\(\int \sec ^5(e+f x) (a+b \sec ^2(e+f x))^{3/2} \, dx\) [241]

Optimal result

Integrand size = 25, antiderivative size = 450 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a+b) \left (a^2-16 a b-16 b^2\right ) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{35 b f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 b f}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f} \]

[Out]

-2/35*(a+2*b)*(a^2-4*a*b-4*b^2)*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/b^2/f+2/35*(a+2*b)*(a^2-4
*a*b-4*b^2)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(cos(f*x+e)^2)^(1/2)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/
2)/b^2/f/(1-a*sin(f*x+e)^2/(a+b))^(1/2)-1/35*(a+b)*(a^2-16*a*b-16*b^2)*EllipticF(sin(f*x+e),(a/(a+b))^(1/2))*(
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)/b/f/(a+b-a*sin(f*
x+e)^2)+1/35*(a^2+11*a*b+8*b^2)*sec(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/b/f+2/35*(4*a+
3*b)*sec(f*x+e)^3*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f+1/7*b*sec(f*x+e)^5*(sec(f*x+e)^2*(a+b
-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4233, 1985, 1986, 424, 541, 538, 437, 435, 432, 430} \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=-\frac {(a+b) \left (a^2-16 a b-16 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{35 b f \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{35 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{35 b^2 f}+\frac {\left (a^2+11 a b+8 b^2\right ) \tan (e+f x) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{35 b f}+\frac {b \tan (e+f x) \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{7 f}+\frac {2 (4 a+3 b) \tan (e+f x) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}}{35 f} \]

[In]

Int[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]

[Out]

(-2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(35*b^2*f) +
(2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e
+ f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(35*b^2*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a + b)*(a^2 - 16*a*
b - 16*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin
[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(35*b*f*(a + b - a*Sin[e + f*x]^2)) + ((a^2 + 11*a*b + 8*b
^2)*Sec[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(35*b*f) + (2*(4*a + 3*b)*Sec[e
+ f*x]^3*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(35*f) + (b*Sec[e + f*x]^5*Sqrt[Sec[e
+ f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(7*f)

Rule 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[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]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[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]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1985

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]

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[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]

Rule 4233

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+\frac {b}{1-x^2}\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (\frac {a+b-a x^2}{1-x^2}\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (\sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {\left (a+b-a x^2\right )^{3/2}}{\left (1-x^2\right )^{9/2}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a+b-a \sin ^2(e+f x)}} \\ & = \frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {-((a+b) (7 a+6 b))+a (7 a+5 b) x^2}{\left (1-x^2\right )^{7/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{7 f \sqrt {a+b-a \sin ^2(e+f x)}} \\ & = \frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {-3 b (a+b) (9 a+8 b)+6 a b (4 a+3 b) x^2}{\left (1-x^2\right )^{5/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f \sqrt {a+b-a \sin ^2(e+f x)}} \\ & = \frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 b f}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {3 b (a+b) \left (a^2-16 a b-16 b^2\right )+3 a b \left (a^2+11 a b+8 b^2\right ) x^2}{\left (1-x^2\right )^{3/2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{105 b^2 f \sqrt {a+b-a \sin ^2(e+f x)}} \\ & = -\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 b f}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {-3 a b (a+b) \left (2 a^2-5 a b-8 b^2\right )+6 a b (a+2 b) \left (a^2-4 a b-4 b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{105 b^3 f \sqrt {a+b-a \sin ^2(e+f x)}} \\ & = -\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 b f}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f}-\frac {\left ((a+b) \left (a^2-16 a b-16 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b-a x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f \sqrt {a+b-a \sin ^2(e+f x)}}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {a+b-a x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{35 b^2 f \sqrt {a+b-a \sin ^2(e+f x)}} \\ & = -\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 b f}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f}+\frac {\left (2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {a x^2}{a+b}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{35 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {\left ((a+b) \left (a^2-16 a b-16 b^2\right ) \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}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {a x^2}{a+b}}} \, dx,x,\sin (e+f x)\right )}{35 b f \left (a+b-a \sin ^2(e+f x)\right )} \\ & = -\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f}+\frac {2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{35 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a+b) \left (a^2-16 a b-16 b^2\right ) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \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}}}{35 b f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {\left (a^2+11 a b+8 b^2\right ) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 b f}+\frac {2 (4 a+3 b) \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{35 f}+\frac {b \sec ^5(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{7 f} \\ \end{align*}

Mathematica [F]

\[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx \]

[In]

Integrate[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2),x]

[Out]

Integrate[Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2), x]

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 16.32 (sec) , antiderivative size = 11007, normalized size of antiderivative = 24.46

method	result	size
default	\(\text {Expression too large to display}\)	\(11007\)

[In]

int(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.18 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

1/35*((2*(-I*a^4 + 2*I*a^3*b + 12*I*a^2*b^2 + 8*I*a*b^3)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^6 - (-I*a^
4 + 16*I*a^2*b^2 + 32*I*a*b^3 + 16*I*b^4)*sqrt(a)*cos(f*x + e)^6)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a
)*elliptic_e(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f*x + e))), (a^2 + 8*a
*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + (2*(I*a^4 - 2*I*a^3*b - 12*I*a^2*b^2 - 8*I*a*b^3)*s
qrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^6 - (I*a^4 - 16*I*a^2*b^2 - 32*I*a*b^3 - 16*I*b^4)*sqrt(a)*cos(f*x +
e)^6)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2
*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) +
(2*(-I*a^3*b - 11*I*a^2*b^2 - 8*I*a*b^3)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^6 - (2*I*a^4 + I*a^3*b - 1
9*I*a^2*b^2 - 34*I*a*b^3 - 16*I*b^4)*sqrt(a)*cos(f*x + e)^6)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*ell
iptic_f(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f*x + e))), (a^2 + 8*a*b +
8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + (2*(I*a^3*b + 11*I*a^2*b^2 + 8*I*a*b^3)*sqrt(a)*sqrt((a*
b + b^2)/a^2)*cos(f*x + e)^6 - (-2*I*a^4 - I*a^3*b + 19*I*a^2*b^2 + 34*I*a*b^3 + 16*I*b^4)*sqrt(a)*cos(f*x + e
)^6)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_f(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b
)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - (2
*(a^4 - 2*a^3*b - 12*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 - (a^3*b + 11*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^4 - 5*a*b
^3 - 2*(4*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^2*
f*cos(f*x + e)^6)

Sympy [F]

\[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \sec ^{5}{\left (e + f x \right )}\, dx \]

[In]

integrate(sec(f*x+e)**5*(a+b*sec(f*x+e)**2)**(3/2),x)

[Out]

Integral((a + b*sec(e + f*x)**2)**(3/2)*sec(e + f*x)**5, x)

Maxima [F]

\[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^5, x)

Giac [F]

\[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^5, x)

Mupad [F(-1)]

Timed out. \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^5} \,d x \]

[In]

int((a + b/cos(e + f*x)^2)^(3/2)/cos(e + f*x)^5,x)

[Out]

int((a + b/cos(e + f*x)^2)^(3/2)/cos(e + f*x)^5, x)