3.64 \(\int \frac{\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=105 \[ -\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) (6-5 \sec (c+d x))}{30 a d}+\frac{\tan ^3(c+d x) (8-5 \sec (c+d x))}{24 a d}-\frac{\tan (c+d x) (16-5 \sec (c+d x))}{16 a d}+\frac{x}{a} \]

[Out]

x/a - (5*ArcTanh[Sin[c + d*x]])/(16*a*d) - ((16 - 5*Sec[c + d*x])*Tan[c + d*x])/(16*a*d) + ((8 - 5*Sec[c + d*x
])*Tan[c + d*x]^3)/(24*a*d) - ((6 - 5*Sec[c + d*x])*Tan[c + d*x]^5)/(30*a*d)

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Rubi [A]  time = 0.144001, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3888, 3881, 3770} \[ -\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{\tan ^5(c+d x) (6-5 \sec (c+d x))}{30 a d}+\frac{\tan ^3(c+d x) (8-5 \sec (c+d x))}{24 a d}-\frac{\tan (c+d x) (16-5 \sec (c+d x))}{16 a d}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^8/(a + a*Sec[c + d*x]),x]

[Out]

x/a - (5*ArcTanh[Sin[c + d*x]])/(16*a*d) - ((16 - 5*Sec[c + d*x])*Tan[c + d*x])/(16*a*d) + ((8 - 5*Sec[c + d*x
])*Tan[c + d*x]^3)/(24*a*d) - ((6 - 5*Sec[c + d*x])*Tan[c + d*x]^5)/(30*a*d)

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3881

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(e*(e*Cot[
c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)
*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{\int (-a+a \sec (c+d x)) \tan ^6(c+d x) \, dx}{a^2}\\ &=-\frac{(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}-\frac{\int (-6 a+5 a \sec (c+d x)) \tan ^4(c+d x) \, dx}{6 a^2}\\ &=\frac{(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac{(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}+\frac{\int (-24 a+15 a \sec (c+d x)) \tan ^2(c+d x) \, dx}{24 a^2}\\ &=-\frac{(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac{(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac{(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}-\frac{\int (-48 a+15 a \sec (c+d x)) \, dx}{48 a^2}\\ &=\frac{x}{a}-\frac{(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac{(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac{(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}-\frac{5 \int \sec (c+d x) \, dx}{16 a}\\ &=\frac{x}{a}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac{(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac{(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}\\ \end{align*}

Mathematica [B]  time = 0.901728, size = 301, normalized size = 2.87 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (2400 \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec ^6(c+d x) (450 \sin (2 c+d x)-3360 \sin (c+2 d x)+2160 \sin (3 c+2 d x)-25 \sin (2 c+3 d x)-25 \sin (4 c+3 d x)-1488 \sin (3 c+4 d x)+720 \sin (5 c+4 d x)+165 \sin (4 c+5 d x)+165 \sin (6 c+5 d x)-368 \sin (5 c+6 d x)+2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+3680 \sin (c)+450 \sin (d x))\right )}{3840 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^8/(a + a*Sec[c + d*x]),x]

[Out]

(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(2400*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[
(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]^6*(2400*d*x*Cos[c] + 1800*d*x*Cos[c + 2*d*x] + 1800*d*x*Cos[3*c + 2*d*x]
+ 720*d*x*Cos[3*c + 4*d*x] + 720*d*x*Cos[5*c + 4*d*x] + 120*d*x*Cos[5*c + 6*d*x] + 120*d*x*Cos[7*c + 6*d*x] +
3680*Sin[c] + 450*Sin[d*x] + 450*Sin[2*c + d*x] - 3360*Sin[c + 2*d*x] + 2160*Sin[3*c + 2*d*x] - 25*Sin[2*c + 3
*d*x] - 25*Sin[4*c + 3*d*x] - 1488*Sin[3*c + 4*d*x] + 720*Sin[5*c + 4*d*x] + 165*Sin[4*c + 5*d*x] + 165*Sin[6*
c + 5*d*x] - 368*Sin[5*c + 6*d*x])))/(3840*a*d*(1 + Sec[c + d*x]))

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Maple [B]  time = 0.093, size = 312, normalized size = 3. \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{6\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-6}}+{\frac{7}{10\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-{\frac{3}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}-{\frac{5}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{9}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{21}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{5}{16\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{6\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-6}}+{\frac{7}{10\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-5}}+{\frac{3}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}-{\frac{5}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{9}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{21}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{5}{16\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^8/(a+a*sec(d*x+c)),x)

[Out]

2/d/a*arctan(tan(1/2*d*x+1/2*c))-1/6/a/d/(tan(1/2*d*x+1/2*c)+1)^6+7/10/a/d/(tan(1/2*d*x+1/2*c)+1)^5-3/4/a/d/(t
an(1/2*d*x+1/2*c)+1)^4-5/12/a/d/(tan(1/2*d*x+1/2*c)+1)^3+9/16/a/d/(tan(1/2*d*x+1/2*c)+1)^2+21/16/a/d/(tan(1/2*
d*x+1/2*c)+1)-5/16/a/d*ln(tan(1/2*d*x+1/2*c)+1)+1/6/a/d/(tan(1/2*d*x+1/2*c)-1)^6+7/10/a/d/(tan(1/2*d*x+1/2*c)-
1)^5+3/4/a/d/(tan(1/2*d*x+1/2*c)-1)^4-5/12/a/d/(tan(1/2*d*x+1/2*c)-1)^3-9/16/a/d/(tan(1/2*d*x+1/2*c)-1)^2+21/1
6/a/d/(tan(1/2*d*x+1/2*c)-1)+5/16/a/d*ln(tan(1/2*d*x+1/2*c)-1)

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Maxima [B]  time = 1.74257, size = 444, normalized size = 4.23 \begin{align*} -\frac{\frac{2 \,{\left (\frac{165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1095 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3138 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{5118 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{1945 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a - \frac{6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac{a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac{480 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{75 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac{75 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^8/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

-1/240*(2*(165*sin(d*x + c)/(cos(d*x + c) + 1) - 1095*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3138*sin(d*x + c)^
5/(cos(d*x + c) + 1)^5 - 5118*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 1945*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 -
315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a - 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a*sin(d*x + c)^4
/(cos(d*x + c) + 1)^4 - 20*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 -
6*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 480*arctan(sin(d*x + c)
/(cos(d*x + c) + 1))/a + 75*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a - 75*log(sin(d*x + c)/(cos(d*x + c) + 1
) - 1)/a)/d

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Fricas [A]  time = 1.22981, size = 354, normalized size = 3.37 \begin{align*} \frac{480 \, d x \cos \left (d x + c\right )^{6} - 75 \, \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) + 75 \, \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (368 \, \cos \left (d x + c\right )^{5} - 165 \, \cos \left (d x + c\right )^{4} - 176 \, \cos \left (d x + c\right )^{3} + 130 \, \cos \left (d x + c\right )^{2} + 48 \, \cos \left (d x + c\right ) - 40\right )} \sin \left (d x + c\right )}{480 \, a d \cos \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^8/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/480*(480*d*x*cos(d*x + c)^6 - 75*cos(d*x + c)^6*log(sin(d*x + c) + 1) + 75*cos(d*x + c)^6*log(-sin(d*x + c)
+ 1) - 2*(368*cos(d*x + c)^5 - 165*cos(d*x + c)^4 - 176*cos(d*x + c)^3 + 130*cos(d*x + c)^2 + 48*cos(d*x + c)
- 40)*sin(d*x + c))/(a*d*cos(d*x + c)^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{8}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**8/(a+a*sec(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**8/(sec(c + d*x) + 1), x)/a

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Giac [A]  time = 12.2761, size = 201, normalized size = 1.91 \begin{align*} \frac{\frac{240 \,{\left (d x + c\right )}}{a} - \frac{75 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} + \frac{75 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 1945 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 5118 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 3138 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1095 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 165 \, \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} - 1\right )}^{6} a}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^8/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

1/240*(240*(d*x + c)/a - 75*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a + 75*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a + 2
*(315*tan(1/2*d*x + 1/2*c)^11 - 1945*tan(1/2*d*x + 1/2*c)^9 + 5118*tan(1/2*d*x + 1/2*c)^7 - 3138*tan(1/2*d*x +
1/2*c)^5 + 1095*tan(1/2*d*x + 1/2*c)^3 - 165*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^6*a))/d