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(375844500*d*tan(c + d*x)**3 + 6765201
00*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) + 145800*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(
375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) + 81182412*d) + 29160*log
(tan(c + d*x)**2 + 1)/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*d*tan(c + d*x) +
81182412*d) + 10625*tan(c + d*x)**3/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 405912060*d*t
an(c + d*x) + 81182412*d) - 206550*tan(c + d*x)/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 4
05912060*d*tan(c + d*x) + 81182412*d) - 269025/(375844500*d*tan(c + d*x)**3 + 676520100*d*tan(c + d*x)**2 + 40
5912060*d*tan(c + d*x) + 81182412*d), Ne(d, 0)), (x/(5*tan(c) + 3)**4, True))

________________________________________________________________________________________

Giac [A]  time = 1.29466, size = 113, normalized size = 1.28 \begin{align*} -\frac{483 \, d x + 483 \, c - \frac{25 \,{\left (6600 \, \tan \left (d x + c\right )^{3} + 11625 \, \tan \left (d x + c\right )^{2} + 4221 \, \tan \left (d x + c\right ) - 2192\right )}}{{\left (5 \, \tan \left (d x + c\right ) + 3\right )}^{3}} - 360 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 720 \, \log \left ({\left | 5 \, \tan \left (d x + c\right ) + 3 \right |}\right )}{1002252 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1002252*(483*d*x + 483*c - 25*(6600*tan(d*x + c)^3 + 11625*tan(d*x + c)^2 + 4221*tan(d*x + c) - 2192)/(5*ta
n(d*x + c) + 3)^3 - 360*log(tan(d*x + c)^2 + 1) + 720*log(abs(5*tan(d*x + c) + 3)))/d