Integrand size = 19, antiderivative size = 118 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\frac {a \log (\cos (c+d x))}{d}-\frac {a \sec (c+d x)}{d}+\frac {3 a \sec ^2(c+d x)}{2 d}+\frac {a \sec ^3(c+d x)}{d}-\frac {3 a \sec ^4(c+d x)}{4 d}-\frac {3 a \sec ^5(c+d x)}{5 d}+\frac {a \sec ^6(c+d x)}{6 d}+\frac {a \sec ^7(c+d x)}{7 d} \] Output:
a*ln(cos(d*x+c))/d-a*sec(d*x+c)/d+3/2*a*sec(d*x+c)^2/d+a*sec(d*x+c)^3/d-3/ 4*a*sec(d*x+c)^4/d-3/5*a*sec(d*x+c)^5/d+1/6*a*sec(d*x+c)^6/d+1/7*a*sec(d*x +c)^7/d
Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=-\frac {a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{d}-\frac {3 a \sec ^5(c+d x)}{5 d}+\frac {a \sec ^7(c+d x)}{7 d}+\frac {a \left (12 \log (\cos (c+d x))+18 \sec ^2(c+d x)-9 \sec ^4(c+d x)+2 \sec ^6(c+d x)\right )}{12 d} \] Input:
Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^7,x]
Output:
-((a*Sec[c + d*x])/d) + (a*Sec[c + d*x]^3)/d - (3*a*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^7)/(7*d) + (a*(12*Log[Cos[c + d*x]] + 18*Sec[c + d*x]^2 - 9*Sec[c + d*x]^4 + 2*Sec[c + d*x]^6))/(12*d)
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, 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.
| \(\displaystyle \int \tan ^7(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\cot \left (c+d x+\frac {\pi }{2}\right )^7 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {\int a^7 (1-\cos (c+d x))^3 (\cos (c+d x)+1)^4 \sec ^8(c+d x)d\cos (c+d x)}{a^6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int (1-\cos (c+d x))^3 (\cos (c+d x)+1)^4 \sec ^8(c+d x)d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a \int \left (\sec ^8(c+d x)+\sec ^7(c+d x)-3 \sec ^6(c+d x)-3 \sec ^5(c+d x)+3 \sec ^4(c+d x)+3 \sec ^3(c+d x)-\sec ^2(c+d x)-\sec (c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (-\frac {1}{7} \sec ^7(c+d x)-\frac {1}{6} \sec ^6(c+d x)+\frac {3}{5} \sec ^5(c+d x)+\frac {3}{4} \sec ^4(c+d x)-\sec ^3(c+d x)-\frac {3}{2} \sec ^2(c+d x)+\sec (c+d x)-\log (\cos (c+d x))\right )}{d}\) |
Input:
Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^7,x]
Output:
-((a*(-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 - S ec[c + d*x]^7/7))/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 = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69
| method | result | size |
| derivativedivides | \(\frac {a \left (\frac {\sec \left (d x +c \right )^{7}}{7}+\frac {\sec \left (d x +c \right )^{6}}{6}-\frac {3 \sec \left (d x +c \right )^{5}}{5}-\frac {3 \sec \left (d x +c \right )^{4}}{4}+\sec \left (d x +c \right )^{3}+\frac {3 \sec \left (d x +c \right )^{2}}{2}-\sec \left (d x +c \right )-\ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(82\) |
| default | \(\frac {a \left (\frac {\sec \left (d x +c \right )^{7}}{7}+\frac {\sec \left (d x +c \right )^{6}}{6}-\frac {3 \sec \left (d x +c \right )^{5}}{5}-\frac {3 \sec \left (d x +c \right )^{4}}{4}+\sec \left (d x +c \right )^{3}+\frac {3 \sec \left (d x +c \right )^{2}}{2}-\sec \left (d x +c \right )-\ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(82\) |
| parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{6}}{6}-\frac {\tan \left (d x +c \right )^{4}}{4}+\frac {\tan \left (d x +c \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {a \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {3 \sec \left (d x +c \right )^{5}}{5}+\sec \left (d x +c \right )^{3}-\sec \left (d x +c \right )\right )}{d}\) | \(93\) |
| risch | \(-i a x -\frac {2 i a c}{d}-\frac {2 a \left (105 \,{\mathrm e}^{13 i \left (d x +c \right )}-315 \,{\mathrm e}^{12 i \left (d x +c \right )}+210 \,{\mathrm e}^{11 i \left (d x +c \right )}-945 \,{\mathrm e}^{10 i \left (d x +c \right )}+903 \,{\mathrm e}^{9 i \left (d x +c \right )}-1820 \,{\mathrm e}^{8 i \left (d x +c \right )}+636 \,{\mathrm e}^{7 i \left (d x +c \right )}-1820 \,{\mathrm e}^{6 i \left (d x +c \right )}+903 \,{\mathrm e}^{5 i \left (d x +c \right )}-945 \,{\mathrm e}^{4 i \left (d x +c \right )}+210 \,{\mathrm e}^{3 i \left (d x +c \right )}-315 \,{\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(195\) |
Input:
int((a+a*sec(d*x+c))*tan(d*x+c)^7,x,method=_RETURNVERBOSE)
Output:
a/d*(1/7*sec(d*x+c)^7+1/6*sec(d*x+c)^6-3/5*sec(d*x+c)^5-3/4*sec(d*x+c)^4+s ec(d*x+c)^3+3/2*sec(d*x+c)^2-sec(d*x+c)-ln(sec(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\frac {420 \, a \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 420 \, a \cos \left (d x + c\right )^{6} + 630 \, a \cos \left (d x + c\right )^{5} + 420 \, a \cos \left (d x + c\right )^{4} - 315 \, a \cos \left (d x + c\right )^{3} - 252 \, a \cos \left (d x + c\right )^{2} + 70 \, a \cos \left (d x + c\right ) + 60 \, a}{420 \, d \cos \left (d x + c\right )^{7}} \] Input:
integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="fricas")
Output:
1/420*(420*a*cos(d*x + c)^7*log(-cos(d*x + c)) - 420*a*cos(d*x + c)^6 + 63 0*a*cos(d*x + c)^5 + 420*a*cos(d*x + c)^4 - 315*a*cos(d*x + c)^3 - 252*a*c os(d*x + c)^2 + 70*a*cos(d*x + c) + 60*a)/(d*cos(d*x + c)^7)
Time = 1.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.25 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{7 d} + \frac {a \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {6 a \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} - \frac {a \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {8 a \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {16 a \sec {\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\left (c \right )} + a\right ) \tan ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate((a+a*sec(d*x+c))*tan(d*x+c)**7,x)
Output:
Piecewise((-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*tan(c + d*x)**6*sec(c + d *x)/(7*d) + a*tan(c + d*x)**6/(6*d) - 6*a*tan(c + d*x)**4*sec(c + d*x)/(35 *d) - a*tan(c + d*x)**4/(4*d) + 8*a*tan(c + d*x)**2*sec(c + d*x)/(35*d) + a*tan(c + d*x)**2/(2*d) - 16*a*sec(c + d*x)/(35*d), Ne(d, 0)), (x*(a*sec(c ) + a)*tan(c)**7, True))
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\frac {420 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac {420 \, a \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{5} - 420 \, a \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{3} + 252 \, a \cos \left (d x + c\right )^{2} - 70 \, a \cos \left (d x + c\right ) - 60 \, a}{\cos \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="maxima")
Output:
1/420*(420*a*log(cos(d*x + c)) - (420*a*cos(d*x + c)^6 - 630*a*cos(d*x + c )^5 - 420*a*cos(d*x + c)^4 + 315*a*cos(d*x + c)^3 + 252*a*cos(d*x + c)^2 - 70*a*cos(d*x + c) - 60*a)/cos(d*x + c)^7)/d
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\frac {420 \, a \log \left ({\left | \cos \left (d x + c\right ) \right |}\right ) - \frac {420 \, a \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{5} - 420 \, a \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{3} + 252 \, a \cos \left (d x + c\right )^{2} - 70 \, a \cos \left (d x + c\right ) - 60 \, a}{\cos \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="giac")
Output:
1/420*(420*a*log(abs(cos(d*x + c))) - (420*a*cos(d*x + c)^6 - 630*a*cos(d* x + c)^5 - 420*a*cos(d*x + c)^4 + 315*a*cos(d*x + c)^3 + 252*a*cos(d*x + c )^2 - 70*a*cos(d*x + c) - 60*a)/cos(d*x + c)^7)/d
Time = 16.81 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.73 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {224\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {166\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {42\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {32\,a}{35}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \] Input:
int(tan(c + d*x)^7*(a + a/cos(c + d*x)),x)
Output:
((32*a)/35 - (42*a*tan(c/2 + (d*x)/2)^2)/5 + (166*a*tan(c/2 + (d*x)/2)^4)/ 5 - (224*a*tan(c/2 + (d*x)/2)^6)/3 + (128*a*tan(c/2 + (d*x)/2)^8)/3 - 14*a *tan(c/2 + (d*x)/2)^10 + 2*a*tan(c/2 + (d*x)/2)^12)/(d*(7*tan(c/2 + (d*x)/ 2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d *x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1)) - (2*a*atanh(tan(c/2 + (d*x)/2)^2))/d
Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx=\frac {a \left (-210 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right )+60 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{6}-72 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{4}+96 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}-192 \sec \left (d x +c \right )+70 \tan \left (d x +c \right )^{6}-105 \tan \left (d x +c \right )^{4}+210 \tan \left (d x +c \right )^{2}\right )}{420 d} \] Input:
int((a+a*sec(d*x+c))*tan(d*x+c)^7,x)
Output:
(a*( - 210*log(tan(c + d*x)**2 + 1) + 60*sec(c + d*x)*tan(c + d*x)**6 - 72 *sec(c + d*x)*tan(c + d*x)**4 + 96*sec(c + d*x)*tan(c + d*x)**2 - 192*sec( c + d*x) + 70*tan(c + d*x)**6 - 105*tan(c + d*x)**4 + 210*tan(c + d*x)**2) )/(420*d)