3.591 \(\int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=79 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]

[Out]

-arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))*2^(1/2)/f/a
^(1/2)/(c-d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2782, 208} \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x
]])])/(Sqrt[a]*Sqrt[c - d]*f))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2782

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c
+ d*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx &=-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{f}\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} \sqrt {c-d} f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 4.10, size = 283, normalized size = 3.58 \[ \frac {\log \left (\tan \left (\frac {1}{2} (e+f x)\right )+1\right )-\log \left ((d-c) \tan \left (\frac {1}{2} (e+f x)\right )+2 \sqrt {c-d} \sqrt {\frac {1}{\cos (e+f x)+1}} \sqrt {c+d \sin (e+f x)}+c-d\right )}{f \sqrt {a (\sin (e+f x)+1)} \sqrt {c+d \sin (e+f x)} \left (\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2 \tan \left (\frac {1}{2} (e+f x)\right )+2}-\frac {\frac {\sqrt {c-d} \left (\frac {1}{\cos (e+f x)+1}\right )^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt {c+d \sin (e+f x)}}-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{(d-c) \tan \left (\frac {1}{2} (e+f x)\right )+2 \sqrt {c-d} \sqrt {\frac {1}{\cos (e+f x)+1}} \sqrt {c+d \sin (e+f x)}+c-d}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Log[1 + Tan[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]
+ (-c + d)*Tan[(e + f*x)/2]])/(f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]]*(Sec[(e + f*x)/2]^2/(2 +
2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d
*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)
]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2])))

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 475, normalized size = 6.01 \[ \left [\frac {\sqrt {2} \log \left (\frac {{\left (c^{2} - 14 \, c d + 17 \, d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (13 \, c^{2} - 22 \, c d - 3 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - \frac {4 \, \sqrt {2} {\left ({\left (c^{2} - 4 \, c d + 3 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} + 8 \, c d - 4 \, d^{2} - {\left (3 \, c^{2} - 4 \, c d + d^{2}\right )} \cos \left (f x + e\right ) + {\left (4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left (c^{2} - 4 \, c d + 3 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {a c - a d}} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left (9 \, c^{2} - 14 \, c d + 9 \, d^{2}\right )} \cos \left (f x + e\right ) + {\left ({\left (c^{2} - 14 \, c d + 17 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} + 2 \, {\left (7 \, c^{2} - 18 \, c d + 7 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 4}\right )}{4 \, \sqrt {a c - a d} f}, \frac {\sqrt {2} \sqrt {-\frac {1}{a c - a d}} \arctan \left (-\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left ({\left (c - 3 \, d\right )} \sin \left (f x + e\right ) - 3 \, c + d\right )} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {1}{a c - a d}}}{4 \, {\left (d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + c \cos \left (f x + e\right )\right )}}\right )}{2 \, f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(2)*log(((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^3 - (13*c^2 - 22*c*d - 3*d^2)*cos(f*x + e)^2 - 4*sqrt(2
)*((c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^2 - 4*c^2 + 8*c*d - 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e) + (4*c^2
- 8*c*d + 4*d^2 + (c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x +
e) + c)/sqrt(a*c - a*d) - 4*c^2 - 8*c*d - 4*d^2 - 2*(9*c^2 - 14*c*d + 9*d^2)*cos(f*x + e) + ((c^2 - 14*c*d + 1
7*d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 + 2*(7*c^2 - 18*c*d + 7*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*
x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4))/(sqrt(
a*c - a*d)*f), 1/2*sqrt(2)*sqrt(-1/(a*c - a*d))*arctan(-1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*
x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/(d*cos(f*x + e)*sin(f*x + e) + c*cos(f*x + e))
)/f]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.23, size = 191, normalized size = 2.42 \[ -\frac {\left (1-\cos \left (f x +e \right )+\sin \left (f x +e \right )\right ) \sqrt {c +d \sin \left (f x +e \right )}\, \ln \left (\frac {2 \sqrt {2 c -2 d}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )-2 c +2 d}{1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}\right ) \sqrt {2}}{f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sin \left (f x +e \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {2 c -2 d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/f*(1-cos(f*x+e)+sin(f*x+e))*(c+d*sin(f*x+e))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x
+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))/(
a*(1+sin(f*x+e)))^(1/2)/sin(f*x+e)*2^(1/2)/((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)/(2*c-2*d)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a*(sin(e + f*x) + 1))*sqrt(c + d*sin(e + f*x))), x)

________________________________________________________________________________________