Integrand size = 23, antiderivative size = 120 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2} d}-\frac {a^3 \tan (c+d x)}{(a+b)^4 d}+\frac {a^2 \tan ^3(c+d x)}{3 (a+b)^3 d}-\frac {a \tan ^5(c+d x)}{5 (a+b)^2 d}+\frac {\tan ^7(c+d x)}{7 (a+b) d} \] Output:
a^(7/2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))/(a+b)^(9/2)/d-a^3*tan(d*x+c )/(a+b)^4/d+1/3*a^2*tan(d*x+c)^3/(a+b)^3/d-1/5*a*tan(d*x+c)^5/(a+b)^2/d+1/ 7*tan(d*x+c)^7/(a+b)/d
Time = 4.89 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.22 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2} d}+\frac {\left (-176 a^3-122 a^2 b-66 a b^2-15 b^3+\left (122 a^3+254 a^2 b+177 a b^2+45 b^3\right ) \sec ^2(c+d x)-3 (a+b)^2 (22 a+15 b) \sec ^4(c+d x)+15 (a+b)^3 \sec ^6(c+d x)\right ) \tan (c+d x)}{105 (a+b)^4 d} \] Input:
Integrate[Tan[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]
Output:
(a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(9/2)*d) + ( (-176*a^3 - 122*a^2*b - 66*a*b^2 - 15*b^3 + (122*a^3 + 254*a^2*b + 177*a*b ^2 + 45*b^3)*Sec[c + d*x]^2 - 3*(a + b)^2*(22*a + 15*b)*Sec[c + d*x]^4 + 1 5*(a + b)^3*Sec[c + d*x]^6)*Tan[c + d*x])/(105*(a + b)^4*d)
Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3674, 254, 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.
| \(\displaystyle \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^8}{a+b \sin (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3674 |
| \(\displaystyle \frac {\int \frac {\tan ^8(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {\int \left (\frac {\tan ^6(c+d x)}{a+b}-\frac {a \tan ^4(c+d x)}{(a+b)^2}+\frac {a^2 \tan ^2(c+d x)}{(a+b)^3}+\frac {a^4}{(a+b)^4 \left ((a+b) \tan ^2(c+d x)+a\right )}-\frac {a^3}{(a+b)^4}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2}}-\frac {a^3 \tan (c+d x)}{(a+b)^4}+\frac {a^2 \tan ^3(c+d x)}{3 (a+b)^3}+\frac {\tan ^7(c+d x)}{7 (a+b)}-\frac {a \tan ^5(c+d x)}{5 (a+b)^2}}{d}\) |
Input:
Int[Tan[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]
Output:
((a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(9/2) - (a^3 *Tan[c + d*x])/(a + b)^4 + (a^2*Tan[c + d*x]^3)/(3*(a + b)^3) - (a*Tan[c + d*x]^5)/(5*(a + b)^2) + Tan[c + d*x]^7/(7*(a + b)))/d
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*((d_.)*tan[(e_.) + (f_.) *(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[f f/f Subst[Int[(d*ff*x)^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(p + 1) ), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m}, x] && IntegerQ [p]
Time = 6.55 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) \tan \left (d x +c \right )^{7}}{7}-\frac {a \left (a^{2}+2 a b +b^{2}\right ) \tan \left (d x +c \right )^{5}}{5}+\frac {a^{2} \tan \left (d x +c \right )^{3} \left (a +b \right )}{3}-a^{3} \tan \left (d x +c \right )}{\left (a +b \right )^{4}}+\frac {a^{4} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{4} \sqrt {a \left (a +b \right )}}}{d}\) | \(120\) |
| default | \(\frac {\frac {\frac {\left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) \tan \left (d x +c \right )^{7}}{7}-\frac {a \left (a^{2}+2 a b +b^{2}\right ) \tan \left (d x +c \right )^{5}}{5}+\frac {a^{2} \tan \left (d x +c \right )^{3} \left (a +b \right )}{3}-a^{3} \tan \left (d x +c \right )}{\left (a +b \right )^{4}}+\frac {a^{4} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{4} \sqrt {a \left (a +b \right )}}}{d}\) | \(120\) |
| risch | \(-\frac {2 i \left (15 b^{3}+176 a^{3}+122 a^{2} b +66 b^{2} a +210 a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+2870 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+840 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+525 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+2436 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+315 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+812 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+1890 a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+1400 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+420 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1722 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+1176 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+224 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+42 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3080 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+3080 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+1260 a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+420 a^{3} {\mathrm e}^{12 i \left (d x +c \right )}+630 a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+420 a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{105 d \left (a +b \right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {\sqrt {-a \left (a +b \right )}\, a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 \left (a +b \right )^{5} d}-\frac {\sqrt {-a \left (a +b \right )}\, a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 \left (a +b \right )^{5} d}\) | \(462\) |
Input:
int(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a+b)^4*(1/7*(a+b)*(a^2+2*a*b+b^2)*tan(d*x+c)^7-1/5*a*(a^2+2*a*b+b^ 2)*tan(d*x+c)^5+1/3*a^2*tan(d*x+c)^3*(a+b)-a^3*tan(d*x+c))+1/(a+b)^4*a^4/( a*(a+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (106) = 212\).
Time = 0.13 (sec) , antiderivative size = 602, normalized size of antiderivative = 5.02 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [\frac {105 \, a^{3} \sqrt {-\frac {a}{a + b}} \cos \left (d x + c\right )^{7} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a}{a + b}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) - 4 \, {\left ({\left (176 \, a^{3} + 122 \, a^{2} b + 66 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - {\left (122 \, a^{3} + 254 \, a^{2} b + 177 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 15 \, a^{3} - 45 \, a^{2} b - 45 \, a b^{2} - 15 \, b^{3} + 3 \, {\left (22 \, a^{3} + 59 \, a^{2} b + 52 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d \cos \left (d x + c\right )^{7}}, -\frac {105 \, a^{3} \sqrt {\frac {a}{a + b}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a}{a + b}}}{2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{7} + 2 \, {\left ({\left (176 \, a^{3} + 122 \, a^{2} b + 66 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - {\left (122 \, a^{3} + 254 \, a^{2} b + 177 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 15 \, a^{3} - 45 \, a^{2} b - 45 \, a b^{2} - 15 \, b^{3} + 3 \, {\left (22 \, a^{3} + 59 \, a^{2} b + 52 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d \cos \left (d x + c\right )^{7}}\right ] \] Input:
integrate(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="fricas")
Output:
[1/420*(105*a^3*sqrt(-a/(a + b))*cos(d*x + c)^7*log(((8*a^2 + 8*a*b + b^2) *cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + 3*a *b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)* cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((176*a^3 + 122*a^2*b + 66*a*b^2 + 15*b^3)*cos(d*x + c)^6 - (122*a^3 + 254*a^2*b + 177*a*b^2 + 45*b^3)*cos( d*x + c)^4 - 15*a^3 - 45*a^2*b - 45*a*b^2 - 15*b^3 + 3*(22*a^3 + 59*a^2*b + 52*a*b^2 + 15*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^4 + 4*a^3*b + 6*a^2 *b^2 + 4*a*b^3 + b^4)*d*cos(d*x + c)^7), -1/210*(105*a^3*sqrt(a/(a + b))*a rctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c)^7 + 2*((176*a^3 + 122*a^2*b + 66*a*b^2 + 15 *b^3)*cos(d*x + c)^6 - (122*a^3 + 254*a^2*b + 177*a*b^2 + 45*b^3)*cos(d*x + c)^4 - 15*a^3 - 45*a^2*b - 45*a*b^2 - 15*b^3 + 3*(22*a^3 + 59*a^2*b + 52 *a*b^2 + 15*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cos(d*x + c)^7)]
Timed out. \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(tan(d*x+c)**8/(a+b*sin(d*x+c)**2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.50 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {105 \, a^{4} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{7} - 21 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \tan \left (d x + c\right )^{5} - 105 \, a^{3} \tan \left (d x + c\right ) + 35 \, {\left (a^{3} + a^{2} b\right )} \tan \left (d x + c\right )^{3}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}}}{105 \, d} \] Input:
integrate(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="maxima")
Output:
1/105*(105*a^4*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))/((a^4 + 4*a^3* b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*a)) + (15*(a^3 + 3*a^2*b + 3*a *b^2 + b^3)*tan(d*x + c)^7 - 21*(a^3 + 2*a^2*b + a*b^2)*tan(d*x + c)^5 - 1 05*a^3*tan(d*x + c) + 35*(a^3 + a^2*b)*tan(d*x + c)^3)/(a^4 + 4*a^3*b + 6* a^2*b^2 + 4*a*b^3 + b^4))/d
Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (106) = 212\).
Time = 0.74 (sec) , antiderivative size = 543, normalized size of antiderivative = 4.52 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^{4} \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )}{{\left (a^{4} d + 4 \, a^{3} b d + 6 \, a^{2} b^{2} d + 4 \, a b^{3} d + b^{4} d\right )} \sqrt {a^{2} + a b}} + \frac {15 \, a^{6} d^{6} \tan \left (d x + c\right )^{7} + 90 \, a^{5} b d^{6} \tan \left (d x + c\right )^{7} + 225 \, a^{4} b^{2} d^{6} \tan \left (d x + c\right )^{7} + 300 \, a^{3} b^{3} d^{6} \tan \left (d x + c\right )^{7} + 225 \, a^{2} b^{4} d^{6} \tan \left (d x + c\right )^{7} + 90 \, a b^{5} d^{6} \tan \left (d x + c\right )^{7} + 15 \, b^{6} d^{6} \tan \left (d x + c\right )^{7} - 21 \, a^{6} d^{6} \tan \left (d x + c\right )^{5} - 105 \, a^{5} b d^{6} \tan \left (d x + c\right )^{5} - 210 \, a^{4} b^{2} d^{6} \tan \left (d x + c\right )^{5} - 210 \, a^{3} b^{3} d^{6} \tan \left (d x + c\right )^{5} - 105 \, a^{2} b^{4} d^{6} \tan \left (d x + c\right )^{5} - 21 \, a b^{5} d^{6} \tan \left (d x + c\right )^{5} + 35 \, a^{6} d^{6} \tan \left (d x + c\right )^{3} + 140 \, a^{5} b d^{6} \tan \left (d x + c\right )^{3} + 210 \, a^{4} b^{2} d^{6} \tan \left (d x + c\right )^{3} + 140 \, a^{3} b^{3} d^{6} \tan \left (d x + c\right )^{3} + 35 \, a^{2} b^{4} d^{6} \tan \left (d x + c\right )^{3} - 105 \, a^{6} d^{6} \tan \left (d x + c\right ) - 315 \, a^{5} b d^{6} \tan \left (d x + c\right ) - 315 \, a^{4} b^{2} d^{6} \tan \left (d x + c\right ) - 105 \, a^{3} b^{3} d^{6} \tan \left (d x + c\right )}{105 \, {\left (a^{7} d^{7} + 7 \, a^{6} b d^{7} + 21 \, a^{5} b^{2} d^{7} + 35 \, a^{4} b^{3} d^{7} + 35 \, a^{3} b^{4} d^{7} + 21 \, a^{2} b^{5} d^{7} + 7 \, a b^{6} d^{7} + b^{7} d^{7}\right )}} \] Input:
integrate(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm="giac")
Output:
a^4*arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b))/((a^4*d + 4* a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d)*sqrt(a^2 + a*b)) + 1/105*(15*a^ 6*d^6*tan(d*x + c)^7 + 90*a^5*b*d^6*tan(d*x + c)^7 + 225*a^4*b^2*d^6*tan(d *x + c)^7 + 300*a^3*b^3*d^6*tan(d*x + c)^7 + 225*a^2*b^4*d^6*tan(d*x + c)^ 7 + 90*a*b^5*d^6*tan(d*x + c)^7 + 15*b^6*d^6*tan(d*x + c)^7 - 21*a^6*d^6*t an(d*x + c)^5 - 105*a^5*b*d^6*tan(d*x + c)^5 - 210*a^4*b^2*d^6*tan(d*x + c )^5 - 210*a^3*b^3*d^6*tan(d*x + c)^5 - 105*a^2*b^4*d^6*tan(d*x + c)^5 - 21 *a*b^5*d^6*tan(d*x + c)^5 + 35*a^6*d^6*tan(d*x + c)^3 + 140*a^5*b*d^6*tan( d*x + c)^3 + 210*a^4*b^2*d^6*tan(d*x + c)^3 + 140*a^3*b^3*d^6*tan(d*x + c) ^3 + 35*a^2*b^4*d^6*tan(d*x + c)^3 - 105*a^6*d^6*tan(d*x + c) - 315*a^5*b* d^6*tan(d*x + c) - 315*a^4*b^2*d^6*tan(d*x + c) - 105*a^3*b^3*d^6*tan(d*x + c))/(a^7*d^7 + 7*a^6*b*d^7 + 21*a^5*b^2*d^7 + 35*a^4*b^3*d^7 + 35*a^3*b^ 4*d^7 + 21*a^2*b^5*d^7 + 7*a*b^6*d^7 + b^7*d^7)
Time = 34.67 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^7}{7\,d\,\left (a+b\right )}+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d\,{\left (a+b\right )}^3}+\frac {a^{7/2}\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{9/2}}\right )}{d\,{\left (a+b\right )}^{9/2}}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d\,{\left (a+b\right )}^2}-\frac {a^3\,\mathrm {tan}\left (c+d\,x\right )}{d\,{\left (a+b\right )}^4} \] Input:
int(tan(c + d*x)^8/(a + b*sin(c + d*x)^2),x)
Output:
tan(c + d*x)^7/(7*d*(a + b)) + (a^2*tan(c + d*x)^3)/(3*d*(a + b)^3) + (a^( 7/2)*atan((tan(c + d*x)*(2*a + 2*b)*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2 *b^2))/(2*a^(1/2)*(a + b)^(9/2))))/(d*(a + b)^(9/2)) - (a*tan(c + d*x)^5)/ (5*d*(a + b)^2) - (a^3*tan(c + d*x))/(d*(a + b)^4)
Time = 2.52 (sec) , antiderivative size = 2952, normalized size of antiderivative = 24.60 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx =\text {Too large to display} \] Input:
int(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x)
Output:
( - 210*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b))) *cos(c + d*x)*sin(c + d*x)**6*a**2 + 630*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt( 2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2 *sqrt(b)*sqrt(a + b) + a + 2*b)))*cos(c + d*x)*sin(c + d*x)**4*a**2 - 630* sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((ta n((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*cos(c + d*x)*sin(c + d*x)**2*a**2 + 210*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b )*sqrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b) *sqrt(a + b) + a + 2*b)))*cos(c + d*x)*a**2 + 210*sqrt(a)*sqrt(2*sqrt(b)*s qrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sq rt(a + b) + a + 2*b)))*cos(c + d*x)*sin(c + d*x)**6*a**3 + 210*sqrt(a)*sqr t(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt (2*sqrt(b)*sqrt(a + b) + a + 2*b)))*cos(c + d*x)*sin(c + d*x)**6*a**2*b - 630*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan((c + d*x)/2)*a )/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*cos(c + d*x)*sin(c + d* x)**4*a**3 - 630*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan(( c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*cos(c + d* x)*sin(c + d*x)**4*a**2*b + 630*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2 *b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a +...