Integrand size = 25, antiderivative size = 275 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=-\frac {(3 a+8 b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}+\frac {8 (a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a (5 a+8 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a+2 b) \sin ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x)}{3 f} \]
-1/3*(3*a+8*b)*cos(f*x+e)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/f+8/3*(a+2*b )*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b* sin(f*x+e)^2)^(1/2)/f/(1+b*sin(f*x+e)^2/a)^(1/2)-1/3*a*(5*a+8*b)*EllipticF (sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(1+b*sin(f*x+e)^ 2/a)^(1/2)/f/(a+b*sin(f*x+e)^2)^(1/2)-(a+2*b)*sin(f*x+e)^2*(a+b*sin(f*x+e) ^2)^(1/2)*tan(f*x+e)/f+1/3*(a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^3/f
Time = 3.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.77 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=\frac {32 a (a+2 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-4 a (5 a+8 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\frac {\left (32 a^2+108 a b+18 b^2+\left (64 a^2+160 a b+17 b^2\right ) \cos (2 (e+f x))-2 b (6 a+17 b) \cos (4 (e+f x))-b^2 \cos (6 (e+f x))\right ) \sec ^2(e+f x) \tan (e+f x)}{4 \sqrt {2}}}{12 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(32*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 4*a*(5*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF [e + f*x, -(b/a)] - ((32*a^2 + 108*a*b + 18*b^2 + (64*a^2 + 160*a*b + 17*b ^2)*Cos[2*(e + f*x)] - 2*b*(6*a + 17*b)*Cos[4*(e + f*x)] - b^2*Cos[6*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(4*Sqrt[2]))/(12*f*Sqrt[2*a + b - b*Co s[2*(e + f*x)]])
Time = 0.54 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3675, 369, 27, 439, 25, 444, 27, 399, 323, 321, 330, 327}
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.
| \(\displaystyle \int \tan ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (e+f x)^4 \left (a+b \sin (e+f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sin ^4(e+f x) \left (b \sin ^2(e+f x)+a\right )^{3/2}}{\left (1-\sin ^2(e+f x)\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {1}{3} \int \frac {3 \sin ^2(e+f x) \sqrt {b \sin ^2(e+f x)+a} \left (2 b \sin ^2(e+f x)+a\right )}{\left (1-\sin ^2(e+f x)\right )^{3/2}}d\sin (e+f x)\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\int \frac {\sin ^2(e+f x) \sqrt {b \sin ^2(e+f x)+a} \left (2 b \sin ^2(e+f x)+a\right )}{\left (1-\sin ^2(e+f x)\right )^{3/2}}d\sin (e+f x)\right )}{f}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\int -\frac {\sin ^2(e+f x) \left (b (3 a+8 b) \sin ^2(e+f x)+2 a (a+3 b)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\int \frac {\sin ^2(e+f x) \left (b (3 a+8 b) \sin ^2(e+f x)+2 a (a+3 b)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {b \left (8 b (a+2 b) \sin ^2(e+f x)+a (3 a+8 b)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{3 b}-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \int \frac {8 b (a+2 b) \sin ^2(e+f x)+a (3 a+8 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (8 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-a (5 a+8 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)\right )-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (8 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (8 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {8 (a+2 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {8 (a+2 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (5 a+8 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {1}{3} (3 a+8 b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}+\frac {\sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \sin ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{\sqrt {1-\sin ^2(e+f x)}}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(-(((a + 2*b)*Sin[e + f*x]^3*Sqrt[a + b *Sin[e + f*x]^2])/Sqrt[1 - Sin[e + f*x]^2]) - ((3*a + 8*b)*Sin[e + f*x]*Sq rt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/3 + (Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2))/(3*(1 - Sin[e + f*x]^2)^(3/2)) + ((8*(a + 2*b)* EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/Sqrt[1 + (b*Sin[e + f*x]^2)/a] - (a*(5*a + 8*b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/Sqrt[a + b*Sin[e + f*x]^2])/3))/f
3.6.5.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 4.98 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(-\frac {\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b^{2} \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b \left (3 a +7 b \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (4 a^{2}+13 a b +9 b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a \left (5 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +8 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -16 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a^{2}+2 a b +b^{2}\right ) \sin \left (f x +e \right )}{3 \left (\sin \left (f x +e \right )-1\right ) \sqrt {-\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(419\) |
-1/3*((-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b^2*cos(f*x+e)^6*sin(f*x+ e)+(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b*(3*a+7*b)*cos(f*x+e)^4*sin (f*x+e)-(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(4*a^2+13*a*b+9*b^2)*co s(f*x+e)^2*sin(f*x+e)-(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(-b/a*cos (f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*a*(5*EllipticF(sin(f*x+e),(- 1/a*b)^(1/2))*a+8*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-8*EllipticE(sin(f *x+e),(-1/a*b)^(1/2))*a-16*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*b)*cos(f*x +e)^2+(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(a^2+2*a*b+b^2)*sin(f*x+e ))/(sin(f*x+e)-1)/(-(a+b*sin(f*x+e)^2)*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2 )/(1+sin(f*x+e))/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=\text {Timed out} \]
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{4} \,d x } \]
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^4(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]