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

Optimal. Leaf size=158 \[ \frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f} \]

[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)

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Rubi [A]
time = 0.17, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2967, 2895} \begin {gather*} \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}} F\left (\text {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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {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}} F\left (\sin ^{-1}\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*}

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Mathematica [A]
time = 23.75, size = 198, normalized size = 1.25 \begin {gather*} \frac {4 a^2 \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{a-b}} F\left (\sin ^{-1}\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)}} \end {gather*}

Antiderivative was successfully verified.

[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]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(649\) vs. \(2(142)=284\).
time = 3.54, size = 650, normalized size = 4.11

method	result	size
default	\(-\frac {\left (\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}\, \sqrt {\frac {\cos \left (f x +e \right ) a +\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-b \sin \left (f x +e \right )-a}{\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )}}\, \sqrt {\frac {a \left (-1+\cos \left (f x +e \right )\right )}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {-a^{2}+b^{2}}+\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}\, \sqrt {\frac {\cos \left (f x +e \right ) a +\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-b \sin \left (f x +e \right )-a}{\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )}}\, \sqrt {\frac {a \left (-1+\cos \left (f x +e \right )\right )}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) b -\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {2}\, b +\sin \left (f x +e \right ) \sqrt {2}\, b -\cos \left (f x +e \right ) \sqrt {2}\, a +\sqrt {2}\, a \right ) \sin \left (f x +e \right ) \sqrt {2}}{2 f \left (-1+\cos \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {d \sin \left (f x +e \right )}\, \sqrt {a +b \sin \left (f x +e \right )}}\)	\(650\)

Verification of antiderivative is not currently implemented for this CAS.

[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*(sin(f*x+e)*cos(f*x+e)*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f*x+e)/(b+(-a^2+b
^2)^(1/2)))^(1/2)*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2)/sin(f*x+e))^(1/2
)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticF((((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+
e)-cos(f*x+e)*a+a)/sin(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^2+b^2)^(1/2)+sin(f*x+e)*cos(f*x+e)*(((-a^2+b^2)^(1/2)*sin(f*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(f
*x+e)/(b+(-a^2+b^2)^(1/2)))^(1/2)*((cos(f*x+e)*a+(-a^2+b^2)^(1/2)*sin(f*x+e)-b*sin(f*x+e)-a)/(-a^2+b^2)^(1/2)/
sin(f*x+e))^(1/2)*(a*(-1+cos(f*x+e))/(b+(-a^2+b^2)^(1/2))/sin(f*x+e))^(1/2)*EllipticF((((-a^2+b^2)^(1/2)*sin(f
*x+e)+b*sin(f*x+e)-cos(f*x+e)*a+a)/sin(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))*b-sin(f*x+e)*cos(f*x+e)*2^(1/2)*b+sin(f*x+e)*2^(1/2)*b-cos(f*x+e)*2^(1/2)*a+2^(1/2)*a)*
sin(f*x+e)/(-1+cos(f*x+e))/cos(f*x+e)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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