3.10 \(\int (a+a \sec (c+d x)) \tan ^8(c+d x) \, dx\)

Optimal. Leaf size=129 \[ \frac{35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac{\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac{\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac{\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x \]

[Out]

a*x + (35*a*ArcTanh[Sin[c + d*x]])/(128*d) - ((128*a + 35*a*Sec[c + d*x])*Tan[c + d*x])/(128*d) + ((64*a + 35*
a*Sec[c + d*x])*Tan[c + d*x]^3)/(192*d) - ((48*a + 35*a*Sec[c + d*x])*Tan[c + d*x]^5)/(240*d) + ((8*a + 7*a*Se
c[c + d*x])*Tan[c + d*x]^7)/(56*d)

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Rubi [A]  time = 0.128695, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3881, 3770} \[ \frac{35 a \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{\tan ^7(c+d x) (7 a \sec (c+d x)+8 a)}{56 d}-\frac{\tan ^5(c+d x) (35 a \sec (c+d x)+48 a)}{240 d}+\frac{\tan ^3(c+d x) (35 a \sec (c+d x)+64 a)}{192 d}-\frac{\tan (c+d x) (35 a \sec (c+d x)+128 a)}{128 d}+a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*x + (35*a*ArcTanh[Sin[c + d*x]])/(128*d) - ((128*a + 35*a*Sec[c + d*x])*Tan[c + d*x])/(128*d) + ((64*a + 35*
a*Sec[c + d*x])*Tan[c + d*x]^3)/(192*d) - ((48*a + 35*a*Sec[c + d*x])*Tan[c + d*x]^5)/(240*d) + ((8*a + 7*a*Se
c[c + d*x])*Tan[c + d*x]^7)/(56*d)

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 (a+a \sec (c+d x)) \tan ^8(c+d x) \, dx &=\frac{(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}-\frac{1}{8} \int (8 a+7 a \sec (c+d x)) \tan ^6(c+d x) \, dx\\ &=-\frac{(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac{(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}+\frac{1}{48} \int (48 a+35 a \sec (c+d x)) \tan ^4(c+d x) \, dx\\ &=\frac{(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac{(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac{(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}-\frac{1}{192} \int (192 a+105 a \sec (c+d x)) \tan ^2(c+d x) \, dx\\ &=-\frac{(128 a+35 a \sec (c+d x)) \tan (c+d x)}{128 d}+\frac{(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac{(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac{(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}+\frac{1}{384} \int (384 a+105 a \sec (c+d x)) \, dx\\ &=a x-\frac{(128 a+35 a \sec (c+d x)) \tan (c+d x)}{128 d}+\frac{(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac{(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac{(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}+\frac{1}{128} (35 a) \int \sec (c+d x) \, dx\\ &=a x+\frac{35 a \tanh ^{-1}(\sin (c+d x))}{128 d}-\frac{(128 a+35 a \sec (c+d x)) \tan (c+d x)}{128 d}+\frac{(64 a+35 a \sec (c+d x)) \tan ^3(c+d x)}{192 d}-\frac{(48 a+35 a \sec (c+d x)) \tan ^5(c+d x)}{240 d}+\frac{(8 a+7 a \sec (c+d x)) \tan ^7(c+d x)}{56 d}\\ \end{align*}

Mathematica [A]  time = 1.74826, size = 115, normalized size = 0.89 \[ \frac{a \left (13440 \tan ^{-1}(\tan (c+d x))+3675 \tanh ^{-1}(\sin (c+d x))-\frac{1}{32} (223232 \cos (c+d x)+75915 \cos (2 (c+d x))+147968 \cos (3 (c+d x))+12950 \cos (4 (c+d x))+47616 \cos (5 (c+d x))+9765 \cos (6 (c+d x))+11264 \cos (7 (c+d x))+18970) \tan (c+d x) \sec ^7(c+d x)\right )}{13440 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(13440*ArcTan[Tan[c + d*x]] + 3675*ArcTanh[Sin[c + d*x]] - ((18970 + 223232*Cos[c + d*x] + 75915*Cos[2*(c +
d*x)] + 147968*Cos[3*(c + d*x)] + 12950*Cos[4*(c + d*x)] + 47616*Cos[5*(c + d*x)] + 9765*Cos[6*(c + d*x)] + 1
1264*Cos[7*(c + d*x)])*Sec[c + d*x]^7*Tan[c + d*x])/32))/(13440*d)

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Maple [A]  time = 0.046, size = 227, normalized size = 1.8 \begin{align*}{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{a\tan \left ( dx+c \right ) }{d}}+ax+{\frac{ac}{d}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{48\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{64\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,a \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{128\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}a}{128\,d}}-{\frac{7\,a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{35\,a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{35\,a\sin \left ( dx+c \right ) }{128\,d}}+{\frac{35\,a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{128\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7/d*a*tan(d*x+c)^7-1/5/d*a*tan(d*x+c)^5+1/3/d*a*tan(d*x+c)^3-1/d*a*tan(d*x+c)+a*x+1/d*a*c+1/8/d*a*sin(d*x+c)
^9/cos(d*x+c)^8-1/48/d*a*sin(d*x+c)^9/cos(d*x+c)^6+1/64/d*a*sin(d*x+c)^9/cos(d*x+c)^4-5/128/d*a*sin(d*x+c)^9/c
os(d*x+c)^2-5/128/d*sin(d*x+c)^7*a-7/128/d*a*sin(d*x+c)^5-35/384/d*a*sin(d*x+c)^3-35/128/d*a*sin(d*x+c)+35/128
/d*a*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 1.7106, size = 221, normalized size = 1.71 \begin{align*} \frac{256 \,{\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a + 35 \, a{\left (\frac{2 \,{\left (279 \, \sin \left (d x + c\right )^{7} - 511 \, \sin \left (d x + c\right )^{5} + 385 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} + 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{26880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/26880*(256*(15*tan(d*x + c)^7 - 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 105*d*x + 105*c - 105*tan(d*x + c))*
a + 35*a*(2*(279*sin(d*x + c)^7 - 511*sin(d*x + c)^5 + 385*sin(d*x + c)^3 - 105*sin(d*x + c))/(sin(d*x + c)^8
- 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) + 105*log(sin(d*x + c) + 1) - 105*log(sin(d*x +
c) - 1)))/d

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Fricas [A]  time = 1.03179, size = 466, normalized size = 3.61 \begin{align*} \frac{26880 \, a d x \cos \left (d x + c\right )^{8} + 3675 \, a \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3675 \, a \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (22528 \, a \cos \left (d x + c\right )^{7} + 9765 \, a \cos \left (d x + c\right )^{6} - 15616 \, a \cos \left (d x + c\right )^{5} - 11410 \, a \cos \left (d x + c\right )^{4} + 8448 \, a \cos \left (d x + c\right )^{3} + 7000 \, a \cos \left (d x + c\right )^{2} - 1920 \, a \cos \left (d x + c\right ) - 1680 \, a\right )} \sin \left (d x + c\right )}{26880 \, d \cos \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/26880*(26880*a*d*x*cos(d*x + c)^8 + 3675*a*cos(d*x + c)^8*log(sin(d*x + c) + 1) - 3675*a*cos(d*x + c)^8*log(
-sin(d*x + c) + 1) - 2*(22528*a*cos(d*x + c)^7 + 9765*a*cos(d*x + c)^6 - 15616*a*cos(d*x + c)^5 - 11410*a*cos(
d*x + c)^4 + 8448*a*cos(d*x + c)^3 + 7000*a*cos(d*x + c)^2 - 1920*a*cos(d*x + c) - 1680*a)*sin(d*x + c))/(d*co
s(d*x + c)^8)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 12.0087, size = 235, normalized size = 1.82 \begin{align*} \frac{13440 \,{\left (d x + c\right )} a + 3675 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3675 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (9765 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 83825 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 321013 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 724649 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1078359 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 508683 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 140175 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 17115 \, a \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 )}^{8}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(13440*(d*x + c)*a + 3675*a*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3675*a*log(abs(tan(1/2*d*x + 1/2*c) -
1)) + 2*(9765*a*tan(1/2*d*x + 1/2*c)^15 - 83825*a*tan(1/2*d*x + 1/2*c)^13 + 321013*a*tan(1/2*d*x + 1/2*c)^11
- 724649*a*tan(1/2*d*x + 1/2*c)^9 + 1078359*a*tan(1/2*d*x + 1/2*c)^7 - 508683*a*tan(1/2*d*x + 1/2*c)^5 + 14017
5*a*tan(1/2*d*x + 1/2*c)^3 - 17115*a*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^8)/d