Integrand size = 21, antiderivative size = 135 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{a d}-\frac {\sec (c+d x)}{a d}-\frac {3 \sec ^2(c+d x)}{2 a d}+\frac {\sec ^3(c+d x)}{a d}+\frac {3 \sec ^4(c+d x)}{4 a d}-\frac {3 \sec ^5(c+d x)}{5 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^7(c+d x)}{7 a d} \] Output:
-ln(cos(d*x+c))/a/d-sec(d*x+c)/a/d-3/2*sec(d*x+c)^2/a/d+sec(d*x+c)^3/a/d+3 /4*sec(d*x+c)^4/a/d-3/5*sec(d*x+c)^5/a/d-1/6*sec(d*x+c)^6/a/d+1/7*sec(d*x+ c)^7/a/d
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {(35 \cos (c+d x) (104+105 \log (\cos (c+d x)))+3 (212+602 \cos (2 (c+d x))+140 \cos (4 (c+d x))+210 \cos (5 (c+d x))+70 \cos (6 (c+d x))+245 \cos (5 (c+d x)) \log (\cos (c+d x))+35 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (3 (c+d x)) (6+7 \log (\cos (c+d x))))) \sec ^7(c+d x)}{6720 a d} \] Input:
Integrate[Tan[c + d*x]^9/(a + a*Sec[c + d*x]),x]
Output:
-1/6720*((35*Cos[c + d*x]*(104 + 105*Log[Cos[c + d*x]]) + 3*(212 + 602*Cos [2*(c + d*x)] + 140*Cos[4*(c + d*x)] + 210*Cos[5*(c + d*x)] + 70*Cos[6*(c + d*x)] + 245*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 35*Cos[7*(c + d*x)]*Log [Cos[c + d*x]] + 105*Cos[3*(c + d*x)]*(6 + 7*Log[Cos[c + d*x]])))*Sec[c + d*x]^7)/(a*d)
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tan ^9(c+d x)}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cot \left (c+d x+\frac {\pi }{2}\right )^9}{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^9}{\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {\int a^7 (1-\cos (c+d x))^4 (\cos (c+d x)+1)^3 \sec ^8(c+d x)d\cos (c+d x)}{a^8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int (1-\cos (c+d x))^4 (\cos (c+d x)+1)^3 \sec ^8(c+d x)d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (\sec ^8(c+d x)-\sec ^7(c+d x)-3 \sec ^6(c+d x)+3 \sec ^5(c+d x)+3 \sec ^4(c+d x)-3 \sec ^3(c+d x)-\sec ^2(c+d x)+\sec (c+d x)\right )d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} \sec ^7(c+d x)+\frac {1}{6} \sec ^6(c+d x)+\frac {3}{5} \sec ^5(c+d x)-\frac {3}{4} \sec ^4(c+d x)-\sec ^3(c+d x)+\frac {3}{2} \sec ^2(c+d x)+\sec (c+d x)+\log (\cos (c+d x))}{a d}\) |
Input:
Int[Tan[c + d*x]^9/(a + a*Sec[c + d*x]),x]
Output:
-((Log[Cos[c + d*x]] + Sec[c + d*x] + (3*Sec[c + d*x]^2)/2 - Sec[c + d*x]^ 3 - (3*Sec[c + d*x]^4)/4 + (3*Sec[c + d*x]^5)/5 + Sec[c + d*x]^6/6 - Sec[c + d*x]^7/7)/(a*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[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] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.54 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{6 \cos \left (d x +c \right )^{6}}-\frac {3}{5 \cos \left (d x +c \right )^{5}}-\frac {1}{\cos \left (d x +c \right )}+\frac {3}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{\cos \left (d x +c \right )^{3}}-\frac {3}{2 \cos \left (d x +c \right )^{2}}+\frac {1}{7 \cos \left (d x +c \right )^{7}}-\ln \left (\cos \left (d x +c \right )\right )}{d a}\) | \(86\) |
| default | \(\frac {-\frac {1}{6 \cos \left (d x +c \right )^{6}}-\frac {3}{5 \cos \left (d x +c \right )^{5}}-\frac {1}{\cos \left (d x +c \right )}+\frac {3}{4 \cos \left (d x +c \right )^{4}}+\frac {1}{\cos \left (d x +c \right )^{3}}-\frac {3}{2 \cos \left (d x +c \right )^{2}}+\frac {1}{7 \cos \left (d x +c \right )^{7}}-\ln \left (\cos \left (d x +c \right )\right )}{d a}\) | \(86\) |
| risch | \(\frac {i x}{a}+\frac {2 i c}{d a}-\frac {2 \left (105 \,{\mathrm e}^{13 i \left (d x +c \right )}+315 \,{\mathrm e}^{12 i \left (d x +c \right )}+210 \,{\mathrm e}^{11 i \left (d x +c \right )}+945 \,{\mathrm e}^{10 i \left (d x +c \right )}+903 \,{\mathrm e}^{9 i \left (d x +c \right )}+1820 \,{\mathrm e}^{8 i \left (d x +c \right )}+636 \,{\mathrm e}^{7 i \left (d x +c \right )}+1820 \,{\mathrm e}^{6 i \left (d x +c \right )}+903 \,{\mathrm e}^{5 i \left (d x +c \right )}+945 \,{\mathrm e}^{4 i \left (d x +c \right )}+210 \,{\mathrm e}^{3 i \left (d x +c \right )}+315 \,{\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{105 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d a}\) | \(204\) |
Input:
int(tan(d*x+c)^9/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(-1/6/cos(d*x+c)^6-3/5/cos(d*x+c)^5-1/cos(d*x+c)+3/4/cos(d*x+c)^4+1/ cos(d*x+c)^3-3/2/cos(d*x+c)^2+1/7/cos(d*x+c)^7-ln(cos(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {420 \, \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) + 420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{420 \, a d \cos \left (d x + c\right )^{7}} \] Input:
integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="fricas")
Output:
-1/420*(420*cos(d*x + c)^7*log(-cos(d*x + c)) + 420*cos(d*x + c)^6 + 630*c os(d*x + c)^5 - 420*cos(d*x + c)^4 - 315*cos(d*x + c)^3 + 252*cos(d*x + c) ^2 + 70*cos(d*x + c) - 60)/(a*d*cos(d*x + c)^7)
\[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\tan ^{9}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(tan(d*x+c)**9/(a+a*sec(d*x+c)),x)
Output:
Integral(tan(c + d*x)**9/(sec(c + d*x) + 1), x)/a
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {420 \, \log \left (\cos \left (d x + c\right )\right )}{a} + \frac {420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{a \cos \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="maxima")
Output:
-1/420*(420*log(cos(d*x + c))/a + (420*cos(d*x + c)^6 + 630*cos(d*x + c)^5 - 420*cos(d*x + c)^4 - 315*cos(d*x + c)^3 + 252*cos(d*x + c)^2 + 70*cos(d *x + c) - 60)/(a*cos(d*x + c)^7))/d
Time = 0.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.65 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{\cos \left (d x + c\right )^{7}} + 420 \, \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{420 \, a d} \] Input:
integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="giac")
Output:
-1/420*((420*cos(d*x + c)^6 + 630*cos(d*x + c)^5 - 420*cos(d*x + c)^4 - 31 5*cos(d*x + c)^3 + 252*cos(d*x + c)^2 + 70*cos(d*x + c) - 60)/cos(d*x + c) ^7 + 420*log(abs(cos(d*x + c))))/(a*d)
Time = 15.98 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.54 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {32}{35}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-21\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \] Input:
int(tan(c + d*x)^9/(a + a/cos(c + d*x)),x)
Output:
(2*atanh(tan(c/2 + (d*x)/2)^2))/(a*d) - ((26*tan(c/2 + (d*x)/2)^4)/5 - (22 *tan(c/2 + (d*x)/2)^2)/5 + (32*tan(c/2 + (d*x)/2)^6)/3 - (128*tan(c/2 + (d *x)/2)^8)/3 + 14*tan(c/2 + (d*x)/2)^10 - 2*tan(c/2 + (d*x)/2)^12 + 32/35)/ (d*(a - 7*a*tan(c/2 + (d*x)/2)^2 + 21*a*tan(c/2 + (d*x)/2)^4 - 35*a*tan(c/ 2 + (d*x)/2)^6 + 35*a*tan(c/2 + (d*x)/2)^8 - 21*a*tan(c/2 + (d*x)/2)^10 + 7*a*tan(c/2 + (d*x)/2)^12 - a*tan(c/2 + (d*x)/2)^14))
Time = 0.17 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.38 \[ \int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6}-1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4}+1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}-420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )-420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6}+1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}-1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}+420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6}+1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}-1260 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+420 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+577 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-1101 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+786 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-192 \cos \left (d x +c \right )-420 \sin \left (d x +c \right )^{6}+840 \sin \left (d x +c \right )^{4}-672 \sin \left (d x +c \right )^{2}+192}{420 \cos \left (d x +c \right ) a d \left (\sin \left (d x +c \right )^{6}-3 \sin \left (d x +c \right )^{4}+3 \sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(tan(d*x+c)^9/(a+a*sec(d*x+c)),x)
Output:
(420*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6 - 1260*cos( c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4 + 1260*cos(c + d*x)* log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2 - 420*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1) - 420*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d* x)**6 + 1260*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 - 1260 *cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 + 420*cos(c + d*x) *log(tan((c + d*x)/2) - 1) - 420*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*si n(c + d*x)**6 + 1260*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)** 4 - 1260*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 420*cos( c + d*x)*log(tan((c + d*x)/2) + 1) + 577*cos(c + d*x)*sin(c + d*x)**6 - 11 01*cos(c + d*x)*sin(c + d*x)**4 + 786*cos(c + d*x)*sin(c + d*x)**2 - 192*c os(c + d*x) - 420*sin(c + d*x)**6 + 840*sin(c + d*x)**4 - 672*sin(c + d*x) **2 + 192)/(420*cos(c + d*x)*a*d*(sin(c + d*x)**6 - 3*sin(c + d*x)**4 + 3* sin(c + d*x)**2 - 1))