\(\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx\) [451]

Optimal result

Integrand size = 25, antiderivative size = 315 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=-\frac {3 a^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} \sqrt {b} f}+\frac {3 a^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} \sqrt {b} f}-\frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)} \left (\sqrt {a}+\sqrt {a} \tan (e+f x)\right )}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} \sqrt {b} f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}} \] Output:

-3/8*a^(5/2)*arctan(1-2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*cos( 
f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)*2^(1/2)/b^(1/2)/f 
+3/8*a^(5/2)*arctan(1+2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*cos( 
f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)*2^(1/2)/b^(1/2)/f 
-3/8*a^(5/2)*arctanh(2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/(b*cos(f*x+e))^( 
1/2)/(a^(1/2)+a^(1/2)*tan(f*x+e)))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/ 
2)*2^(1/2)/b^(1/2)/f-1/2*a*b*(a*sin(f*x+e))^(3/2)/f/(b*sec(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.50 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=-\frac {a^2 \cot (e+f x) \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \left (4 \sin ^2(e+f x)-3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{8 f} \] Input:

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

Output:

-1/8*(a^2*Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]*(4*Sin[e 
+ f*x]^2 - 3*Sqrt[2]*ArcTan[(-1 + Sqrt[Tan[e + f*x]^2])/(Sqrt[2]*(Tan[e + 
f*x]^2)^(1/4))]*(Tan[e + f*x]^2)^(1/4) + 3*Sqrt[2]*ArcTanh[(Sqrt[2]*(Tan[e 
 + f*x]^2)^(1/4))/(1 + Sqrt[Tan[e + f*x]^2])]*(Tan[e + f*x]^2)^(1/4)))/f
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3063, 3042, 3065, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int (a \sin (e+f x))^{5/2} \sqrt {b \sec (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x))^{5/2} \sqrt {b \sec (e+f x)}dx\)

\(\Big \downarrow \) 3063

\(\displaystyle \frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3065

\(\displaystyle \frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3054

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {a \tan (e+f x)}{b \left (\tan ^2(e+f x) a^2+a^2\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\int \frac {\tan (e+f x) a+a}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}+\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} b}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {a} b}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}\)

Input:

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

Output:

(3*a^3*b*Sqrt[b*Cos[e + f*x]]*((-(ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e 
 + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]/(Sqrt[2]*Sqrt[a]*Sqrt[b])) + Arc 
Tan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x 
]])]/(Sqrt[2]*Sqrt[a]*Sqrt[b]))/(2*b) - (-1/2*Log[a - (Sqrt[2]*Sqrt[a]*Sqr 
t[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + a*Tan[e + f*x]]/(Sqrt[2] 
*Sqrt[a]*Sqrt[b]) + Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[a*Sin[e + f*x]]) 
/Sqrt[b*Cos[e + f*x]] + a*Tan[e + f*x]]/(2*Sqrt[2]*Sqrt[a]*Sqrt[b]))/(2*b) 
)*Sqrt[b*Sec[e + f*x]])/(2*f) - (a*b*(a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[b*S 
ec[e + f*x]])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3054

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f)   Subst[Int[x^(k 
*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + 
 f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] 
&& LtQ[m, 1]
 

rule 3063

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(-a)*b*(a*Sin[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n 
- 1)/(f*(m - n))), x] + Simp[a^2*((m - 1)/(m - n))   Int[(a*Sin[e + f*x])^( 
m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] 
 && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
 

rule 3065

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n   Int[(a*Sin[e 
+ f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && Int 
egerQ[m - 1/2] && IntegerQ[n - 1/2]
 

Maple [A] (warning: unable to verify)

Time = 13.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {\sqrt {2}\, a^{2} \left (3 \ln \left (-\frac {\cos \left (f x +e \right ) \cot \left (f x +e \right )+2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-2 \cot \left (f x +e \right )-\sin \left (f x +e \right )-2 \cos \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )-3 \ln \left (\frac {-\cos \left (f x +e \right ) \cot \left (f x +e \right )+2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+2 \cot \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )-\csc \left (f x +e \right )-2}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )+\left (4 \cos \left (f x +e \right )+4\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\right ) \sqrt {a \sin \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} \sqrt {b \sec \left (f x +e \right )}\, \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{256 f \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\)	\(438\)

Input:

int((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/256/f*2^(1/2)*a^2*(3*ln(-(cos(f*x+e)*cot(f*x+e)+2*(-2*sin(f*x+e)/(cos(f 
*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)-2*cot(f*x+e)-sin(f*x+e)-2*cos(f*x+ 
e)+csc(f*x+e)+2)/(cos(f*x+e)-1))-3*ln((-cos(f*x+e)*cot(f*x+e)+2*(-2*sin(f* 
x+e)/(cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)+2*cot(f*x+e)+sin(f*x+e) 
+2*cos(f*x+e)-csc(f*x+e)-2)/(cos(f*x+e)-1))-6*arctan(((-2*sin(f*x+e)/(cos( 
f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))-6*a 
rctan(((-2*sin(f*x+e)/(cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sin(f*x+e)+cos(f* 
x+e)-1)/(cos(f*x+e)-1))+(4*cos(f*x+e)+4)*sin(f*x+e)*(-2*sin(f*x+e)/(cos(f* 
x+e)+1)^2*cos(f*x+e))^(1/2))*(a*sin(f*x+e))^(1/2)*cos(f*x+e)*sin(f*x+e)^3* 
(b*sec(f*x+e))^(1/2)/(-sin(f*x+e)/(cos(f*x+e)+1)^2*cos(f*x+e))^(1/2)*sec(1 
/2*f*x+1/2*e)^5*csc(1/2*f*x+1/2*e)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (241) = 482\).

Time = 0.14 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.66 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=-\frac {16 \, \sqrt {a \sin \left (f x + e\right )} a^{2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 6 \, \sqrt {2} \sqrt {a b} a^{2} \arctan \left (-\frac {\sqrt {2} \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a b \cos \left (f x + e\right ) - a b \sin \left (f x + e\right )}\right ) + 3 \, \sqrt {2} \sqrt {a b} a^{2} \arctan \left (\frac {2 \, a b \cos \left (f x + e\right )^{2} - 2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a b + \sqrt {2} \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{2 \, {\left (a b \cos \left (f x + e\right )^{2} + a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a b\right )}}\right ) + 3 \, \sqrt {2} \sqrt {a b} a^{2} \arctan \left (-\frac {2 \, a b \cos \left (f x + e\right )^{2} - 2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a b - \sqrt {2} \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{2 \, {\left (a b \cos \left (f x + e\right )^{2} + a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a b\right )}}\right ) + 3 \, \sqrt {2} \sqrt {a b} a^{2} \log \left (4 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, \sqrt {2} \sqrt {a b} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + a b\right ) - 3 \, \sqrt {2} \sqrt {a b} a^{2} \log \left (4 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \sqrt {2} \sqrt {a b} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + a b\right )}{32 \, f} \] Input:

integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(5/2),x, algorithm="fricas")
 

Output:

-1/32*(16*sqrt(a*sin(f*x + e))*a^2*sqrt(b/cos(f*x + e))*cos(f*x + e)*sin(f 
*x + e) + 6*sqrt(2)*sqrt(a*b)*a^2*arctan(-sqrt(2)*sqrt(a*b)*sqrt(a*sin(f*x 
 + e))*sqrt(b/cos(f*x + e))*cos(f*x + e)/(a*b*cos(f*x + e) - a*b*sin(f*x + 
 e))) + 3*sqrt(2)*sqrt(a*b)*a^2*arctan(1/2*(2*a*b*cos(f*x + e)^2 - 2*a*b*c 
os(f*x + e)*sin(f*x + e) - 2*a*b + sqrt(2)*sqrt(a*b)*sqrt(a*sin(f*x + e))* 
sqrt(b/cos(f*x + e)))/(a*b*cos(f*x + e)^2 + a*b*cos(f*x + e)*sin(f*x + e) 
- a*b)) + 3*sqrt(2)*sqrt(a*b)*a^2*arctan(-1/2*(2*a*b*cos(f*x + e)^2 - 2*a* 
b*cos(f*x + e)*sin(f*x + e) - 2*a*b - sqrt(2)*sqrt(a*b)*sqrt(a*sin(f*x + e 
))*sqrt(b/cos(f*x + e)))/(a*b*cos(f*x + e)^2 + a*b*cos(f*x + e)*sin(f*x + 
e) - a*b)) + 3*sqrt(2)*sqrt(a*b)*a^2*log(4*a*b*cos(f*x + e)*sin(f*x + e) + 
 2*sqrt(2)*sqrt(a*b)*(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e))*sqrt(a*s 
in(f*x + e))*sqrt(b/cos(f*x + e)) + a*b) - 3*sqrt(2)*sqrt(a*b)*a^2*log(4*a 
*b*cos(f*x + e)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*b)*(cos(f*x + e)^2 + cos(f 
*x + e)*sin(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) + a*b))/f
 

Sympy [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate((b*sec(f*x+e))**(1/2)*(a*sin(f*x+e))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((a*sin(e + f*x))^(5/2)*(b/cos(e + f*x))^(1/2),x)
 

Output:

int((a*sin(e + f*x))^(5/2)*(b/cos(e + f*x))^(1/2), x)
 

Reduce [F]

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

int((b*sec(f*x+e))^(1/2)*(a*sin(f*x+e))^(5/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(b)*sqrt(a)*int(sqrt(sin(e + f*x))*sqrt(sec(e + f*x))*sin(e + f*x)**2, 
x)*a**2