3.7.15 \(\int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx\) [615]

Optimal. Leaf size=183 \[ \frac {2 \sqrt {2} \sqrt [4]{-a+b} \sqrt {c \cos (e+f x)} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b} \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}}{\sqrt [4]{-a+b}}\right )\right |-1\right ) \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}}}{\sqrt [4]{a+b} c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}} \]

[Out]

2*(-a+b)^(1/4)*EllipticF((a+b)^(1/4)*((1+cos(f*x+e)+sin(f*x+e))/(1+cos(f*x+e)-sin(f*x+e)))^(1/2)/(-a+b)^(1/4),
I)*2^(1/2)*(c*cos(f*x+e))^(1/2)*((a+b*sin(f*x+e))/(a-b)/(1-sin(f*x+e)))^(1/2)/(a+b)^(1/4)/c/f/((1+cos(f*x+e)+s
in(f*x+e))/(1+cos(f*x+e)-sin(f*x+e)))^(1/2)/(a+b*sin(f*x+e))^(1/2)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(374\) vs. \(2(183)=366\).
time = 0.31, antiderivative size = 374, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2776, 226} \begin {gather*} \frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {c \cos (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1) \left (\frac {\sqrt {a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt {a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right )^2}} \left (\frac {\sqrt {a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt {a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a+b} \sqrt {\frac {\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right )}{c f \sqrt [4]{a+b} \sqrt {\frac {\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1)}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),x]

[Out]

(Sqrt[2]*(a - b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[2*ArcTan[((a + b)^(1/4)*Sqrt[(1 + Cos[e + f*x] + Sin[e +
f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])/(a - b)^(1/4)], 1/2]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e
+ f*x]))]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x])*(1 + (Sqrt[a + b]*(1 + Co
s[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x])))^2)]*(1 + (Sqrt[a + b]*(1 + Cos[e
+ f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x]))))/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e +
f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((
a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x]))])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 2776

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[2*Sq
rt[2]*Sqrt[g*Cos[e + f*x]]*(Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]/(f*g*Sqrt[a + b*Sin[e + f*
x]]*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])), Subst[Int[1/Sqrt[1 + (a + b)*
(x^4/(a - b))], x], x, Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]], x] /; FreeQ
[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {c \cos (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx &=\frac {\left (2 \sqrt {2} \sqrt {c \cos (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(a+b) x^4}{a-b}}} \, dx,x,\sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}\right )}{c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}\\ &=\frac {\sqrt {2} \sqrt [4]{a-b} \sqrt {c \cos (e+f x)} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b} \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}}{\sqrt [4]{a-b}}\right )|\frac {1}{2}\right ) \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\cos (f x) \sin (e)-\cos (e) \sin (f x)) \left (1+\frac {\sqrt {a+b} (1+\cos (e+f x)+\sin (e+f x))}{\sqrt {a-b} (1+\cos (e+f x)-\sin (e+f x))}\right )^2}} \left (1+\frac {\sqrt {a+b} (1+\cos (e+f x)+\sin (e+f x))}{\sqrt {a-b} (1+\cos (e+f x)-\sin (e+f x))}\right )}{\sqrt [4]{a+b} c f \sqrt {\frac {1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a-b) (1-\cos (f x) \sin (e)-\cos (e) \sin (f x))}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.22, size = 117, normalized size = 0.64 \begin {gather*} -\frac {2 c \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {2 (a+b \sin (e+f x))}{(a-b) (-1+\sin (e+f x))}\right ) (-1+\sin (e+f x)) \left (\frac {(a+b) (1+\sin (e+f x))}{(a-b) (-1+\sin (e+f x))}\right )^{3/4} \sqrt {a+b \sin (e+f x)}}{(a+b) f (c \cos (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),x]

[Out]

(-2*c*Hypergeometric2F1[1/2, 3/4, 3/2, (-2*(a + b*Sin[e + f*x]))/((a - b)*(-1 + Sin[e + f*x]))]*(-1 + Sin[e +
f*x])*(((a + b)*(1 + Sin[e + f*x]))/((a - b)*(-1 + Sin[e + f*x])))^(3/4)*Sqrt[a + b*Sin[e + f*x]])/((a + b)*f*
(c*Cos[e + f*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(750\) vs. \(2(163)=326\).
time = 11.44, size = 751, normalized size = 4.10

method	result	size
default	\(\frac {4 \EllipticF \left (\sqrt {\frac {\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}-a \sin \left (f x +e \right )-b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}-b \right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}{\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}+a \sin \left (f x +e \right )+b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}}, \sqrt {\frac {\left (b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}{\left (-a -b +\sqrt {-a^{2}+b^{2}}\right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}}\right ) \sqrt {\frac {\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}-a \sin \left (f x +e \right )-b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}-b \right ) \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}{\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}+a \sin \left (f x +e \right )+b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}}\, \sqrt {\frac {\left (1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right ) \sqrt {-a^{2}+b^{2}}\, a}{\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}+a \sin \left (f x +e \right )+b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b \right ) \left (a -b +\sqrt {-a^{2}+b^{2}}\right )}}\, \sqrt {-\frac {\left (1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right ) \sqrt {-a^{2}+b^{2}}\, a}{\left (\cos \left (f x +e \right ) \sqrt {-a^{2}+b^{2}}+a \sin \left (f x +e \right )+b \cos \left (f x +e \right )+\sqrt {-a^{2}+b^{2}}+b \right ) \left (-a -b +\sqrt {-a^{2}+b^{2}}\right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\sqrt {-a^{2}+b^{2}}\, \cos \left (f x +e \right ) a -\sqrt {-a^{2}+b^{2}}\, \cos \left (f x +e \right ) b -a \sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right ) a^{2}-\cos \left (f x +e \right ) b^{2}+a^{2} \sin \left (f x +e \right )-a b \sin \left (f x +e \right )-b \sqrt {-a^{2}+b^{2}}+a b -b^{2}\right )}{f \sqrt {a +b \sin \left (f x +e \right )}\, \sin \left (f x +e \right )^{4} \sqrt {c \cos \left (f x +e \right )}\, \sqrt {-a^{2}+b^{2}}\, \left (-a +b +\sqrt {-a^{2}+b^{2}}\right )}\)	\(751\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/f*EllipticF(((cos(f*x+e)*(-a^2+b^2)^(1/2)-a*sin(f*x+e)-b*cos(f*x+e)+(-a^2+b^2)^(1/2)-b)/(cos(f*x+e)*(-a^2+b^
2)^(1/2)+a*sin(f*x+e)+b*cos(f*x+e)+(-a^2+b^2)^(1/2)+b)*(-a+b+(-a^2+b^2)^(1/2))/(a-b+(-a^2+b^2)^(1/2)))^(1/2),(
(b+(-a^2+b^2)^(1/2)+a)*(a-b+(-a^2+b^2)^(1/2))/(-a-b+(-a^2+b^2)^(1/2))/(-a+b+(-a^2+b^2)^(1/2)))^(1/2))*((cos(f*
x+e)*(-a^2+b^2)^(1/2)-a*sin(f*x+e)-b*cos(f*x+e)+(-a^2+b^2)^(1/2)-b)/(cos(f*x+e)*(-a^2+b^2)^(1/2)+a*sin(f*x+e)+
b*cos(f*x+e)+(-a^2+b^2)^(1/2)+b)*(-a+b+(-a^2+b^2)^(1/2))/(a-b+(-a^2+b^2)^(1/2)))^(1/2)*((1+cos(f*x+e)+sin(f*x+
e))/(cos(f*x+e)*(-a^2+b^2)^(1/2)+a*sin(f*x+e)+b*cos(f*x+e)+(-a^2+b^2)^(1/2)+b)*(-a^2+b^2)^(1/2)*a/(a-b+(-a^2+b
^2)^(1/2)))^(1/2)*(-(1+cos(f*x+e)-sin(f*x+e))/(cos(f*x+e)*(-a^2+b^2)^(1/2)+a*sin(f*x+e)+b*cos(f*x+e)+(-a^2+b^2
)^(1/2)+b)*(-a^2+b^2)^(1/2)*a/(-a-b+(-a^2+b^2)^(1/2)))^(1/2)*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^2*((-a^2+b^2)^(1
/2)*cos(f*x+e)*a-(-a^2+b^2)^(1/2)*cos(f*x+e)*b-a*(-a^2+b^2)^(1/2)*sin(f*x+e)+cos(f*x+e)*a^2-cos(f*x+e)*b^2+a^2
*sin(f*x+e)-a*b*sin(f*x+e)-b*(-a^2+b^2)^(1/2)+a*b-b^2)/(a+b*sin(f*x+e))^(1/2)/sin(f*x+e)^4/(c*cos(f*x+e))^(1/2
)/(-a^2+b^2)^(1/2)/(-a+b+(-a^2+b^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(1/(c*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(c*cos(f*x + e))*sqrt(b*sin(f*x + e) + a)/(b*c*cos(f*x + e)*sin(f*x + e) + a*c*cos(f*x + e)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c \cos {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(1/2),x)

[Out]

Integral(1/(sqrt(c*cos(e + f*x))*sqrt(a + b*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(1/(c*cos(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*cos(f*x + e))*sqrt(b*sin(f*x + e) + a)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {c\,\cos \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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