Integrand size = 25, antiderivative size = 345 \[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\frac {4 (a-b) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{15 a^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\cos ^2(e+f x) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{5 a f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (8 a^2-7 a b+8 b^2\right ) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{15 a^3 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b \left (4 a^2-3 a b+8 b^2\right ) \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{15 a^3 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
4/15*(a-b)*sin(f*x+e)*(a+b-a*sin(f*x+e)^2)/a^2/f/(sec(f*x+e)^2*(a+b-a*sin( f*x+e)^2))^(1/2)+1/5*cos(f*x+e)^2*sin(f*x+e)*(a+b-a*sin(f*x+e)^2)/a/f/(sec (f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)+1/15*(8*a^2-7*a*b+8*b^2)*EllipticE(s in(f*x+e),(a/(a+b))^(1/2))*(a+b-a*sin(f*x+e)^2)/a^3/f/(cos(f*x+e)^2)^(1/2) /(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/(1-a*sin(f*x+e)^2/(a+b))^(1/2)- 1/15*b*(4*a^2-3*a*b+8*b^2)*EllipticF(sin(f*x+e),(a/(a+b))^(1/2))*(1-a*sin( f*x+e)^2/(a+b))^(1/2)/a^3/f/(cos(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*sin( f*x+e)^2))^(1/2)
\[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx \]
Time = 0.61 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4636, 2057, 2058, 318, 25, 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.
| \(\displaystyle \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (e+f x)^5 \sqrt {a+b \sec (e+f x)^2}}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(e+f x)\right )^2}{\sqrt {a+\frac {b}{1-\sin ^2(e+f x)}}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(e+f x)\right )^2}{\sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\left (1-\sin ^2(e+f x)\right )^{5/2}}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}-\frac {\int -\frac {\sqrt {1-\sin ^2(e+f x)} \left (-4 (a-b) \sin ^2(e+f x)+4 a-b\right )}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int \frac {\sqrt {1-\sin ^2(e+f x)} \left (-4 (a-b) \sin ^2(e+f x)+4 a-b\right )}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {4 (a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}-\frac {\int -\frac {8 a^2-3 b a+4 b^2-\left (8 a^2-7 b a+8 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\int \frac {8 a^2-3 b a+4 b^2-\left (8 a^2-7 b a+8 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 a}+\frac {4 (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (8 a^2-7 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b \left (4 a^2-3 a b+8 b^2\right ) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}}{3 a}+\frac {4 (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (8 a^2-7 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b \left (4 a^2-3 a b+8 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {4 (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (8 a^2-7 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b \left (4 a^2-3 a b+8 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {4 (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (8 a^2-7 a b+8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b \left (4 a^2-3 a b+8 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {4 (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (8 a^2-7 a b+8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b \left (4 a^2-3 a b+8 b^2\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {4 (a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}}{5 a}+\frac {\sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*((Sin[e + f*x]*(1 - Sin[e + f*x]^2)^(3/2)* Sqrt[a + b - a*Sin[e + f*x]^2])/(5*a) + ((4*(a - b)*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*a) + (((8*a^2 - 7*a*b + 8*b^2)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(4*a^2 - 3*a*b + 8* b^2)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2 )/(a + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2]))/(3*a))/(5*a)))/(f*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x]^2)])
3.3.62.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 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_) + (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[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b/(1 - ff^2*x^2)^(n/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 - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 11.26 (sec) , antiderivative size = 8756, normalized size of antiderivative = 25.38
\[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{5}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{5}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\cos \left (f x + e\right )^{5}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\cos ^5(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^5}{\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}} \,d x \]