Integrand size = 21, antiderivative size = 169 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {x}{a^3}+\frac {19 \text {arctanh}(\sin (c+d x))}{16 a^3 d}+\frac {\tan (c+d x)}{a^3 d}-\frac {17 \sec (c+d x) \tan (c+d x)}{16 a^3 d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{8 a^3 d}-\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {3 \sec (c+d x) \tan ^3(c+d x)}{4 a^3 d}+\frac {\sec ^3(c+d x) \tan ^3(c+d x)}{6 a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d} \] Output:
-x/a^3+19/16*arctanh(sin(d*x+c))/a^3/d+tan(d*x+c)/a^3/d-17/16*sec(d*x+c)*t an(d*x+c)/a^3/d-1/8*sec(d*x+c)^3*tan(d*x+c)/a^3/d-1/3*tan(d*x+c)^3/a^3/d+3 /4*sec(d*x+c)*tan(d*x+c)^3/a^3/d+1/6*sec(d*x+c)^3*tan(d*x+c)^3/a^3/d-3/5*t an(d*x+c)^5/a^3/d
Time = 3.03 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.79 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (9120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec ^6(c+d x) (2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+1760 \sin (c)-210 \sin (d x)-210 \sin (2 c+d x)-1440 \sin (c+2 d x)+1200 \sin (3 c+2 d x)+865 \sin (2 c+3 d x)+865 \sin (4 c+3 d x)-1296 \sin (3 c+4 d x)-240 \sin (5 c+4 d x)+435 \sin (4 c+5 d x)+435 \sin (6 c+5 d x)-176 \sin (5 c+6 d x))\right )}{960 a^3 d (1+\sec (c+d x))^3} \] Input:
Integrate[Tan[c + d*x]^10/(a + a*Sec[c + d*x])^3,x]
Output:
-1/960*(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(9120*(Log[Cos[(c + d*x)/2] - Si n[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]^6*(2400*d*x*Cos[c] + 1800*d*x*Cos[c + 2*d*x] + 1800*d*x*Cos[3*c + 2*d*x] + 720*d*x*Cos[3*c + 4*d*x] + 720*d*x*Cos[5*c + 4*d*x] + 120*d*x*Cos [5*c + 6*d*x] + 120*d*x*Cos[7*c + 6*d*x] + 1760*Sin[c] - 210*Sin[d*x] - 21 0*Sin[2*c + d*x] - 1440*Sin[c + 2*d*x] + 1200*Sin[3*c + 2*d*x] + 865*Sin[2 *c + 3*d*x] + 865*Sin[4*c + 3*d*x] - 1296*Sin[3*c + 4*d*x] - 240*Sin[5*c + 4*d*x] + 435*Sin[4*c + 5*d*x] + 435*Sin[6*c + 5*d*x] - 176*Sin[5*c + 6*d* x])))/(a^3*d*(1 + Sec[c + d*x])^3)
Time = 0.54 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4376, 25, 3042, 4374, 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.
| \(\displaystyle \int \frac {\tan ^{10}(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot \left (c+d x+\frac {\pi }{2}\right )^{10}}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int -(a-a \sec (c+d x))^3 \tan ^4(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a-a \sec (c+d x))^3 \tan ^4(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \cot \left (c+d x+\frac {\pi }{2}\right )^4 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3dx}{a^6}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle -\frac {\int \left (a^3 \tan ^4(c+d x)-a^3 \sec ^3(c+d x) \tan ^4(c+d x)+3 a^3 \sec ^2(c+d x) \tan ^4(c+d x)-3 a^3 \sec (c+d x) \tan ^4(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {19 a^3 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {3 a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}-\frac {a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}+\frac {a^3 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac {3 a^3 \tan ^3(c+d x) \sec (c+d x)}{4 d}+\frac {17 a^3 \tan (c+d x) \sec (c+d x)}{16 d}+a^3 x}{a^6}\) |
Input:
Int[Tan[c + d*x]^10/(a + a*Sec[c + d*x])^3,x]
Output:
-((a^3*x - (19*a^3*ArcTanh[Sin[c + d*x]])/(16*d) - (a^3*Tan[c + d*x])/d + (17*a^3*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*Sec[c + d*x]^3*Tan[c + d* x])/(8*d) + (a^3*Tan[c + d*x]^3)/(3*d) - (3*a^3*Sec[c + d*x]*Tan[c + d*x]^ 3)/(4*d) - (a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d) + (3*a^3*Tan[c + d*x] ^5)/(5*d))/a^6)
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Result contains complex when optimal does not.
Time = 2.56 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {x}{a^{3}}+\frac {i \left (435 \,{\mathrm e}^{11 i \left (d x +c \right )}-240 \,{\mathrm e}^{10 i \left (d x +c \right )}+865 \,{\mathrm e}^{9 i \left (d x +c \right )}+1200 \,{\mathrm e}^{8 i \left (d x +c \right )}-210 \,{\mathrm e}^{7 i \left (d x +c \right )}+1760 \,{\mathrm e}^{6 i \left (d x +c \right )}+210 \,{\mathrm e}^{5 i \left (d x +c \right )}+1440 \,{\mathrm e}^{4 i \left (d x +c \right )}-865 \,{\mathrm e}^{3 i \left (d x +c \right )}+1296 \,{\mathrm e}^{2 i \left (d x +c \right )}-435 \,{\mathrm e}^{i \left (d x +c \right )}+176\right )}{120 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d \,a^{3}}+\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d \,a^{3}}\) | \(195\) |
| derivativedivides | \(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {11}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {35}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {19 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {11}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {35}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {19 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}}{d \,a^{3}}\) | \(230\) |
| default | \(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {11}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {35}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {19 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {11}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {11}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {5}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {35}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {19 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}}{d \,a^{3}}\) | \(230\) |
Input:
int(tan(d*x+c)^10/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-x/a^3+1/120*I*(435*exp(11*I*(d*x+c))-240*exp(10*I*(d*x+c))+865*exp(9*I*(d *x+c))+1200*exp(8*I*(d*x+c))-210*exp(7*I*(d*x+c))+1760*exp(6*I*(d*x+c))+21 0*exp(5*I*(d*x+c))+1440*exp(4*I*(d*x+c))-865*exp(3*I*(d*x+c))+1296*exp(2*I *(d*x+c))-435*exp(I*(d*x+c))+176)/d/a^3/(exp(2*I*(d*x+c))+1)^6-19/16/d/a^3 *ln(exp(I*(d*x+c))-I)+19/16/d/a^3*ln(exp(I*(d*x+c))+I)
Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {480 \, d x \cos \left (d x + c\right )^{6} - 285 \, \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) + 285 \, \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (176 \, \cos \left (d x + c\right )^{5} - 435 \, \cos \left (d x + c\right )^{4} + 208 \, \cos \left (d x + c\right )^{3} + 110 \, \cos \left (d x + c\right )^{2} - 144 \, \cos \left (d x + c\right ) + 40\right )} \sin \left (d x + c\right )}{480 \, a^{3} d \cos \left (d x + c\right )^{6}} \] Input:
integrate(tan(d*x+c)^10/(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-1/480*(480*d*x*cos(d*x + c)^6 - 285*cos(d*x + c)^6*log(sin(d*x + c) + 1) + 285*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(176*cos(d*x + c)^5 - 435* cos(d*x + c)^4 + 208*cos(d*x + c)^3 + 110*cos(d*x + c)^2 - 144*cos(d*x + c ) + 40)*sin(d*x + c))/(a^3*d*cos(d*x + c)^6)
\[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\tan ^{10}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(tan(d*x+c)**10/(a+a*sec(d*x+c))**3,x)
Output:
Integral(tan(c + d*x)**10/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (155) = 310\).
Time = 0.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.03 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {95 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {366 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1746 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3135 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {525 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a^{3} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {480 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {285 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {285 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{240 \, d} \] Input:
integrate(tan(d*x+c)^10/(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
-1/240*(2*(45*sin(d*x + c)/(cos(d*x + c) + 1) - 95*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 366*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 1746*sin(d*x + c) ^7/(cos(d*x + c) + 1)^7 - 3135*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 525*s in(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a^3 - 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 6*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^3*sin(d*x + c)^12/(cos(d *x + c) + 1)^12) + 480*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - 285*l og(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^3 + 285*log(sin(d*x + c)/(cos(d* x + c) + 1) - 1)/a^3)/d
Time = 0.52 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {240 \, {\left (d x + c\right )}}{a^{3}} - \frac {285 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {285 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 3135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1746 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 366 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 95 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6} a^{3}}}{240 \, d} \] Input:
integrate(tan(d*x+c)^10/(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
-1/240*(240*(d*x + c)/a^3 - 285*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 + 2 85*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3 + 2*(525*tan(1/2*d*x + 1/2*c)^11 - 3135*tan(1/2*d*x + 1/2*c)^9 + 1746*tan(1/2*d*x + 1/2*c)^7 - 366*tan(1/2 *d*x + 1/2*c)^5 - 95*tan(1/2*d*x + 1/2*c)^3 + 45*tan(1/2*d*x + 1/2*c))/((t an(1/2*d*x + 1/2*c)^2 - 1)^6*a^3))/d
Time = 13.73 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.23 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {19\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^3\,d}-\frac {x}{a^3}-\frac {\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {209\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}+\frac {291\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}-\frac {61\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}-\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \] Input:
int(tan(c + d*x)^10/(a + a/cos(c + d*x))^3,x)
Output:
(19*atanh(tan(c/2 + (d*x)/2)))/(8*a^3*d) - x/a^3 - ((3*tan(c/2 + (d*x)/2)) /8 - (19*tan(c/2 + (d*x)/2)^3)/24 - (61*tan(c/2 + (d*x)/2)^5)/20 + (291*ta n(c/2 + (d*x)/2)^7)/20 - (209*tan(c/2 + (d*x)/2)^9)/8 + (35*tan(c/2 + (d*x )/2)^11)/8)/(d*(15*a^3*tan(c/2 + (d*x)/2)^4 - 6*a^3*tan(c/2 + (d*x)/2)^2 - 20*a^3*tan(c/2 + (d*x)/2)^6 + 15*a^3*tan(c/2 + (d*x)/2)^8 - 6*a^3*tan(c/2 + (d*x)/2)^10 + a^3*tan(c/2 + (d*x)/2)^12 + a^3))
Time = 0.15 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.86 \[ \int \frac {\tan ^{10}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {-176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-240 \cos \left (d x +c \right ) \sin \left (d x +c \right )-285 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6}+855 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}-855 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}+285 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+285 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6}-855 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}+855 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}-285 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-240 \sin \left (d x +c \right )^{6} d x +435 \sin \left (d x +c \right )^{5}+720 \sin \left (d x +c \right )^{4} d x -760 \sin \left (d x +c \right )^{3}-720 \sin \left (d x +c \right )^{2} d x +285 \sin \left (d x +c \right )+240 d x}{240 a^{3} d \left (\sin \left (d x +c \right )^{6}-3 \sin \left (d x +c \right )^{4}+3 \sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(tan(d*x+c)^10/(a+a*sec(d*x+c))^3,x)
Output:
( - 176*cos(c + d*x)*sin(c + d*x)**5 + 560*cos(c + d*x)*sin(c + d*x)**3 - 240*cos(c + d*x)*sin(c + d*x) - 285*log(tan((c + d*x)/2) - 1)*sin(c + d*x) **6 + 855*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 - 855*log(tan((c + d*x )/2) - 1)*sin(c + d*x)**2 + 285*log(tan((c + d*x)/2) - 1) + 285*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6 - 855*log(tan((c + d*x)/2) + 1)*sin(c + d* x)**4 + 855*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 - 285*log(tan((c + d *x)/2) + 1) - 240*sin(c + d*x)**6*d*x + 435*sin(c + d*x)**5 + 720*sin(c + d*x)**4*d*x - 760*sin(c + d*x)**3 - 720*sin(c + d*x)**2*d*x + 285*sin(c + d*x) + 240*d*x)/(240*a**3*d*(sin(c + d*x)**6 - 3*sin(c + d*x)**4 + 3*sin(c + d*x)**2 - 1))