3.37 \(\int (a+a \sec (c+d x))^3 \tan ^9(c+d x) \, dx\)

Optimal. Leaf size=210 \[ \frac {a^3 \sec ^{11}(c+d x)}{11 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]

[Out]

-a^3*ln(cos(d*x+c))/d+3*a^3*sec(d*x+c)/d-1/2*a^3*sec(d*x+c)^2/d-11/3*a^3*sec(d*x+c)^3/d-3/2*a^3*sec(d*x+c)^4/d
+14/5*a^3*sec(d*x+c)^5/d+7/3*a^3*sec(d*x+c)^6/d-6/7*a^3*sec(d*x+c)^7/d-11/8*a^3*sec(d*x+c)^8/d-1/9*a^3*sec(d*x
+c)^9/d+3/10*a^3*sec(d*x+c)^10/d+1/11*a^3*sec(d*x+c)^11/d

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Rubi [A]  time = 0.10, antiderivative size = 210, 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} \[ \frac {a^3 \sec ^{11}(c+d x)}{11 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]

[Out]

-((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/(2*d) - (11*a^3*Sec[c + d*x]^3)/(
3*d) - (3*a^3*Sec[c + d*x]^4)/(2*d) + (14*a^3*Sec[c + d*x]^5)/(5*d) + (7*a^3*Sec[c + d*x]^6)/(3*d) - (6*a^3*Se
c[c + d*x]^7)/(7*d) - (11*a^3*Sec[c + d*x]^8)/(8*d) - (a^3*Sec[c + d*x]^9)/(9*d) + (3*a^3*Sec[c + d*x]^10)/(10
*d) + (a^3*Sec[c + d*x]^11)/(11*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 ^9(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^4 (a+a x)^7}{x^{12}} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^{11}}{x^{12}}+\frac {3 a^{11}}{x^{11}}-\frac {a^{11}}{x^{10}}-\frac {11 a^{11}}{x^9}-\frac {6 a^{11}}{x^8}+\frac {14 a^{11}}{x^7}+\frac {14 a^{11}}{x^6}-\frac {6 a^{11}}{x^5}-\frac {11 a^{11}}{x^4}-\frac {a^{11}}{x^3}+\frac {3 a^{11}}{x^2}+\frac {a^{11}}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \sec ^2(c+d x)}{2 d}-\frac {11 a^3 \sec ^3(c+d x)}{3 d}-\frac {3 a^3 \sec ^4(c+d x)}{2 d}+\frac {14 a^3 \sec ^5(c+d x)}{5 d}+\frac {7 a^3 \sec ^6(c+d x)}{3 d}-\frac {6 a^3 \sec ^7(c+d x)}{7 d}-\frac {11 a^3 \sec ^8(c+d x)}{8 d}-\frac {a^3 \sec ^9(c+d x)}{9 d}+\frac {3 a^3 \sec ^{10}(c+d x)}{10 d}+\frac {a^3 \sec ^{11}(c+d x)}{11 d}\\ \end {align*}

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Mathematica [A]  time = 0.88, size = 214, normalized size = 1.02 \[ -\frac {a^3 \sec ^{11}(c+d x) (-1613260 \cos (2 (c+d x))+960960 \cos (3 (c+d x))-1131504 \cos (4 (c+d x))+314160 \cos (5 (c+d x))-432894 \cos (6 (c+d x))+145530 \cos (7 (c+d x))-106260 \cos (8 (c+d x))+6930 \cos (9 (c+d x))-20790 \cos (10 (c+d x))+1143450 \cos (3 (c+d x)) \log (\cos (c+d x))+571725 \cos (5 (c+d x)) \log (\cos (c+d x))+190575 \cos (7 (c+d x)) \log (\cos (c+d x))+38115 \cos (9 (c+d x)) \log (\cos (c+d x))+3465 \cos (11 (c+d x)) \log (\cos (c+d x))+462 \cos (c+d x) (3465 \log (\cos (c+d x))[phone])}{3548160 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^9,x]

[Out]

-1/3548160*(a^3*(-1151740 - 1613260*Cos[2*(c + d*x)] + 960960*Cos[3*(c + d*x)] - 1131504*Cos[4*(c + d*x)] + 31
4160*Cos[5*(c + d*x)] - 432894*Cos[6*(c + d*x)] + 145530*Cos[7*(c + d*x)] - 106260*Cos[8*(c + d*x)] + 6930*Cos
[9*(c + d*x)] - 20790*Cos[10*(c + d*x)] + 1143450*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + 571725*Cos[5*(c + d*x)]
*Log[Cos[c + d*x]] + 190575*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 38115*Cos[9*(c + d*x)]*Log[Cos[c + d*x]] + 34
65*Cos[11*(c + d*x)]*Log[Cos[c + d*x]] + 462*Cos[c + d*x]*(2606 + 3465*Log[Cos[c + d*x]]))*Sec[c + d*x]^11)/d

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fricas [A]  time = 0.87, size = 169, normalized size = 0.80 \[ -\frac {27720 \, a^{3} \cos \left (d x + c\right )^{11} \log \left (-\cos \left (d x + c\right )\right ) - 83160 \, a^{3} \cos \left (d x + c\right )^{10} + 13860 \, a^{3} \cos \left (d x + c\right )^{9} + 101640 \, a^{3} \cos \left (d x + c\right )^{8} + 41580 \, a^{3} \cos \left (d x + c\right )^{7} - 77616 \, a^{3} \cos \left (d x + c\right )^{6} - 64680 \, a^{3} \cos \left (d x + c\right )^{5} + 23760 \, a^{3} \cos \left (d x + c\right )^{4} + 38115 \, a^{3} \cos \left (d x + c\right )^{3} + 3080 \, a^{3} \cos \left (d x + c\right )^{2} - 8316 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3}}{27720 \, d \cos \left (d x + c\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="fricas")

[Out]

-1/27720*(27720*a^3*cos(d*x + c)^11*log(-cos(d*x + c)) - 83160*a^3*cos(d*x + c)^10 + 13860*a^3*cos(d*x + c)^9
+ 101640*a^3*cos(d*x + c)^8 + 41580*a^3*cos(d*x + c)^7 - 77616*a^3*cos(d*x + c)^6 - 64680*a^3*cos(d*x + c)^5 +
23760*a^3*cos(d*x + c)^4 + 38115*a^3*cos(d*x + c)^3 + 3080*a^3*cos(d*x + c)^2 - 8316*a^3*cos(d*x + c) - 2520*
a^3)/(d*cos(d*x + c)^11)

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giac [A]  time = 19.68, size = 367, normalized size = 1.75 \[ \frac {27720 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 27720 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {153343 \, a^{3} + \frac {1742213 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9043705 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28369275 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {59954070 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {67458930 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {57997170 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {36975510 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {16879995 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {5213945 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {976261 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {83711 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{11}}}{27720 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="giac")

[Out]

1/27720*(27720*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 27720*a^3*log(abs(-(cos(d*x + c) - 1
)/(cos(d*x + c) + 1) - 1)) + (153343*a^3 + 1742213*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9043705*a^3*(co
s(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 28369275*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 59954070*a^3
*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 67458930*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 57997170
*a^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 36975510*a^3*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 1687
9995*a^3*(cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 + 5213945*a^3*(cos(d*x + c) - 1)^9/(cos(d*x + c) + 1)^9 + 9
76261*a^3*(cos(d*x + c) - 1)^10/(cos(d*x + c) + 1)^10 + 83711*a^3*(cos(d*x + c) - 1)^11/(cos(d*x + c) + 1)^11)
/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^11)/d

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maple [A]  time = 0.87, size = 351, normalized size = 1.67 \[ \frac {4352 a^{3} \cos \left (d x +c \right )}{3465 d}+\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{99 d \cos \left (d x +c \right )^{9}}-\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{693 d \cos \left (d x +c \right )^{7}}+\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{1155 d \cos \left (d x +c \right )^{5}}-\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{693 d \cos \left (d x +c \right )^{3}}+\frac {34 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{99 d \cos \left (d x +c \right )}+\frac {a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{11 d \cos \left (d x +c \right )^{11}}+\frac {3 a^{3} \left (\sin ^{10}\left (d x +c \right )\right )}{10 d \cos \left (d x +c \right )^{10}}+\frac {34 a^{3} \cos \left (d x +c \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{99 d}+\frac {272 a^{3} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{693 d}+\frac {544 a^{3} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{1155 d}+\frac {2176 a^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3465 d}+\frac {a^{3} \left (\tan ^{8}\left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x)

[Out]

4352/3465*a^3*cos(d*x+c)/d+34/99/d*a^3*sin(d*x+c)^10/cos(d*x+c)^9-34/693/d*a^3*sin(d*x+c)^10/cos(d*x+c)^7+34/1
155/d*a^3*sin(d*x+c)^10/cos(d*x+c)^5-34/693/d*a^3*sin(d*x+c)^10/cos(d*x+c)^3+34/99/d*a^3*sin(d*x+c)^10/cos(d*x
+c)+1/11/d*a^3*sin(d*x+c)^10/cos(d*x+c)^11+3/10/d*a^3*sin(d*x+c)^10/cos(d*x+c)^10+34/99/d*a^3*cos(d*x+c)*sin(d
*x+c)^8+272/693/d*a^3*cos(d*x+c)*sin(d*x+c)^6+544/1155/d*a^3*cos(d*x+c)*sin(d*x+c)^4+2176/3465/d*a^3*cos(d*x+c
)*sin(d*x+c)^2+1/8/d*a^3*tan(d*x+c)^8-1/6/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

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maxima [A]  time = 0.65, size = 162, normalized size = 0.77 \[ -\frac {27720 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {83160 \, a^{3} \cos \left (d x + c\right )^{10} - 13860 \, a^{3} \cos \left (d x + c\right )^{9} - 101640 \, a^{3} \cos \left (d x + c\right )^{8} - 41580 \, a^{3} \cos \left (d x + c\right )^{7} + 77616 \, a^{3} \cos \left (d x + c\right )^{6} + 64680 \, a^{3} \cos \left (d x + c\right )^{5} - 23760 \, a^{3} \cos \left (d x + c\right )^{4} - 38115 \, a^{3} \cos \left (d x + c\right )^{3} - 3080 \, a^{3} \cos \left (d x + c\right )^{2} + 8316 \, a^{3} \cos \left (d x + c\right ) + 2520 \, a^{3}}{\cos \left (d x + c\right )^{11}}}{27720 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm="maxima")

[Out]

-1/27720*(27720*a^3*log(cos(d*x + c)) - (83160*a^3*cos(d*x + c)^10 - 13860*a^3*cos(d*x + c)^9 - 101640*a^3*cos
(d*x + c)^8 - 41580*a^3*cos(d*x + c)^7 + 77616*a^3*cos(d*x + c)^6 + 64680*a^3*cos(d*x + c)^5 - 23760*a^3*cos(d
*x + c)^4 - 38115*a^3*cos(d*x + c)^3 - 3080*a^3*cos(d*x + c)^2 + 8316*a^3*cos(d*x + c) + 2520*a^3)/cos(d*x + c
)^11)/d

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mupad [B]  time = 5.26, size = 337, normalized size = 1.60 \[ \frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-22\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+\frac {332\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{3}-\frac {1012\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {10456\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{15}-\frac {5192\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}+\frac {8164\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {3676\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {10090\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{63}-\frac {9334\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{315}+\frac {8704\,a^3}{3465}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+330\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+462\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-330\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+165\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-55\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^9*(a + a/cos(c + d*x))^3,x)

[Out]

(2*a^3*atanh(tan(c/2 + (d*x)/2)^2))/d - ((10090*a^3*tan(c/2 + (d*x)/2)^4)/63 - (9334*a^3*tan(c/2 + (d*x)/2)^2)
/315 - (3676*a^3*tan(c/2 + (d*x)/2)^6)/7 + (8164*a^3*tan(c/2 + (d*x)/2)^8)/7 - (5192*a^3*tan(c/2 + (d*x)/2)^10
)/5 + (10456*a^3*tan(c/2 + (d*x)/2)^12)/15 - (1012*a^3*tan(c/2 + (d*x)/2)^14)/3 + (332*a^3*tan(c/2 + (d*x)/2)^
16)/3 - 22*a^3*tan(c/2 + (d*x)/2)^18 + 2*a^3*tan(c/2 + (d*x)/2)^20 + (8704*a^3)/3465)/(d*(11*tan(c/2 + (d*x)/2
)^2 - 55*tan(c/2 + (d*x)/2)^4 + 165*tan(c/2 + (d*x)/2)^6 - 330*tan(c/2 + (d*x)/2)^8 + 462*tan(c/2 + (d*x)/2)^1
0 - 462*tan(c/2 + (d*x)/2)^12 + 330*tan(c/2 + (d*x)/2)^14 - 165*tan(c/2 + (d*x)/2)^16 + 55*tan(c/2 + (d*x)/2)^
18 - 11*tan(c/2 + (d*x)/2)^20 + tan(c/2 + (d*x)/2)^22 - 1))

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sympy [A]  time = 51.80, size = 439, normalized size = 2.09 \[ \begin {cases} \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{11 d} + \frac {3 a^{3} \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{3 d} + \frac {a^{3} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac {8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{99 d} - \frac {3 a^{3} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {8 a^{3} \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{21 d} - \frac {a^{3} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac {16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{231 d} + \frac {3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {16 a^{3} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{1155 d} - \frac {3 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {64 a^{3} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{105 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {128 a^{3} \sec ^{3}{\left (c + d x \right )}}{3465 d} + \frac {3 a^{3} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {128 a^{3} \sec {\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{3} \tan ^{9}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**9,x)

[Out]

Piecewise((a**3*log(tan(c + d*x)**2 + 1)/(2*d) + a**3*tan(c + d*x)**8*sec(c + d*x)**3/(11*d) + 3*a**3*tan(c +
d*x)**8*sec(c + d*x)**2/(10*d) + a**3*tan(c + d*x)**8*sec(c + d*x)/(3*d) + a**3*tan(c + d*x)**8/(8*d) - 8*a**3
*tan(c + d*x)**6*sec(c + d*x)**3/(99*d) - 3*a**3*tan(c + d*x)**6*sec(c + d*x)**2/(10*d) - 8*a**3*tan(c + d*x)*
*6*sec(c + d*x)/(21*d) - a**3*tan(c + d*x)**6/(6*d) + 16*a**3*tan(c + d*x)**4*sec(c + d*x)**3/(231*d) + 3*a**3
*tan(c + d*x)**4*sec(c + d*x)**2/(10*d) + 16*a**3*tan(c + d*x)**4*sec(c + d*x)/(35*d) + a**3*tan(c + d*x)**4/(
4*d) - 64*a**3*tan(c + d*x)**2*sec(c + d*x)**3/(1155*d) - 3*a**3*tan(c + d*x)**2*sec(c + d*x)**2/(10*d) - 64*a
**3*tan(c + d*x)**2*sec(c + d*x)/(105*d) - a**3*tan(c + d*x)**2/(2*d) + 128*a**3*sec(c + d*x)**3/(3465*d) + 3*
a**3*sec(c + d*x)**2/(10*d) + 128*a**3*sec(c + d*x)/(105*d), Ne(d, 0)), (x*(a*sec(c) + a)**3*tan(c)**9, True))

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