3.38 \(\int (a+a \sec (c+d x))^3 \tan ^7(c+d x) \, dx\)

Optimal. Leaf size=137 \[ \frac {a^3 \sec ^9(c+d x)}{9 d}+\frac {3 a^3 \sec ^8(c+d x)}{8 d}-\frac {4 a^3 \sec ^6(c+d x)}{3 d}-\frac {6 a^3 \sec ^5(c+d x)}{5 d}+\frac {3 a^3 \sec ^4(c+d x)}{2 d}+\frac {8 a^3 \sec ^3(c+d x)}{3 d}-\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \log (\cos (c+d x))}{d} \]

[Out]

a^3*ln(cos(d*x+c))/d-3*a^3*sec(d*x+c)/d+8/3*a^3*sec(d*x+c)^3/d+3/2*a^3*sec(d*x+c)^4/d-6/5*a^3*sec(d*x+c)^5/d-4
/3*a^3*sec(d*x+c)^6/d+3/8*a^3*sec(d*x+c)^8/d+1/9*a^3*sec(d*x+c)^9/d

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Rubi [A]  time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {a^3 \sec ^9(c+d x)}{9 d}+\frac {3 a^3 \sec ^8(c+d x)}{8 d}-\frac {4 a^3 \sec ^6(c+d x)}{3 d}-\frac {6 a^3 \sec ^5(c+d x)}{5 d}+\frac {3 a^3 \sec ^4(c+d x)}{2 d}+\frac {8 a^3 \sec ^3(c+d x)}{3 d}-\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^7,x]

[Out]

(a^3*Log[Cos[c + d*x]])/d - (3*a^3*Sec[c + d*x])/d + (8*a^3*Sec[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]^4)/(2*
d) - (6*a^3*Sec[c + d*x]^5)/(5*d) - (4*a^3*Sec[c + d*x]^6)/(3*d) + (3*a^3*Sec[c + d*x]^8)/(8*d) + (a^3*Sec[c +
d*x]^9)/(9*d)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps

\begin {align*} \int (a+a \sec (c+d x))^3 \tan ^7(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^3 (a+a x)^6}{x^{10}} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^9}{x^{10}}+\frac {3 a^9}{x^9}-\frac {8 a^9}{x^7}-\frac {6 a^9}{x^6}+\frac {6 a^9}{x^5}+\frac {8 a^9}{x^4}-\frac {3 a^9}{x^2}-\frac {a^9}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac {a^3 \log (\cos (c+d x))}{d}-\frac {3 a^3 \sec (c+d x)}{d}+\frac {8 a^3 \sec ^3(c+d x)}{3 d}+\frac {3 a^3 \sec ^4(c+d x)}{2 d}-\frac {6 a^3 \sec ^5(c+d x)}{5 d}-\frac {4 a^3 \sec ^6(c+d x)}{3 d}+\frac {3 a^3 \sec ^8(c+d x)}{8 d}+\frac {a^3 \sec ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 110, normalized size = 0.80 \[ \frac {a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (40 \sec ^9(c+d x)+135 \sec ^8(c+d x)-480 \sec ^6(c+d x)-432 \sec ^5(c+d x)+540 \sec ^4(c+d x)+960 \sec ^3(c+d x)-1080 \sec (c+d x)+360 \log (\cos (c+d x))\right )}{2880 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^7,x]

[Out]

(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(360*Log[Cos[c + d*x]] - 1080*Sec[c + d*x] + 960*Sec[c + d*x]^3 +
540*Sec[c + d*x]^4 - 432*Sec[c + d*x]^5 - 480*Sec[c + d*x]^6 + 135*Sec[c + d*x]^8 + 40*Sec[c + d*x]^9))/(2880
*d)

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fricas [A]  time = 0.83, size = 117, normalized size = 0.85 \[ \frac {360 \, a^{3} \cos \left (d x + c\right )^{9} \log \left (-\cos \left (d x + c\right )\right ) - 1080 \, a^{3} \cos \left (d x + c\right )^{8} + 960 \, a^{3} \cos \left (d x + c\right )^{6} + 540 \, a^{3} \cos \left (d x + c\right )^{5} - 432 \, a^{3} \cos \left (d x + c\right )^{4} - 480 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right ) + 40 \, a^{3}}{360 \, d \cos \left (d x + c\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^7,x, algorithm="fricas")

[Out]

1/360*(360*a^3*cos(d*x + c)^9*log(-cos(d*x + c)) - 1080*a^3*cos(d*x + c)^8 + 960*a^3*cos(d*x + c)^6 + 540*a^3*
cos(d*x + c)^5 - 432*a^3*cos(d*x + c)^4 - 480*a^3*cos(d*x + c)^3 + 135*a^3*cos(d*x + c) + 40*a^3)/(d*cos(d*x +
c)^9)

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giac [B]  time = 28.27, size = 317, normalized size = 2.31 \[ -\frac {2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {14297 \, a^{3} + \frac {133713 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {560052 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1384068 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1594782 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1336734 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {781956 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {302004 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {69201 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {7129 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{9}}}{2520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^7,x, algorithm="giac")

[Out]

-1/2520*(2520*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 2520*a^3*log(abs(-(cos(d*x + c) - 1)/
(cos(d*x + c) + 1) - 1)) + (14297*a^3 + 133713*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 560052*a^3*(cos(d*x
+ c) - 1)^2/(cos(d*x + c) + 1)^2 + 1384068*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 1594782*a^3*(cos(d
*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 1336734*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 781956*a^3*(cos(
d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 302004*a^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 69201*a^3*(cos(d
*x + c) - 1)^8/(cos(d*x + c) + 1)^8 + 7129*a^3*(cos(d*x + c) - 1)^9/(cos(d*x + c) + 1)^9)/((cos(d*x + c) - 1)/
(cos(d*x + c) + 1) + 1)^9)/d

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maple [B]  time = 0.81, size = 288, normalized size = 2.10 \[ \frac {a^{3} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {4 a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{9 d \cos \left (d x +c \right )^{7}}-\frac {4 a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{45 d \cos \left (d x +c \right )^{5}}+\frac {4 a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{45 d \cos \left (d x +c \right )^{3}}-\frac {4 a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{9 d \cos \left (d x +c \right )}-\frac {64 a^{3} \cos \left (d x +c \right )}{45 d}-\frac {4 a^{3} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{9 d}-\frac {8 a^{3} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{15 d}-\frac {32 a^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{45 d}+\frac {3 a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{8}}+\frac {a^{3} \left (\sin ^{8}\left (d x +c \right )\right )}{9 d \cos \left (d x +c \right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*tan(d*x+c)^7,x)

[Out]

1/6/d*a^3*tan(d*x+c)^6-1/4*a^3*tan(d*x+c)^4/d+1/2*a^3*tan(d*x+c)^2/d+a^3*ln(cos(d*x+c))/d+4/9/d*a^3*sin(d*x+c)
^8/cos(d*x+c)^7-4/45/d*a^3*sin(d*x+c)^8/cos(d*x+c)^5+4/45/d*a^3*sin(d*x+c)^8/cos(d*x+c)^3-4/9/d*a^3*sin(d*x+c)
^8/cos(d*x+c)-64/45*a^3*cos(d*x+c)/d-4/9/d*a^3*cos(d*x+c)*sin(d*x+c)^6-8/15/d*a^3*cos(d*x+c)*sin(d*x+c)^4-32/4
5/d*a^3*cos(d*x+c)*sin(d*x+c)^2+3/8/d*a^3*sin(d*x+c)^8/cos(d*x+c)^8+1/9/d*a^3*sin(d*x+c)^8/cos(d*x+c)^9

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maxima [A]  time = 0.36, size = 110, normalized size = 0.80 \[ \frac {360 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1080 \, a^{3} \cos \left (d x + c\right )^{8} - 960 \, a^{3} \cos \left (d x + c\right )^{6} - 540 \, a^{3} \cos \left (d x + c\right )^{5} + 432 \, a^{3} \cos \left (d x + c\right )^{4} + 480 \, a^{3} \cos \left (d x + c\right )^{3} - 135 \, a^{3} \cos \left (d x + c\right ) - 40 \, a^{3}}{\cos \left (d x + c\right )^{9}}}{360 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^7,x, algorithm="maxima")

[Out]

1/360*(360*a^3*log(cos(d*x + c)) - (1080*a^3*cos(d*x + c)^8 - 960*a^3*cos(d*x + c)^6 - 540*a^3*cos(d*x + c)^5
+ 432*a^3*cos(d*x + c)^4 + 480*a^3*cos(d*x + c)^3 - 135*a^3*cos(d*x + c) - 40*a^3)/cos(d*x + c)^9)/d

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mupad [B]  time = 5.17, size = 278, normalized size = 2.03 \[ \frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {218\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-174\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1382\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}-\frac {1558\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {602\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {138\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {128\,a^3}{45}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^7*(a + a/cos(c + d*x))^3,x)

[Out]

((602*a^3*tan(c/2 + (d*x)/2)^4)/5 - (138*a^3*tan(c/2 + (d*x)/2)^2)/5 - (1558*a^3*tan(c/2 + (d*x)/2)^6)/5 + (13
82*a^3*tan(c/2 + (d*x)/2)^8)/5 - 174*a^3*tan(c/2 + (d*x)/2)^10 + (218*a^3*tan(c/2 + (d*x)/2)^12)/3 - 18*a^3*ta
n(c/2 + (d*x)/2)^14 + 2*a^3*tan(c/2 + (d*x)/2)^16 + (128*a^3)/45)/(d*(9*tan(c/2 + (d*x)/2)^2 - 36*tan(c/2 + (d
*x)/2)^4 + 84*tan(c/2 + (d*x)/2)^6 - 126*tan(c/2 + (d*x)/2)^8 + 126*tan(c/2 + (d*x)/2)^10 - 84*tan(c/2 + (d*x)
/2)^12 + 36*tan(c/2 + (d*x)/2)^14 - 9*tan(c/2 + (d*x)/2)^16 + tan(c/2 + (d*x)/2)^18 - 1)) - (2*a^3*atanh(tan(c
/2 + (d*x)/2)^2))/d

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sympy [A]  time = 22.57, size = 350, normalized size = 2.55 \[ \begin {cases} - \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{9 d} + \frac {3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac {3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{7 d} + \frac {a^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {2 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{21 d} - \frac {3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac {18 a^{3} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} - \frac {a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {8 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{105 d} + \frac {3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{8 d} + \frac {24 a^{3} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {16 a^{3} \sec ^{3}{\left (c + d x \right )}}{315 d} - \frac {3 a^{3} \sec ^{2}{\left (c + d x \right )}}{8 d} - \frac {48 a^{3} \sec {\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{3} \tan ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**7,x)

[Out]

Piecewise((-a**3*log(tan(c + d*x)**2 + 1)/(2*d) + a**3*tan(c + d*x)**6*sec(c + d*x)**3/(9*d) + 3*a**3*tan(c +
d*x)**6*sec(c + d*x)**2/(8*d) + 3*a**3*tan(c + d*x)**6*sec(c + d*x)/(7*d) + a**3*tan(c + d*x)**6/(6*d) - 2*a**
3*tan(c + d*x)**4*sec(c + d*x)**3/(21*d) - 3*a**3*tan(c + d*x)**4*sec(c + d*x)**2/(8*d) - 18*a**3*tan(c + d*x)
**4*sec(c + d*x)/(35*d) - a**3*tan(c + d*x)**4/(4*d) + 8*a**3*tan(c + d*x)**2*sec(c + d*x)**3/(105*d) + 3*a**3
*tan(c + d*x)**2*sec(c + d*x)**2/(8*d) + 24*a**3*tan(c + d*x)**2*sec(c + d*x)/(35*d) + a**3*tan(c + d*x)**2/(2
*d) - 16*a**3*sec(c + d*x)**3/(315*d) - 3*a**3*sec(c + d*x)**2/(8*d) - 48*a**3*sec(c + d*x)/(35*d), Ne(d, 0)),
(x*(a*sec(c) + a)**3*tan(c)**7, True))

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