Integrand size = 25, antiderivative size = 222 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\frac {4 b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 a (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2} \tan (e+f x)}{f} \]
4/3*b*cos(f*x+e)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/f-1/3*(7*a+8*b)*Ellip ticE(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)+4/3*a*(a+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+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)/f
Time = 3.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\frac {-8 a (7 a+8 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+32 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+\sqrt {2} \left (24 a^2+40 a b+13 b^2-4 b (2 a+3 b) \cos (2 (e+f x))-b^2 \cos (4 (e+f x))\right ) \tan (e+f x)}{24 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(-8*a*(7*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x , -(b/a)] + 32*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[ e + f*x, -(b/a)] + Sqrt[2]*(24*a^2 + 40*a*b + 13*b^2 - 4*b*(2*a + 3*b)*Cos [2*(e + f*x)] - b^2*Cos[4*(e + f*x)])*Tan[e + f*x])/(24*f*Sqrt[2*a + b - b *Cos[2*(e + f*x)]])
Time = 0.42 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3675, 369, 403, 25, 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.
| \(\displaystyle \int \tan ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (e+f x)^2 \left (a+b \sin (e+f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sin ^2(e+f x) \left (b \sin ^2(e+f x)+a\right )^{3/2}}{\left (1-\sin ^2(e+f x)\right )^{3/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}-\int \frac {\sqrt {b \sin ^2(e+f x)+a} \left (4 b \sin ^2(e+f x)+a\right )}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \int -\frac {b (7 a+8 b) \sin ^2(e+f x)+a (3 a+4 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (-\frac {1}{3} \int \frac {b (7 a+8 b) \sin ^2(e+f x)+a (3 a+4 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (4 a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)-(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)\right )+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}-(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)\right )+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}-(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)\right )+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}\right )+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {1}{3} \left (\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}\right )+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{\sqrt {1-\sin ^2(e+f x)}}+\frac {4}{3} b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {a+b \sin ^2(e+f x)}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((4*b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x ]^2]*Sqrt[a + b*Sin[e + f*x]^2])/3 + (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^ (3/2))/Sqrt[1 - Sin[e + f*x]^2] + (-(((7*a + 8*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] ) + (4*a*(a + 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
3.6.6.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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 = 5.14 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.75
| method | result | size |
| default | \(\frac {\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-b^{2} \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+4 a b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )-7 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -2 b \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a -2 b^{2} \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \sin \left (f x +e \right ) a^{2}+6 a b \sin \left (f x +e \right )+3 b^{2} \sin \left (f x +e \right )\right )}{3 \sqrt {-\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(389\) |
1/3*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(-b^2*cos(f*x+e)^4*sin(f*x+ e)+4*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin( f*x+e),(-1/a*b)^(1/2))*a^2+4*a*b*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+( a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))-7*(cos(f*x+e)^2)^(1/2)* (-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2 -8*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f* x+e),(-1/a*b)^(1/2))*a*b-2*b*cos(f*x+e)^2*sin(f*x+e)*a-2*b^2*cos(f*x+e)^2* sin(f*x+e)+3*sin(f*x+e)*a^2+6*a*b*sin(f*x+e)+3*b^2*sin(f*x+e))/(-(a+b*sin( f*x+e)^2)*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)/cos(f*x+e)/(a+b*sin(f*x+e)^ 2)^(1/2)/f
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{2} \,d x } \]
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\int \left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx \]
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{2} \,d x } \]
\[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^2(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]