\(\int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx\) [1478]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2967, 2895} \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{\sqrt {d} f} \]

[In]

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

[Out]

(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]
*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f)

Rule 2895

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]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 2967

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] + Dist[2*a*(m/(g^2*(2*m + 1))), Int[(g*Cos[e + f*x])^(p + 2)*((a + b*Sin[e + f*x])^(m - 1)/Sq
rt[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}+\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}-\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.45 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {4 a^2 \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {a+b \sin (e+f x)}{a (-1+\sin (e+f x))}}\right ),\frac {2 a}{a-b}\right ) \sec (e+f x) \sqrt {-\frac {(a+b) \sin (e+f x) (a+b \sin (e+f x))}{a^2 (-1+\sin (e+f x))^2}} \sin ^4\left (\frac {1}{4} (2 e-\pi +2 f x)\right )+(a+b) (a+b \sin (e+f x)) \tan (e+f x)}{(a+b) f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \]

[In]

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

[Out]

(4*a^2*Sqrt[-(((a + b)*Cot[(2*e - Pi + 2*f*x)/4]^2)/(a - b))]*EllipticF[ArcSin[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*Sq
rt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(949\) vs. \(2(142)=284\).

Time = 2.59 (sec) , antiderivative size = 950, normalized size of antiderivative = 6.01

method	result	size
default	\(\frac {\left (\sqrt {-a^{2}+b^{2}}\, \sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\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}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{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 \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, \cos \left (f x +e \right )+\sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\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}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{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 \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}\, \sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\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}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{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 \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}+\sqrt {-\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )-\sqrt {-a^{2}+b^{2}}+b}{\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}}}}\, F\left (\sqrt {\frac {a \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{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 \csc \left (f x +e \right )-a \cot \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, b -\sqrt {2}\, b \cos \left (f x +e \right )+\tan \left (f x +e \right ) \sqrt {2}\, a +\sec \left (f x +e \right ) \sqrt {2}\, b \right ) \sqrt {2}}{2 f \sqrt {a +b \sin \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\)	\(950\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

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