\(\int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx\) [1474]

Optimal result

Integrand size = 35, antiderivative size = 158 \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f} \] Output:

sec(f*x+e)*(d*sin(f*x+e))^(1/2)*(a+b*sin(f*x+e))^(1/2)/d/f-(a+b)^(1/2)*(a* 
(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*EllipticF(d^(1/ 
2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),(-(a+b)/(a-b))^ 
(1/2))*tan(f*x+e)/d^(1/2)/f
 

Mathematica [A] (warning: unable to verify)

Time = 6.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {4 a^2 \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {a+b \sin (e+f x)}{a (-1+\sin (e+f x))}}\right ),\frac {2 a}{a-b}\right ) \sec (e+f x) \sqrt {-\frac {(a+b) \sin (e+f x) (a+b \sin (e+f x))}{a^2 (-1+\sin (e+f x))^2}} \sin ^4\left (\frac {1}{4} (2 e-\pi +2 f x)\right )+(a+b) (a+b \sin (e+f x)) \tan (e+f x)}{(a+b) f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \] Input:

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

Output:

(4*a^2*Sqrt[-(((a + b)*Cot[(2*e - Pi + 2*f*x)/4]^2)/(a - b))]*EllipticF[Ar 
cSin[Sqrt[-((a + b*Sin[e + f*x])/(a*(-1 + Sin[e + f*x])))]], (2*a)/(a - b) 
]*Sec[e + f*x]*Sqrt[-(((a + b)*Sin[e + f*x]*(a + b*Sin[e + f*x]))/(a^2*(-1 
 + Sin[e + f*x])^2))]*Sin[(2*e - Pi + 2*f*x)/4]^4 + (a + b)*(a + b*Sin[e + 
 f*x])*Tan[e + f*x])/((a + b)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f* 
x]])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3042, 3367, 3042, 3295}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{\cos (e+f x)^2 \sqrt {d \sin (e+f x)}}dx\)

\(\Big \downarrow \) 3367

\(\displaystyle \frac {1}{2} a \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx+\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} a \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx+\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}\)

\(\Big \downarrow \) 3295

\(\displaystyle \frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{\sqrt {d} f}\)

Input:

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

Output:

(Sec[e + f*x]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(d*f) - (Sqrt 
[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/( 
a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*S 
qrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f)
 

Defintions of rubi rules used

rule 3042

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

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3367

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*(g*Cos 
[e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^m/(d*f*g*(2*m 
 + 1))), x] + Simp[2*a*(m/(g^2*(2*m + 1)))   Int[(g*Cos[e + f*x])^(p + 2)*( 
(a + b*Sin[e + f*x])^(m - 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, 
e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(139)=278\).

Time = 3.96 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.34

method	result	size
default	\(\frac {\left (\left (1+\cos \left (f x +e \right )\right ) \sqrt {-a^{2}+b^{2}}\, \sqrt {\frac {a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}-b -a \csc \left (f x +e \right )}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {a \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}-b -a \csc \left (f x +e \right )}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {-\frac {a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}-b -a \csc \left (f x +e \right )}{b +\sqrt {-a^{2}+b^{2}}}}+\left (1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}-b -a \csc \left (f x +e \right )}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {a \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}-b -a \csc \left (f x +e \right )}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {-\frac {a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}-b -a \csc \left (f x +e \right )}{b +\sqrt {-a^{2}+b^{2}}}}\, b +\tan \left (f x +e \right ) \sqrt {2}\, a +\sin \left (f x +e \right ) \tan \left (f x +e \right ) b \sqrt {2}\right ) \sqrt {2}}{2 f \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\)	\(528\)

Input:

int(sec(f*x+e)^2*(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

1/2/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*((1+cos(f*x+e))*(-a^2+b^ 
2)^(1/2)*(1/(-a^2+b^2)^(1/2)*(a*cot(f*x+e)+(-a^2+b^2)^(1/2)-b-a*csc(f*x+e) 
))^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*a*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*Ellipti 
cF((-(a*cot(f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^ 
(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*(-(a*cot( 
f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2)+(1+cos 
(f*x+e))*(1/(-a^2+b^2)^(1/2)*(a*cot(f*x+e)+(-a^2+b^2)^(1/2)-b-a*csc(f*x+e) 
))^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*a*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*Ellipti 
cF((-(a*cot(f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^ 
(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*(-(a*cot( 
f*x+e)-(-a^2+b^2)^(1/2)-b-a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2)*b+tan( 
f*x+e)*2^(1/2)*a+sin(f*x+e)*tan(f*x+e)*b*2^(1/2))*2^(1/2)
 

Fricas [F]

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

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

Output:

integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))*sec(f*x + e)^2/(d*s 
in(f*x + e)), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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