\(\int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx\) [473]

Optimal result

Integrand size = 25, antiderivative size = 354 \[ \int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {3 a^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} b^{5/2} f}+\frac {3 a^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} b^{5/2} f}-\frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)} \left (\sqrt {a}+\sqrt {a} \tan (e+f x)\right )}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} b^{5/2} f}-\frac {a (a \sin (e+f x))^{3/2}}{16 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}} \] Output:

-3/64*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^(5/2)/ 
f+3/64*a^(5/2)*arctan(1+2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*co 
s(f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)*2^(1/2)/b^(5/2) 
/f-3/64*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^(5/2)/f-1/16*a*(a*sin(f*x+e))^(3/2)/b/f/(b*sec(f*x+e))^(1/ 
2)+1/4*(a*sin(f*x+e))^(7/2)/a/b/f/(b*sec(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.47 \[ \int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx=\frac {a^3 \left (4-6 \cos (2 (e+f x))+2 \cos (4 (e+f x))+3 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{64 b f \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}} \] Input:

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

Output:

(a^3*(4 - 6*Cos[2*(e + f*x)] + 2*Cos[4*(e + f*x)] + 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)))/(64*b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a* 
Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.11, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3062, 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.

\(\displaystyle \int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 3062

\(\displaystyle \frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2}dx}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2}dx}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3063

\(\displaystyle \frac {\frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3}{4} a^2 \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3065

\(\displaystyle \frac {\frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3}{4} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}dx-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 3054

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {a \tan (e+f x)}{b \left (\tan ^2(e+f x) a^2+a^2\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\int \frac {\tan (e+f x) a+a}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}+\frac {\int \frac {1}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int \frac {1}{-\frac {a \tan (e+f x)}{b}-1}d\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\int \frac {a-a \tan (e+f x)}{\tan ^2(e+f x) a^2+a^2}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{\sqrt {b} \left (\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}\right )}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {a}-\frac {2 \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}-\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {2} \sqrt {a} b}+\frac {\int \frac {\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{\frac {\tan (e+f x) a}{b}+\frac {a}{b}+\frac {\sqrt {2} \sqrt {a \sin (e+f x)} \sqrt {a}}{\sqrt {b} \sqrt {b \cos (e+f x)}}}d\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}}{2 \sqrt {a} b}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {3 a^3 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+a \tan (e+f x)+a\right )}{2 \sqrt {2} \sqrt {a} \sqrt {b}}}{2 b}\right )}{2 f}-\frac {a b (a \sin (e+f x))^{3/2}}{2 f \sqrt {b \sec (e+f x)}}}{8 b^2}+\frac {(a \sin (e+f x))^{7/2}}{4 a b f \sqrt {b \sec (e+f x)}}\)

Input:

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

Output:

(a*Sin[e + f*x])^(7/2)/(4*a*b*f*Sqrt[b*Sec[e + f*x]]) + ((3*a^3*b*Sqrt[b*C 
os[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])) + ArcTan[1 + (Sqrt[2]* 
Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]/(Sqrt[2]*Sqr 
t[a]*Sqrt[b]))/(2*b) - (-1/2*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]]/(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*Sec[e + f*x]]))/(8 
*b^2)
 

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 3062

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

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 = 4.66 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.28

method	result	size
default	\(-\frac {\sqrt {2}\, a^{2} \left (-3 \ln \left (-\frac {\cos \left (f x +e \right ) \cot \left (f x +e \right )-2 \cot \left (f x +e \right )-2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\csc \left (f x +e \right )-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+2}{-1+\cos \left (f x +e \right )}\right )+3 \ln \left (-\frac {\cos \left (f x +e \right ) \cot \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\csc \left (f x +e \right )-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+2}{-1+\cos \left (f x +e \right )}\right )+6 \arctan \left (\frac {-\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{-1+\cos \left (f x +e \right )}\right )-6 \arctan \left (\frac {\sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{-1+\cos \left (f x +e \right )}\right )+\left (16 \cos \left (f x +e \right )^{3}+16 \cos \left (f x +e \right )^{2}-12 \cos \left (f x +e \right )-12\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\right ) \sqrt {a \sin \left (f x +e \right )}\, \sin \left (f x +e \right )^{6} \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{8} \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{16384 f b \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b \sec \left (f x +e \right )}}\)	\(454\)

Input:

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

Output:

-1/16384/f*2^(1/2)*a^2/b*(-3*ln(-(cos(f*x+e)*cot(f*x+e)-2*cot(f*x+e)-2*(-2 
*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+csc(f*x+e)-2*cos 
(f*x+e)-sin(f*x+e)+2)/(-1+cos(f*x+e)))+3*ln(-(cos(f*x+e)*cot(f*x+e)-2*cot( 
f*x+e)+2*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+csc( 
f*x+e)-2*cos(f*x+e)-sin(f*x+e)+2)/(-1+cos(f*x+e)))+6*arctan((-(-2*sin(f*x+ 
e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(-1+cos(f*x 
+e)))-6*arctan(((-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+ 
e)+cos(f*x+e)-1)/(-1+cos(f*x+e)))+(16*cos(f*x+e)^3+16*cos(f*x+e)^2-12*cos( 
f*x+e)-12)*sin(f*x+e)*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*( 
a*sin(f*x+e))^(1/2)*sin(f*x+e)^6/(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2) 
^(1/2)/(b*sec(f*x+e))^(1/2)*sec(1/2*f*x+1/2*e)^8*csc(1/2*f*x+1/2*e)^6
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.54 \[ \int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {6 \, \sqrt {2} a^{2} b \sqrt {\frac {a}{b}} \arctan \left (-\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {a}{b}} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \cos \left (f x + e\right ) - a \sin \left (f x + e\right )}\right ) + 3 \, \sqrt {2} a^{2} b \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + \sqrt {2} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {a}{b}} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 2 \, a}{2 \, {\left (a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a\right )}}\right ) + 3 \, \sqrt {2} a^{2} b \sqrt {\frac {a}{b}} \arctan \left (-\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - \sqrt {2} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {a}{b}} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 2 \, a}{2 \, {\left (a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a\right )}}\right ) + 3 \, \sqrt {2} a^{2} b \sqrt {\frac {a}{b}} \log \left (2 \, \sqrt {2} {\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 {a}{b}} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 4 \, a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a\right ) - 3 \, \sqrt {2} a^{2} b \sqrt {\frac {a}{b}} \log \left (-2 \, \sqrt {2} {\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 {a}{b}} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 4 \, a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a\right ) + 16 \, {\left (4 \, a^{2} \cos \left (f x + e\right )^{3} - 3 \, a^{2} \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{256 \, b^{2} f} \] Input:

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

Output:

-1/256*(6*sqrt(2)*a^2*b*sqrt(a/b)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e))*sqr 
t(a/b)*sqrt(b/cos(f*x + e))*cos(f*x + e)/(a*cos(f*x + e) - a*sin(f*x + e)) 
) + 3*sqrt(2)*a^2*b*sqrt(a/b)*arctan(1/2*(2*a*cos(f*x + e)^2 - 2*a*cos(f*x 
 + e)*sin(f*x + e) + sqrt(2)*sqrt(a*sin(f*x + e))*sqrt(a/b)*sqrt(b/cos(f*x 
 + e)) - 2*a)/(a*cos(f*x + e)^2 + a*cos(f*x + e)*sin(f*x + e) - a)) + 3*sq 
rt(2)*a^2*b*sqrt(a/b)*arctan(-1/2*(2*a*cos(f*x + e)^2 - 2*a*cos(f*x + e)*s 
in(f*x + e) - sqrt(2)*sqrt(a*sin(f*x + e))*sqrt(a/b)*sqrt(b/cos(f*x + e)) 
- 2*a)/(a*cos(f*x + e)^2 + a*cos(f*x + e)*sin(f*x + e) - a)) + 3*sqrt(2)*a 
^2*b*sqrt(a/b)*log(2*sqrt(2)*(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e))* 
sqrt(a*sin(f*x + e))*sqrt(a/b)*sqrt(b/cos(f*x + e)) + 4*a*cos(f*x + e)*sin 
(f*x + e) + a) - 3*sqrt(2)*a^2*b*sqrt(a/b)*log(-2*sqrt(2)*(cos(f*x + e)^2 
+ cos(f*x + e)*sin(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(a/b)*sqrt(b/cos(f*x 
 + e)) + 4*a*cos(f*x + e)*sin(f*x + e) + a) + 16*(4*a^2*cos(f*x + e)^3 - 3 
*a^2*cos(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))*sin(f*x + e)) 
/(b^2*f)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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