Integrand size = 23, antiderivative size = 267 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\left (5 a^3+60 a^2 b+120 a b^2+64 b^3\right ) x}{16 a^5}-\frac {\sqrt {b} (a+b)^{3/2} (3 a+8 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^5 f}-\frac {\left (33 a^2+82 a b+48 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(9 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {b \left (19 a^2+52 a b+32 b^2\right ) \tan (e+f x)}{16 a^4 f \left (a+b+b \tan ^2(e+f x)\right )} \]
1/16*(5*a^3+60*a^2*b+120*a*b^2+64*b^3)*x/a^5-1/2*(a+b)^(3/2)*(3*a+8*b)*arc tan(b^(1/2)*tan(f*x+e)/(a+b)^(1/2))*b^(1/2)/a^5/f-1/48*(33*a^2+82*a*b+48*b ^2)*cos(f*x+e)*sin(f*x+e)/a^3/f/(a+b+b*tan(f*x+e)^2)+1/24*(9*a+8*b)*cos(f* x+e)^3*sin(f*x+e)/a^2/f/(a+b+b*tan(f*x+e)^2)+1/6*cos(f*x+e)^3*sin(f*x+e)^3 /a/f/(a+b+b*tan(f*x+e)^2)-1/16*b*(19*a^2+52*a*b+32*b^2)*tan(f*x+e)/a^4/f/( a+b+b*tan(f*x+e)^2)
Result contains complex when optimal does not.
Time = 19.91 (sec) , antiderivative size = 2468, normalized size of antiderivative = 9.24 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\text {Result too large to show} \]
-1/512*((a + 2*b + a*Cos[2*e + 2*f*x])^2*Sec[e + f*x]^4*(16*x + ((-a^3 + 6 *a^2*b + 24*a*b^2 + 16*b^3)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*S in[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I *Sin[e])^4]) + ((a^2 + 8*a*b + 8*b^2)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x])) /(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + S in[e]))))/(a^2*(a + b*Sec[e + f*x]^2)^2) + (3*(a + 2*b + a*Cos[2*e + 2*f*x ])^2*Sec[e + f*x]^4*(-64*(a + 2*b)*x + ((a^4 - 16*a^3*b - 144*a^2*b^2 - 25 6*a*b^3 - 128*b^4)*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*S in[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4] )]*(Cos[2*e] - I*Sin[2*e]))/(b*(a + b)^(3/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^ 4]) + (16*a*Cos[2*f*x]*Sin[2*e])/f + (16*a*Cos[2*e]*Sin[2*f*x])/f - ((a^3 + 18*a^2*b + 48*a*b^2 + 32*b^3)*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)])*(Cos[e] - Sin[e])*(Cos[e] + Sin[e]) )))/(4096*a^3*(a + b*Sec[e + f*x]^2)^2) + (3*(a + 2*b + a*Cos[2*e + 2*f*x] )^2*Sec[e + f*x]^4*(((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]]) /(a + b)^(3/2) - (a*Sqrt[b]*Sin[2*(e + f*x)])/((a + b)*(a + 2*b + a*Cos[2* (e + f*x)]))))/(2048*b^(3/2)*f*(a + b*Sec[e + f*x]^2)^2) - ((a + 2*b + a*C os[2*e + 2*f*x])^2*Sec[e + f*x]^4*(-((a*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt [a + b]])/(a + b)^(3/2)) + (Sqrt[b]*(a + 2*b)*Sin[2*(e + f*x)])/((a + b...
Time = 0.53 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4620, 372, 440, 402, 27, 402, 27, 397, 216, 218}
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.
| \(\displaystyle \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (e+f x)^6}{\left (a+b \sec (e+f x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\tan ^6(e+f x)}{\left (\tan ^2(e+f x)+1\right )^4 \left (b \tan ^2(e+f x)+a+b\right )^2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\int \frac {\tan ^2(e+f x) \left ((b-6 (a+b)) \tan ^2(e+f x)+3 (a+b)\right )}{\left (\tan ^2(e+f x)+1\right )^3 \left (b \tan ^2(e+f x)+a+b\right )^2}d\tan (e+f x)}{6 a}}{f}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\int \frac {(a+b) (9 a+8 b)-\left (24 a^2+65 b a+40 b^2\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right )^2 \left (b \tan ^2(e+f x)+a+b\right )^2}d\tan (e+f x)}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\int \frac {3 \left ((a+b) \left (5 a^2+22 b a+16 b^2\right )-b \left (33 a^2+82 b a+48 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )^2}d\tan (e+f x)}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 \int \frac {(a+b) \left (5 a^2+22 b a+16 b^2\right )-b \left (33 a^2+82 b a+48 b^2\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )^2}d\tan (e+f x)}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 \left (\frac {\int \frac {2 (a+b) \left ((a+b) \left (5 a^2+36 b a+32 b^2\right )-b \left (19 a^2+52 b a+32 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{2 a (a+b)}-\frac {b \left (19 a^2+52 a b+32 b^2\right ) \tan (e+f x)}{a \left (a+b \tan ^2(e+f x)+b\right )}\right )}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 \left (\frac {\int \frac {(a+b) \left (5 a^2+36 b a+32 b^2\right )-b \left (19 a^2+52 b a+32 b^2\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{a}-\frac {b \left (19 a^2+52 a b+32 b^2\right ) \tan (e+f x)}{a \left (a+b \tan ^2(e+f x)+b\right )}\right )}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 \left (\frac {\frac {\left (5 a^3+60 a^2 b+120 a b^2+64 b^3\right ) \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)}{a}-\frac {8 b (a+b)^2 (3 a+8 b) \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a}}{a}-\frac {b \left (19 a^2+52 a b+32 b^2\right ) \tan (e+f x)}{a \left (a+b \tan ^2(e+f x)+b\right )}\right )}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 \left (\frac {\frac {\left (5 a^3+60 a^2 b+120 a b^2+64 b^3\right ) \arctan (\tan (e+f x))}{a}-\frac {8 b (a+b)^2 (3 a+8 b) \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a}}{a}-\frac {b \left (19 a^2+52 a b+32 b^2\right ) \tan (e+f x)}{a \left (a+b \tan ^2(e+f x)+b\right )}\right )}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\tan ^3(e+f x)}{6 a \left (\tan ^2(e+f x)+1\right )^3 \left (a+b \tan ^2(e+f x)+b\right )}-\frac {\frac {\frac {\left (33 a^2+82 a b+48 b^2\right ) \tan (e+f x)}{2 a \left (\tan ^2(e+f x)+1\right ) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 \left (\frac {\frac {\left (5 a^3+60 a^2 b+120 a b^2+64 b^3\right ) \arctan (\tan (e+f x))}{a}-\frac {8 \sqrt {b} (a+b)^{3/2} (3 a+8 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a}}{a}-\frac {b \left (19 a^2+52 a b+32 b^2\right ) \tan (e+f x)}{a \left (a+b \tan ^2(e+f x)+b\right )}\right )}{2 a}}{4 a}-\frac {(9 a+8 b) \tan (e+f x)}{4 a \left (\tan ^2(e+f x)+1\right )^2 \left (a+b \tan ^2(e+f x)+b\right )}}{6 a}}{f}\) |
(Tan[e + f*x]^3/(6*a*(1 + Tan[e + f*x]^2)^3*(a + b + b*Tan[e + f*x]^2)) - (-1/4*((9*a + 8*b)*Tan[e + f*x])/(a*(1 + Tan[e + f*x]^2)^2*(a + b + b*Tan[ e + f*x]^2)) + (((33*a^2 + 82*a*b + 48*b^2)*Tan[e + f*x])/(2*a*(1 + Tan[e + f*x]^2)*(a + b + b*Tan[e + f*x]^2)) - (3*((((5*a^3 + 60*a^2*b + 120*a*b^ 2 + 64*b^3)*ArcTan[Tan[e + f*x]])/a - (8*Sqrt[b]*(a + b)^(3/2)*(3*a + 8*b) *ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/a)/a - (b*(19*a^2 + 52*a*b + 32*b^2)*Tan[e + f*x])/(a*(a + b + b*Tan[e + f*x]^2))))/(2*a))/(4*a))/(6*a) )/f
3.1.47.3.1 Defintions of rubi rules used
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Time = 5.73 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {-\frac {\left (a +b \right )^{2} b \left (\frac {a \tan \left (f x +e \right )}{2 a +2 b +2 b \tan \left (f x +e \right )^{2}}+\frac {\left (3 a +8 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \sqrt {\left (a +b \right ) b}}\right )}{a^{5}}+\frac {\frac {\left (-\frac {9}{4} a^{2} b -\frac {3}{2} a \,b^{2}-\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )^{5}+\left (-4 a^{2} b -3 a \,b^{2}-\frac {5}{6} a^{3}\right ) \tan \left (f x +e \right )^{3}+\left (-\frac {5}{16} a^{3}-\frac {7}{4} a^{2} b -\frac {3}{2} a \,b^{2}\right ) \tan \left (f x +e \right )}{\left (1+\tan \left (f x +e \right )^{2}\right )^{3}}+\frac {\left (5 a^{3}+60 a^{2} b +120 a \,b^{2}+64 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{5}}}{f}\) | \(204\) |
| default | \(\frac {-\frac {\left (a +b \right )^{2} b \left (\frac {a \tan \left (f x +e \right )}{2 a +2 b +2 b \tan \left (f x +e \right )^{2}}+\frac {\left (3 a +8 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \sqrt {\left (a +b \right ) b}}\right )}{a^{5}}+\frac {\frac {\left (-\frac {9}{4} a^{2} b -\frac {3}{2} a \,b^{2}-\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )^{5}+\left (-4 a^{2} b -3 a \,b^{2}-\frac {5}{6} a^{3}\right ) \tan \left (f x +e \right )^{3}+\left (-\frac {5}{16} a^{3}-\frac {7}{4} a^{2} b -\frac {3}{2} a \,b^{2}\right ) \tan \left (f x +e \right )}{\left (1+\tan \left (f x +e \right )^{2}\right )^{3}}+\frac {\left (5 a^{3}+60 a^{2} b +120 a \,b^{2}+64 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{5}}}{f}\) | \(204\) |
| risch | \(\frac {5 x}{16 a^{2}}+\frac {15 x b}{4 a^{3}}+\frac {15 x \,b^{2}}{2 a^{4}}+\frac {4 x \,b^{3}}{a^{5}}-\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )}}{128 a^{2} f}-\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} b}{2 a^{3} f}-\frac {15 i {\mathrm e}^{-2 i \left (f x +e \right )}}{128 a^{2} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )}}{128 a^{2} f}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )} b}{32 a^{3} f}+\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b^{2}}{8 a^{4} f}+\frac {i {\mathrm e}^{2 i \left (f x +e \right )} b}{2 a^{3} f}+\frac {15 i {\mathrm e}^{2 i \left (f x +e \right )}}{128 a^{2} f}-\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b^{2}}{8 a^{4} f}-\frac {i b \left (a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+4 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+5 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+2 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}+a^{3}+2 a^{2} b +a \,b^{2}\right )}{a^{5} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )} b}{32 a^{3} f}-\frac {3 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b -b^{2}}-a -2 b}{a}\right )}{4 f \,a^{3}}-\frac {11 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b -b^{2}}-a -2 b}{a}\right ) b}{4 f \,a^{4}}-\frac {2 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b -b^{2}}-a -2 b}{a}\right ) b^{2}}{f \,a^{5}}+\frac {3 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b -b^{2}}+a +2 b}{a}\right )}{4 f \,a^{3}}+\frac {11 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b -b^{2}}+a +2 b}{a}\right ) b}{4 f \,a^{4}}+\frac {2 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b -b^{2}}+a +2 b}{a}\right ) b^{2}}{f \,a^{5}}-\frac {\sin \left (6 f x +6 e \right )}{192 a^{2} f}\) | \(708\) |
1/f*(-(a+b)^2*b/a^5*(1/2*a*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)+1/2*(3*a+8*b)/( (a+b)*b)^(1/2)*arctan(b*tan(f*x+e)/((a+b)*b)^(1/2)))+1/a^5*(((-9/4*a^2*b-3 /2*a*b^2-11/16*a^3)*tan(f*x+e)^5+(-4*a^2*b-3*a*b^2-5/6*a^3)*tan(f*x+e)^3+( -5/16*a^3-7/4*a^2*b-3/2*a*b^2)*tan(f*x+e))/(1+tan(f*x+e)^2)^3+1/16*(5*a^3+ 60*a^2*b+120*a*b^2+64*b^3)*arctan(tan(f*x+e))))
Time = 0.33 (sec) , antiderivative size = 674, normalized size of antiderivative = 2.52 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [\frac {3 \, {\left (5 \, a^{4} + 60 \, a^{3} b + 120 \, a^{2} b^{2} + 64 \, a b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{3} b + 60 \, a^{2} b^{2} + 120 \, a b^{3} + 64 \, b^{4}\right )} f x + 6 \, {\left (3 \, a^{2} b + 11 \, a b^{2} + 8 \, b^{3} + {\left (3 \, a^{3} + 11 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a b - b^{2}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b - b^{2}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) - {\left (8 \, a^{4} \cos \left (f x + e\right )^{7} - 2 \, {\left (13 \, a^{4} + 8 \, a^{3} b\right )} \cos \left (f x + e\right )^{5} + {\left (33 \, a^{4} + 82 \, a^{3} b + 48 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (19 \, a^{3} b + 52 \, a^{2} b^{2} + 32 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (a^{6} f \cos \left (f x + e\right )^{2} + a^{5} b f\right )}}, \frac {3 \, {\left (5 \, a^{4} + 60 \, a^{3} b + 120 \, a^{2} b^{2} + 64 \, a b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{3} b + 60 \, a^{2} b^{2} + 120 \, a b^{3} + 64 \, b^{4}\right )} f x + 12 \, {\left (3 \, a^{2} b + 11 \, a b^{2} + 8 \, b^{3} + {\left (3 \, a^{3} + 11 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a b + b^{2}} \arctan \left (\frac {{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{2 \, \sqrt {a b + b^{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) - {\left (8 \, a^{4} \cos \left (f x + e\right )^{7} - 2 \, {\left (13 \, a^{4} + 8 \, a^{3} b\right )} \cos \left (f x + e\right )^{5} + {\left (33 \, a^{4} + 82 \, a^{3} b + 48 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (19 \, a^{3} b + 52 \, a^{2} b^{2} + 32 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (a^{6} f \cos \left (f x + e\right )^{2} + a^{5} b f\right )}}\right ] \]
[1/48*(3*(5*a^4 + 60*a^3*b + 120*a^2*b^2 + 64*a*b^3)*f*x*cos(f*x + e)^2 + 3*(5*a^3*b + 60*a^2*b^2 + 120*a*b^3 + 64*b^4)*f*x + 6*(3*a^2*b + 11*a*b^2 + 8*b^3 + (3*a^3 + 11*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*lo g(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - (8*a^4*co s(f*x + e)^7 - 2*(13*a^4 + 8*a^3*b)*cos(f*x + e)^5 + (33*a^4 + 82*a^3*b + 48*a^2*b^2)*cos(f*x + e)^3 + 3*(19*a^3*b + 52*a^2*b^2 + 32*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^6*f*cos(f*x + e)^2 + a^5*b*f), 1/48*(3*(5*a^4 + 60* a^3*b + 120*a^2*b^2 + 64*a*b^3)*f*x*cos(f*x + e)^2 + 3*(5*a^3*b + 60*a^2*b ^2 + 120*a*b^3 + 64*b^4)*f*x + 12*(3*a^2*b + 11*a*b^2 + 8*b^3 + (3*a^3 + 1 1*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*c os(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - (8*a^4*c os(f*x + e)^7 - 2*(13*a^4 + 8*a^3*b)*cos(f*x + e)^5 + (33*a^4 + 82*a^3*b + 48*a^2*b^2)*cos(f*x + e)^3 + 3*(19*a^3*b + 52*a^2*b^2 + 32*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^6*f*cos(f*x + e)^2 + a^5*b*f)]
Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (19 \, a^{2} b + 52 \, a b^{2} + 32 \, b^{3}\right )} \tan \left (f x + e\right )^{7} + {\left (33 \, a^{3} + 253 \, a^{2} b + 516 \, a b^{2} + 288 \, b^{3}\right )} \tan \left (f x + e\right )^{5} + {\left (40 \, a^{3} + 319 \, a^{2} b + 564 \, a b^{2} + 288 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{3} + 41 \, a^{2} b + 68 \, a b^{2} + 32 \, b^{3}\right )} \tan \left (f x + e\right )}{a^{4} b \tan \left (f x + e\right )^{8} + {\left (a^{5} + 4 \, a^{4} b\right )} \tan \left (f x + e\right )^{6} + a^{5} + a^{4} b + 3 \, {\left (a^{5} + 2 \, a^{4} b\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{5} + 4 \, a^{4} b\right )} \tan \left (f x + e\right )^{2}} - \frac {3 \, {\left (5 \, a^{3} + 60 \, a^{2} b + 120 \, a b^{2} + 64 \, b^{3}\right )} {\left (f x + e\right )}}{a^{5}} + \frac {24 \, {\left (3 \, a^{3} b + 14 \, a^{2} b^{2} + 19 \, a b^{3} + 8 \, b^{4}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a^{5}}}{48 \, f} \]
-1/48*((3*(19*a^2*b + 52*a*b^2 + 32*b^3)*tan(f*x + e)^7 + (33*a^3 + 253*a^ 2*b + 516*a*b^2 + 288*b^3)*tan(f*x + e)^5 + (40*a^3 + 319*a^2*b + 564*a*b^ 2 + 288*b^3)*tan(f*x + e)^3 + 3*(5*a^3 + 41*a^2*b + 68*a*b^2 + 32*b^3)*tan (f*x + e))/(a^4*b*tan(f*x + e)^8 + (a^5 + 4*a^4*b)*tan(f*x + e)^6 + a^5 + a^4*b + 3*(a^5 + 2*a^4*b)*tan(f*x + e)^4 + (3*a^5 + 4*a^4*b)*tan(f*x + e)^ 2) - 3*(5*a^3 + 60*a^2*b + 120*a*b^2 + 64*b^3)*(f*x + e)/a^5 + 24*(3*a^3*b + 14*a^2*b^2 + 19*a*b^3 + 8*b^4)*arctan(b*tan(f*x + e)/sqrt((a + b)*b))/( sqrt((a + b)*b)*a^5))/f
Time = 0.36 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {3 \, {\left (5 \, a^{3} + 60 \, a^{2} b + 120 \, a b^{2} + 64 \, b^{3}\right )} {\left (f x + e\right )}}{a^{5}} - \frac {24 \, {\left (3 \, a^{3} b + 14 \, a^{2} b^{2} + 19 \, a b^{3} + 8 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{\sqrt {a b + b^{2}} a^{5}} - \frac {24 \, {\left (a^{2} b \tan \left (f x + e\right ) + 2 \, a b^{2} \tan \left (f x + e\right ) + b^{3} \tan \left (f x + e\right )\right )}}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )} a^{4}} - \frac {33 \, a^{2} \tan \left (f x + e\right )^{5} + 108 \, a b \tan \left (f x + e\right )^{5} + 72 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} + 192 \, a b \tan \left (f x + e\right )^{3} + 144 \, b^{2} \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right ) + 84 \, a b \tan \left (f x + e\right ) + 72 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3} a^{4}}}{48 \, f} \]
1/48*(3*(5*a^3 + 60*a^2*b + 120*a*b^2 + 64*b^3)*(f*x + e)/a^5 - 24*(3*a^3* b + 14*a^2*b^2 + 19*a*b^3 + 8*b^4)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))/(sqrt(a*b + b^2)*a^5) - 24*(a^2*b* tan(f*x + e) + 2*a*b^2*tan(f*x + e) + b^3*tan(f*x + e))/((b*tan(f*x + e)^2 + a + b)*a^4) - (33*a^2*tan(f*x + e)^5 + 108*a*b*tan(f*x + e)^5 + 72*b^2* tan(f*x + e)^5 + 40*a^2*tan(f*x + e)^3 + 192*a*b*tan(f*x + e)^3 + 144*b^2* tan(f*x + e)^3 + 15*a^2*tan(f*x + e) + 84*a*b*tan(f*x + e) + 72*b^2*tan(f* x + e))/((tan(f*x + e)^2 + 1)^3*a^4))/f
Time = 20.40 (sec) , antiderivative size = 1461, normalized size of antiderivative = 5.47 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
(atanh((75*b^3*tan(e + f*x)*(- 3*a*b^3 - a^3*b - b^4 - 3*a^2*b^2)^(1/2))/( 256*((211*a*b^4)/128 + (811*b^5)/256 + (75*a^2*b^3)/256 + (41*b^6)/(16*a) + (3*b^7)/(4*a^2))) + (17*b^4*tan(e + f*x)*(- 3*a*b^3 - a^3*b - b^4 - 3*a^ 2*b^2)^(1/2))/(16*((811*a*b^5)/256 + (41*b^6)/16 + (211*a^2*b^4)/128 + (75 *a^3*b^3)/256 + (3*b^7)/(4*a))) + (3*b^5*tan(e + f*x)*(- 3*a*b^3 - a^3*b - b^4 - 3*a^2*b^2)^(1/2))/(4*((41*a*b^6)/16 + (3*b^7)/4 + (811*a^2*b^5)/256 + (211*a^3*b^4)/128 + (75*a^4*b^3)/256)))*(-b*(a + b)^3)^(1/2)*(3*a + 8*b ))/(2*a^5*f) - (atan(((((((8*a^10*b^5 + 17*a^11*b^4 + (41*a^12*b^3)/4 + (5 *a^13*b^2)/4)/a^12 - (tan(e + f*x)*(2048*a^10*b^3 + 1024*a^11*b^2)*(a*b^2* 120i + a^2*b*60i + a^3*5i + b^3*64i))/(4096*a^13))*(a*b^2*120i + a^2*b*60i + a^3*5i + b^3*64i))/(32*a^5) - (tan(e + f*x)*(34816*a*b^8 + 8192*b^9 + 5 9520*a^2*b^7 + 52160*a^3*b^6 + 24640*a^4*b^5 + 5976*a^5*b^4 + 601*a^6*b^3) )/(128*a^8))*(a*b^2*120i + a^2*b*60i + a^3*5i + b^3*64i)*1i)/(32*a^5) - (( (((8*a^10*b^5 + 17*a^11*b^4 + (41*a^12*b^3)/4 + (5*a^13*b^2)/4)/a^12 + (ta n(e + f*x)*(2048*a^10*b^3 + 1024*a^11*b^2)*(a*b^2*120i + a^2*b*60i + a^3*5 i + b^3*64i))/(4096*a^13))*(a*b^2*120i + a^2*b*60i + a^3*5i + b^3*64i))/(3 2*a^5) + (tan(e + f*x)*(34816*a*b^8 + 8192*b^9 + 59520*a^2*b^7 + 52160*a^3 *b^6 + 24640*a^4*b^5 + 5976*a^5*b^4 + 601*a^6*b^3))/(128*a^8))*(a*b^2*120i + a^2*b*60i + a^3*5i + b^3*64i)*1i)/(32*a^5))/((376*a*b^10 + 64*b^11 + 93 7*a^2*b^9 + (10285*a^3*b^8)/8 + (33701*a^4*b^7)/32 + (8333*a^5*b^6)/16 ...