\(\int \frac {\sin ^{\frac {3}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) [466]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (warning: unable to verify)
Fricas [A] (verification not implemented)
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 23, antiderivative size = 276 \[ \int \frac {\sin ^{\frac {3}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\sqrt {b} \arctan \left (1-\frac {\sqrt {2} \sqrt {b \cos (e+f x)}}{\sqrt {b} \sqrt {\sin (e+f x)}}\right )}{4 \sqrt {2} f \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {\sqrt {b} \arctan \left (1+\frac {\sqrt {2} \sqrt {b \cos (e+f x)}}{\sqrt {b} \sqrt {\sin (e+f x)}}\right )}{4 \sqrt {2} f \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b \cos (e+f x)}}{\left (\sqrt {b}+\sqrt {b} \cot (e+f x)\right ) \sqrt {\sin (e+f x)}}\right )}{4 \sqrt {2} f \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \sqrt {\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.53 \[ \int \frac {\sin ^{\frac {3}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {b \left (-4 \sin ^2(e+f x)+\sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \tan ^2(e+f x)^{3/4}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \tan ^2(e+f x)^{3/4}\right )}{8 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3063, 3042, 3065, 3042, 3055, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3063

\(\displaystyle \frac {1}{4} \int \frac {1}{\sqrt {b \sec (e+f x)} \sqrt {\sin (e+f x)}}dx-\frac {b \sqrt {\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \frac {1}{\sqrt {b \sec (e+f x)} \sqrt {\sin (e+f x)}}dx-\frac {b \sqrt {\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3065

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3055

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

-1/2*(b*((-(ArcTan[1 - (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/(Sqrt[b]*Sqrt[Sin[e 
+ f*x]])]/(Sqrt[2]*Sqrt[b])) + ArcTan[1 + (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/( 
Sqrt[b]*Sqrt[Sin[e + f*x]])]/(Sqrt[2]*Sqrt[b]))/2 + (Log[b + b*Cot[e + f*x 
] - (Sqrt[2]*Sqrt[b]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[e + f*x]]]/(2*Sqrt[2]* 
Sqrt[b]) - Log[b + b*Cot[e + f*x] + (Sqrt[2]*Sqrt[b]*Sqrt[b*Cos[e + f*x]]) 
/Sqrt[Sin[e + f*x]]]/(2*Sqrt[2]*Sqrt[b]))/2))/(f*Sqrt[b*Cos[e + f*x]]*Sqrt 
[b*Sec[e + f*x]]) - (b*Sqrt[Sin[e + f*x]])/(2*f*(b*Sec[e + f*x])^(3/2))
 

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 3055
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), 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*Cos[e + f*x])^(1/k)/(b*Sin[ 
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 [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(450\) vs. \(2(214)=428\).

Time = 0.44 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.63

method result size
default \(\frac {\sqrt {2}\, \left (-4 \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}}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+\left (2-2 \cos \left (f x +e \right )\right ) \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 (-1+\cos \left (f x +e \right )\right ) \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 )+\left (1-\cos \left (f x +e \right )\right ) \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 )+\left (-2+2 \cos \left (f x +e \right )\right ) \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 )\right ) \sin \left (f x +e \right )^{\frac {3}{2}} \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{128 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}}}\, \sqrt {b \sec \left (f x +e \right )}}\) \(451\)

Input:

int(sin(f*x+e)^(3/2)/(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/128/f*2^(1/2)*(-4*(-2*sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos( 
f*x+e)*sin(f*x+e)^2+(2-2*cos(f*x+e))*arctan((-(-2*sin(f*x+e)*cos(f*x+e)/(c 
os(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(-1+cos(f*x+e)))+(-1+cos(f* 
x+e))*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)))+(1-cos(f*x+e))*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*c 
os(f*x+e)-sin(f*x+e)+2)/(-1+cos(f*x+e)))+(-2+2*cos(f*x+e))*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+co 
s(f*x+e))))*sin(f*x+e)^(3/2)/(-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)^3*csc(1/2*f*x+1/2*e)^3
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.51 \[ \int \frac {\sin ^{\frac {3}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {16 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} \sqrt {\sin \left (f x + e\right )} - 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )}}{2 \, \sqrt {b} \sqrt {\sin \left (f x + e\right )}}\right ) + \sqrt {2} \sqrt {b} \arctan \left (\frac {2 \, \cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) + \frac {\sqrt {2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{\sqrt {b}} - 2}{2 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 1\right )}}\right ) + \sqrt {2} \sqrt {b} \arctan \left (-\frac {2 \, \cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - \frac {\sqrt {2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{\sqrt {b}} - 2}{2 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 1\right )}}\right ) - \sqrt {2} \sqrt {b} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{\sqrt {b}} + 4 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 1\right ) + \sqrt {2} \sqrt {b} \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{\sqrt {b}} + 4 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 1\right )}{32 \, b f} \] Input:

integrate(sin(f*x+e)^(3/2)/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

-1/32*(16*sqrt(b/cos(f*x + e))*cos(f*x + e)^2*sqrt(sin(f*x + e)) - 2*sqrt( 
2)*sqrt(b)*arctan(-1/2*sqrt(2)*sqrt(b/cos(f*x + e))*(cos(f*x + e) - sin(f* 
x + e))/(sqrt(b)*sqrt(sin(f*x + e)))) + sqrt(2)*sqrt(b)*arctan(1/2*(2*cos( 
f*x + e)^2 - 2*cos(f*x + e)*sin(f*x + e) + sqrt(2)*sqrt(b/cos(f*x + e))*sq 
rt(sin(f*x + e))/sqrt(b) - 2)/(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e) 
- 1)) + sqrt(2)*sqrt(b)*arctan(-1/2*(2*cos(f*x + e)^2 - 2*cos(f*x + e)*sin 
(f*x + e) - sqrt(2)*sqrt(b/cos(f*x + e))*sqrt(sin(f*x + e))/sqrt(b) - 2)/( 
cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e) - 1)) - sqrt(2)*sqrt(b)*log(2*s 
qrt(2)*(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e))*sqrt(b/cos(f*x + e))*s 
qrt(sin(f*x + e))/sqrt(b) + 4*cos(f*x + e)*sin(f*x + e) + 1) + sqrt(2)*sqr 
t(b)*log(-2*sqrt(2)*(cos(f*x + e)^2 + cos(f*x + e)*sin(f*x + e))*sqrt(b/co 
s(f*x + e))*sqrt(sin(f*x + e))/sqrt(b) + 4*cos(f*x + e)*sin(f*x + e) + 1)) 
/(b*f)
 

Sympy [F]

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

integrate(sin(f*x+e)**(3/2)/(b*sec(f*x+e))**(1/2),x)
 

Output:

Integral(sin(e + f*x)**(3/2)/sqrt(b*sec(e + f*x)), x)
 

Maxima [F]

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

integrate(sin(f*x+e)^(3/2)/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(sin(f*x+e)^(3/2)/(b*sec(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(sin(e + f*x)^(3/2)/(b/cos(e + f*x))^(1/2),x)
 

Output:

int(sin(e + f*x)^(3/2)/(b/cos(e + f*x))^(1/2), x)
 

Reduce [F]

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

int(sin(f*x+e)^(3/2)/(b*sec(f*x+e))^(1/2),x)
 

Output:

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