\(\int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx\) [321]

Optimal result

Integrand size = 26, antiderivative size = 108 \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx=\frac {2 a^2 (1+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1+\sec (e+f x)\right ) \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}} \]

[Out]

2*a^2*(1+4*n)*hypergeom([1/2, 1-n],[3/2],1+sec(f*x+e))*tan(f*x+e)/f/(1+2*n)/(a-a*sec(f*x+e))^(1/2)+2*a^2*(-sec
(f*x+e))^n*tan(f*x+e)/f/(1+2*n)/(a-a*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3899, 21, 3891, 67} \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx=\frac {2 a^2 (4 n+1) \tan (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},\sec (e+f x)+1\right )}{f (2 n+1) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt {a-a \sec (e+f x)}} \]

[In]

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

[Out]

(2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Se
c[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 3891

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[a^2*d*(
Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])), Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x]
, x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]

Rule 3899

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-b^2)
*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*(m + n - 1))), x] + Dist[b/(m + n - 1), Int[
(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /;
FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}}-\frac {(2 a) \int \frac {(-\sec (e+f x))^n \left (-a \left (\frac {1}{2}+2 n\right )+a \left (\frac {1}{2}+2 n\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}} \, dx}{1+2 n} \\ & = \frac {2 a^2 (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {(a (1+4 n)) \int (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \, dx}{1+2 n} \\ & = \frac {2 a^2 (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {\left (a^3 (1+4 n) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(-x)^{-1+n}}{\sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{f (1+2 n) \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^2 (1+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1+\sec (e+f x)\right ) \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt {a-a \sec (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.32 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.93 \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx=\frac {2^{-\frac {3}{2}+n} e^{-\frac {1}{2} i (e+f x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{\frac {1}{2}+n} \left (1+e^{2 i (e+f x)}\right )^{\frac {1}{2}+n} \csc ^3\left (\frac {1}{2} (e+f x)\right ) \left (-\frac {\operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {3}{2}+n,\frac {2+n}{2},-e^{2 i (e+f x)}\right )}{n}+\frac {3 e^{i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {3}{2}+n,\frac {3+n}{2},-e^{2 i (e+f x)}\right )}{1+n}-\frac {3 e^{2 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+n,\frac {2+n}{2},\frac {4+n}{2},-e^{2 i (e+f x)}\right )}{2+n}+\frac {e^{3 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+n,\frac {3+n}{2},\frac {5+n}{2},-e^{2 i (e+f x)}\right )}{3+n}\right ) (-\sec (e+f x))^n \sec ^{-\frac {3}{2}-n}(e+f x) (a-a \sec (e+f x))^{3/2}}{f} \]

[In]

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

[Out]

(2^(-3/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1/2 + n)*(1 + E^((2*I)*(e + f*x)))^(1/2 + n)*Csc[(e
+ f*x)/2]^3*(-(Hypergeometric2F1[n/2, 3/2 + n, (2 + n)/2, -E^((2*I)*(e + f*x))]/n) + (3*E^(I*(e + f*x))*Hyper
geometric2F1[(1 + n)/2, 3/2 + n, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) - (3*E^((2*I)*(e + f*x))*Hypergeome
tric2F1[3/2 + n, (2 + n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))])/(2 + n) + (E^((3*I)*(e + f*x))*Hypergeometric2F1
[3/2 + n, (3 + n)/2, (5 + n)/2, -E^((2*I)*(e + f*x))])/(3 + n))*(-Sec[e + f*x])^n*Sec[e + f*x]^(-3/2 - n)*(a -
a*Sec[e + f*x])^(3/2))/(E^((I/2)*(e + f*x))*f)

Maple [F]

\[\int \left (-\sec \left (f x +e \right )\right )^{n} \left (a -a \sec \left (f x +e \right )\right )^{\frac {3}{2}}d x\]

[In]

int((-sec(f*x+e))^n*(a-a*sec(f*x+e))^(3/2),x)

[Out]

int((-sec(f*x+e))^n*(a-a*sec(f*x+e))^(3/2),x)

Fricas [F]

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

[In]

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

[Out]

integral(-(a*sec(f*x + e) - a)*sqrt(-a*sec(f*x + e) + a)*(-sec(f*x + e))^n, x)

Sympy [F]

\[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx=\int \left (- \sec {\left (e + f x \right )}\right )^{n} \left (- a \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((-sec(f*x+e))**n*(a-a*sec(f*x+e))**(3/2),x)

[Out]

Integral((-sec(e + f*x))**n*(-a*(sec(e + f*x) - 1))**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((-a*sec(f*x + e) + a)^(3/2)*(-sec(f*x + e))^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((-a*sec(f*x + e) + a)^(3/2)*(-sec(f*x + e))^n, x)

Mupad [F(-1)]

Timed out. \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx=\int {\left (a-\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (-\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

[In]

int((a - a/cos(e + f*x))^(3/2)*(-1/cos(e + f*x))^n,x)

[Out]

int((a - a/cos(e + f*x))^(3/2)*(-1/cos(e + f*x))^n, x)