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+35/128*a*arctanh(sin(d*x+c))/d-1/128*(128*a+35*a*sec(d*x+c))*tan(d*x+c)/d+1/192*(64*a+35*a*sec(d*x+c))*tan
(d*x+c)^3/d-1/240*(48*a+35*a*sec(d*x+c))*tan(d*x+c)^5/d+1/56*(8*a+7*a*sec(d*x+c))*tan(d*x+c)^7/d

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Rubi [A]  time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, 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 3770

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

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]

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*}

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Mathematica [A]  time = 1.89, 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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fricas [A]  time = 0.76, size = 156, normalized size = 1.21 \[ \frac {26880 \, a d x \cos \left (d x + c\right )^{8} + 3675 \, a \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3675 \, a \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (22528 \, a \cos \left (d x + c\right )^{7} + 9765 \, a \cos \left (d x + c\right )^{6} - 15616 \, a \cos \left (d x + c\right )^{5} - 11410 \, a \cos \left (d x + c\right )^{4} + 8448 \, a \cos \left (d x + c\right )^{3} + 7000 \, a \cos \left (d x + c\right )^{2} - 1920 \, a \cos \left (d x + c\right ) - 1680 \, a\right )} \sin \left (d x + c\right )}{26880 \, d \cos \left (d x + c\right )^{8}} \]

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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giac [A]  time = 12.22, size = 174, normalized size = 1.35 \[ \frac {13440 \, {\left (d x + c\right )} a + 3675 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3675 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 83825 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 321013 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 724649 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1078359 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 508683 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 140175 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 17115 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{8}}}{13440 \, d} \]

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

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maple [A]  time = 0.54, size = 227, normalized size = 1.76 \[ \frac {a \left (\tan ^{7}\left (d x +c \right )\right )}{7 d}-\frac {a \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \tan \left (d x +c \right )}{d}+a x +\frac {c a}{d}+\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{8}}-\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{48 d \cos \left (d x +c \right )^{6}}+\frac {a \left (\sin ^{9}\left (d x +c \right )\right )}{64 d \cos \left (d x +c \right )^{4}}-\frac {5 a \left (\sin ^{9}\left (d x +c \right )\right )}{128 d \cos \left (d x +c \right )^{2}}-\frac {5 a \left (\sin ^{7}\left (d x +c \right )\right )}{128 d}-\frac {7 a \left (\sin ^{5}\left (d x +c \right )\right )}{128 d}-\frac {35 a \left (\sin ^{3}\left (d x +c \right )\right )}{384 d}-\frac {35 a \sin \left (d x +c \right )}{128 d}+\frac {35 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128 d} \]

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*a*tan(d*x+c)^7/d-1/5*a*tan(d*x+c)^5/d+1/3*a*tan(d*x+c)^3/d-a*tan(d*x+c)/d+a*x+1/d*c*a+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/cos
(d*x+c)^2-5/128*a*sin(d*x+c)^7/d-7/128*a*sin(d*x+c)^5/d-35/384*a*sin(d*x+c)^3/d-35/128*a*sin(d*x+c)/d+35/128/d
*a*ln(sec(d*x+c)+tan(d*x+c))

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maxima [A]  time = 0.53, size = 164, normalized size = 1.27 \[ \frac {256 \, {\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a + 35 \, a {\left (\frac {2 \, {\left (279 \, \sin \left (d x + c\right )^{7} - 511 \, \sin \left (d x + c\right )^{5} + 385 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} + 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{26880 \, d} \]

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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mupad [B]  time = 2.47, size = 242, normalized size = 1.88 \[ a\,x-\frac {-\frac {93\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {2395\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}-\frac {45859\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{960}+\frac {724649\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6720}-\frac {359453\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2240}+\frac {24223\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}-\frac {1335\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\frac {163\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {35\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - ((163*a*tan(c/2 + (d*x)/2))/64 - (1335*a*tan(c/2 + (d*x)/2)^3)/64 + (24223*a*tan(c/2 + (d*x)/2)^5)/320 -
(359453*a*tan(c/2 + (d*x)/2)^7)/2240 + (724649*a*tan(c/2 + (d*x)/2)^9)/6720 - (45859*a*tan(c/2 + (d*x)/2)^11)
/960 + (2395*a*tan(c/2 + (d*x)/2)^13)/192 - (93*a*tan(c/2 + (d*x)/2)^15)/64)/(d*(28*tan(c/2 + (d*x)/2)^4 - 8*t
an(c/2 + (d*x)/2)^2 - 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 - 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/
2 + (d*x)/2)^12 - 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) + (35*a*atanh(tan(c/2 + (d*x)/2)))/(64
*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \tan ^{8}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \tan ^{8}{\left (c + d x \right )}\, dx\right ) \]

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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