Integrand size = 23, antiderivative size = 154 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 \sqrt {b} (3 a+7 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{9/2} f}-\frac {(a+3 b) \cos (e+f x)}{a^4 f}+\frac {\cos ^3(e+f x)}{3 a^3 f}+\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (9 a+13 b) \cos (e+f x)}{8 a^4 f \left (b+a \cos ^2(e+f x)\right )} \]
-(a+3*b)*cos(f*x+e)/a^4/f+1/3*cos(f*x+e)^3/a^3/f+1/4*b^2*(a+b)*cos(f*x+e)/ a^4/f/(b+a*cos(f*x+e)^2)^2-1/8*b*(9*a+13*b)*cos(f*x+e)/a^4/f/(b+a*cos(f*x+ e)^2)+5/8*(3*a+7*b)*arctan(cos(f*x+e)*a^(1/2)/b^(1/2))*b^(1/2)/a^(9/2)/f
Result contains complex when optimal does not.
Time = 8.74 (sec) , antiderivative size = 1153, normalized size of antiderivative = 7.49 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x)))^3 \sec ^6(e+f x) \left (3 \left (9 a^4+1920 a b^3+4480 b^4\right ) \arctan \left (\frac {\left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )-\frac {-27 a^{11/2} \sqrt {b} \cos (e+f x)+162 a^{9/2} b^{3/2} \cos (e+f x)+10816 a^{7/2} b^{5/2} \cos (e+f x)+51552 a^{5/2} b^{7/2} \cos (e+f x)+87424 a^{3/2} b^{9/2} \cos (e+f x)+53760 \sqrt {a} b^{11/2} \cos (e+f x)-27 a^{11/2} \sqrt {b} \cos (e+f x) \cos (2 (e+f x))+47936 a^{5/2} b^{7/2} \cos (e+f x) \cos (2 (e+f x))+44800 a^{3/2} b^{9/2} \cos (e+f x) \cos (2 (e+f x))+27 a^{9/2} \sqrt {b} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-216 a^{7/2} b^{3/2} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-3600 a^{5/2} b^{5/2} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-5184 a^{3/2} b^{7/2} \cos (e+f x) (a+2 b+a \cos (2 (e+f x)))-27 a^4 \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2-5760 a b^3 \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2-13440 b^4 \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2+27 a^4 \arctan \left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2+27 a^4 \arctan \left (\frac {\sqrt {a}+\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2-2304 a^{3/2} b^{5/2} \cos (e) \cos (f x) (a+2 b+a \cos (2 (e+f x)))^2+1920 a^{7/2} b^{5/2} \cos (e+f x) \cos (4 (e+f x))+3584 a^{5/2} b^{7/2} \cos (e+f x) \cos (4 (e+f x))-128 a^{7/2} b^{5/2} \cos (e+f x) \cos (6 (e+f x))+2304 a^{3/2} b^{5/2} (a+2 b+a \cos (2 (e+f x)))^2 \sin (e) \sin (f x)+54 a^{9/2} b^{3/2} \csc (e+f x) \sin (4 (e+f x))+3108 a^{7/2} b^{5/2} \csc (e+f x) \sin (4 (e+f x))}{(a+2 b+a \cos (2 (e+f x)))^2}\right )}{24576 a^{9/2} b^{5/2} f \left (a+b \sec ^2(e+f x)\right )^3} \]
((a + 2*b + a*Cos[2*(e + f*x)])^3*Sec[e + f*x]^6*(3*(9*a^4 + 1920*a*b^3 + 4480*b^4)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*S in[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e] )^2]*Tan[(f*x)/2]))/Sqrt[b]] - (-27*a^(11/2)*Sqrt[b]*Cos[e + f*x] + 162*a^ (9/2)*b^(3/2)*Cos[e + f*x] + 10816*a^(7/2)*b^(5/2)*Cos[e + f*x] + 51552*a^ (5/2)*b^(7/2)*Cos[e + f*x] + 87424*a^(3/2)*b^(9/2)*Cos[e + f*x] + 53760*Sq rt[a]*b^(11/2)*Cos[e + f*x] - 27*a^(11/2)*Sqrt[b]*Cos[e + f*x]*Cos[2*(e + f*x)] + 47936*a^(5/2)*b^(7/2)*Cos[e + f*x]*Cos[2*(e + f*x)] + 44800*a^(3/2 )*b^(9/2)*Cos[e + f*x]*Cos[2*(e + f*x)] + 27*a^(9/2)*Sqrt[b]*Cos[e + f*x]* (a + 2*b + a*Cos[2*(e + f*x)]) - 216*a^(7/2)*b^(3/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 3600*a^(5/2)*b^(5/2)*Cos[e + f*x]*(a + 2*b + a*Co s[2*(e + f*x)]) - 5184*a^(3/2)*b^(7/2)*Cos[e + f*x]*(a + 2*b + a*Cos[2*(e + f*x)]) - 27*a^4*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e ])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 57 60*a*b^3*ArcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Si n[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e]) ^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2 - 13440*b^4*A rcTan[((-Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[( f*x)/2] + Cos[e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan...
Time = 0.44 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4621, 360, 25, 2345, 1467, 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.
| \(\displaystyle \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (e+f x)^3}{\left (a+b \sec (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle -\frac {\int \frac {\cos ^6(e+f x) \left (1-\cos ^2(e+f x)\right )}{\left (a \cos ^2(e+f x)+b\right )^3}d\cos (e+f x)}{f}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle -\frac {-\frac {\int -\frac {-4 a^3 \cos ^6(e+f x)+4 a^2 (a+b) \cos ^4(e+f x)-4 a b (a+b) \cos ^2(e+f x)+b^2 (a+b)}{\left (a \cos ^2(e+f x)+b\right )^2}d\cos (e+f x)}{4 a^4}-\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 \left (a \cos ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {-4 a^3 \cos ^6(e+f x)+4 a^2 (a+b) \cos ^4(e+f x)-4 a b (a+b) \cos ^2(e+f x)+b^2 (a+b)}{\left (a \cos ^2(e+f x)+b\right )^2}d\cos (e+f x)}{4 a^4}-\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 \left (a \cos ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {\frac {\frac {b (9 a+13 b) \cos (e+f x)}{2 \left (a \cos ^2(e+f x)+b\right )}-\frac {\int \frac {8 a^2 b \cos ^4(e+f x)-8 a b (a+2 b) \cos ^2(e+f x)+b^2 (7 a+11 b)}{a \cos ^2(e+f x)+b}d\cos (e+f x)}{2 b}}{4 a^4}-\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 \left (a \cos ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {\frac {\frac {b (9 a+13 b) \cos (e+f x)}{2 \left (a \cos ^2(e+f x)+b\right )}-\frac {\int \left (8 a b \cos ^2(e+f x)-8 b (a+3 b)+\frac {5 \left (7 b^3+3 a b^2\right )}{a \cos ^2(e+f x)+b}\right )d\cos (e+f x)}{2 b}}{4 a^4}-\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 \left (a \cos ^2(e+f x)+b\right )^2}}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\frac {b (9 a+13 b) \cos (e+f x)}{2 \left (a \cos ^2(e+f x)+b\right )}-\frac {\frac {5 b^{3/2} (3 a+7 b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {8}{3} a b \cos ^3(e+f x)-8 b (a+3 b) \cos (e+f x)}{2 b}}{4 a^4}-\frac {b^2 (a+b) \cos (e+f x)}{4 a^4 \left (a \cos ^2(e+f x)+b\right )^2}}{f}\) |
-((-1/4*(b^2*(a + b)*Cos[e + f*x])/(a^4*(b + a*Cos[e + f*x]^2)^2) + ((b*(9 *a + 13*b)*Cos[e + f*x])/(2*(b + a*Cos[e + f*x]^2)) - ((5*b^(3/2)*(3*a + 7 *b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/Sqrt[a] - 8*b*(a + 3*b)*Cos[e + f*x] + (8*a*b*Cos[e + f*x]^3)/3)/(2*b))/(4*a^4))/f)
3.1.55.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 5.74 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -3 b \cos \left (f x +e \right )}{a^{4}}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {13}{8} a b \right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a +11 b \right ) \cos \left (f x +e \right )}{8}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a +7 b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(125\) |
| default | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -3 b \cos \left (f x +e \right )}{a^{4}}+\frac {b \left (\frac {\left (-\frac {9}{8} a^{2}-\frac {13}{8} a b \right ) \cos \left (f x +e \right )^{3}-\frac {b \left (7 a +11 b \right ) \cos \left (f x +e \right )}{8}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a +7 b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(125\) |
| risch | \(\frac {{\mathrm e}^{3 i \left (f x +e \right )}}{24 a^{3} f}-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )}}{8 a^{3} f}-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )} b}{2 a^{4} f}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )}}{8 a^{3} f}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )} b}{2 a^{4} f}+\frac {{\mathrm e}^{-3 i \left (f x +e \right )}}{24 a^{3} f}-\frac {b \left (9 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+13 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+27 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}+67 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+44 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}+27 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}+67 a b \,{\mathrm e}^{3 i \left (f x +e \right )}+44 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}+9 a^{2} {\mathrm e}^{i \left (f x +e \right )}+13 a b \,{\mathrm e}^{i \left (f x +e \right )}\right )}{4 a^{4} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}+\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 a^{4} f}+\frac {35 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{16 a^{5} f}-\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 a^{4} f}-\frac {35 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{16 a^{5} f}\) | \(477\) |
1/f*(1/a^4*(1/3*a*cos(f*x+e)^3-cos(f*x+e)*a-3*b*cos(f*x+e))+b/a^4*(((-9/8* a^2-13/8*a*b)*cos(f*x+e)^3-1/8*b*(7*a+11*b)*cos(f*x+e))/(b+a*cos(f*x+e)^2) ^2+5/8*(3*a+7*b)/(a*b)^(1/2)*arctan(a*cos(f*x+e)/(a*b)^(1/2))))
Time = 0.30 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.85 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\left [\frac {16 \, a^{3} \cos \left (f x + e\right )^{7} - 16 \, {\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{5} - 50 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left ({\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 7 \, b^{3} + 2 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 30 \, {\left (3 \, a b^{2} + 7 \, b^{3}\right )} \cos \left (f x + e\right )}{48 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} + 2 \, a^{5} b f \cos \left (f x + e\right )^{2} + a^{4} b^{2} f\right )}}, \frac {8 \, a^{3} \cos \left (f x + e\right )^{7} - 8 \, {\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{5} - 25 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left ({\left (3 \, a^{3} + 7 \, a^{2} b\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 7 \, b^{3} + 2 \, {\left (3 \, a^{2} b + 7 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - 15 \, {\left (3 \, a b^{2} + 7 \, b^{3}\right )} \cos \left (f x + e\right )}{24 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} + 2 \, a^{5} b f \cos \left (f x + e\right )^{2} + a^{4} b^{2} f\right )}}\right ] \]
[1/48*(16*a^3*cos(f*x + e)^7 - 16*(3*a^3 + 7*a^2*b)*cos(f*x + e)^5 - 50*(3 *a^2*b + 7*a*b^2)*cos(f*x + e)^3 + 15*((3*a^3 + 7*a^2*b)*cos(f*x + e)^4 + 3*a*b^2 + 7*b^3 + 2*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a *cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 30*(3*a*b^2 + 7*b^3)*cos(f*x + e))/(a^6*f*cos(f*x + e)^4 + 2*a^5*b*f*co s(f*x + e)^2 + a^4*b^2*f), 1/24*(8*a^3*cos(f*x + e)^7 - 8*(3*a^3 + 7*a^2*b )*cos(f*x + e)^5 - 25*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^3 + 15*((3*a^3 + 7* a^2*b)*cos(f*x + e)^4 + 3*a*b^2 + 7*b^3 + 2*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - 15*(3*a*b^2 + 7*b^3)* cos(f*x + e))/(a^6*f*cos(f*x + e)^4 + 2*a^5*b*f*cos(f*x + e)^2 + a^4*b^2*f )]
Timed out. \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {3 \, {\left ({\left (9 \, a^{2} b + 13 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (7 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (f x + e\right )\right )}}{a^{6} \cos \left (f x + e\right )^{4} + 2 \, a^{5} b \cos \left (f x + e\right )^{2} + a^{4} b^{2}} - \frac {15 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {8 \, {\left (a \cos \left (f x + e\right )^{3} - 3 \, {\left (a + 3 \, b\right )} \cos \left (f x + e\right )\right )}}{a^{4}}}{24 \, f} \]
-1/24*(3*((9*a^2*b + 13*a*b^2)*cos(f*x + e)^3 + (7*a*b^2 + 11*b^3)*cos(f*x + e))/(a^6*cos(f*x + e)^4 + 2*a^5*b*cos(f*x + e)^2 + a^4*b^2) - 15*(3*a*b + 7*b^2)*arctan(a*cos(f*x + e)/sqrt(a*b))/(sqrt(a*b)*a^4) - 8*(a*cos(f*x + e)^3 - 3*(a + 3*b)*cos(f*x + e))/a^4)/f
Time = 0.42 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} f} - \frac {\frac {9 \, a^{2} b \cos \left (f x + e\right )^{3}}{f} + \frac {13 \, a b^{2} \cos \left (f x + e\right )^{3}}{f} + \frac {7 \, a b^{2} \cos \left (f x + e\right )}{f} + \frac {11 \, b^{3} \cos \left (f x + e\right )}{f}}{8 \, {\left (a \cos \left (f x + e\right )^{2} + b\right )}^{2} a^{4}} + \frac {a^{6} f^{17} \cos \left (f x + e\right )^{3} - 3 \, a^{6} f^{17} \cos \left (f x + e\right ) - 9 \, a^{5} b f^{17} \cos \left (f x + e\right )}{3 \, a^{9} f^{18}} \]
5/8*(3*a*b + 7*b^2)*arctan(a*cos(f*x + e)/sqrt(a*b))/(sqrt(a*b)*a^4*f) - 1 /8*(9*a^2*b*cos(f*x + e)^3/f + 13*a*b^2*cos(f*x + e)^3/f + 7*a*b^2*cos(f*x + e)/f + 11*b^3*cos(f*x + e)/f)/((a*cos(f*x + e)^2 + b)^2*a^4) + 1/3*(a^6 *f^17*cos(f*x + e)^3 - 3*a^6*f^17*cos(f*x + e) - 9*a^5*b*f^17*cos(f*x + e) )/(a^9*f^18)
Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {{\cos \left (e+f\,x\right )}^3}{3\,a^3\,f}-\frac {\left (\frac {9\,a^2\,b}{8}+\frac {13\,a\,b^2}{8}\right )\,{\cos \left (e+f\,x\right )}^3+\left (\frac {11\,b^3}{8}+\frac {7\,a\,b^2}{8}\right )\,\cos \left (e+f\,x\right )}{f\,\left (a^6\,{\cos \left (e+f\,x\right )}^4+2\,a^5\,b\,{\cos \left (e+f\,x\right )}^2+a^4\,b^2\right )}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {3\,b}{a^4}+\frac {1}{a^3}\right )}{f}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (3\,a+7\,b\right )}{7\,b^2+3\,a\,b}\right )\,\left (3\,a+7\,b\right )}{8\,a^{9/2}\,f} \]
cos(e + f*x)^3/(3*a^3*f) - (cos(e + f*x)^3*((13*a*b^2)/8 + (9*a^2*b)/8) + cos(e + f*x)*((7*a*b^2)/8 + (11*b^3)/8))/(f*(a^4*b^2 + a^6*cos(e + f*x)^4 + 2*a^5*b*cos(e + f*x)^2)) - (cos(e + f*x)*((3*b)/a^4 + 1/a^3))/f + (5*b^( 1/2)*atan((a^(1/2)*b^(1/2)*cos(e + f*x)*(3*a + 7*b))/(3*a*b + 7*b^2))*(3*a + 7*b))/(8*a^(9/2)*f)