3.6.66 \(\int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx\) [566]

Optimal. Leaf size=105 \[ -\frac {\sqrt {a} (c+d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2849, 2854, 211} \begin {gather*} -\frac {\sqrt {a} (c+d) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]

[Out]

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

Rule 211

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

Rule 2849

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[2*n*((b*c + a*d)
/(b*(2*n + 1))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), 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] && GtQ[n, 0] && IntegerQ[2*n]

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(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 \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {1}{2} (c+d) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx\\ &=-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c+d)) \text {Subst}\left (\int \frac {1}{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 {a} (c+d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.90, size = 350, normalized size = 3.33 \begin {gather*} \frac {\sqrt {a (1+\sin (e+f x))} \left (-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{f}-\frac {i (c+d) \left (\log \left (\frac {e^{-i e} \left (2 \sqrt [4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}+2 \sqrt {d} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right )}{\sqrt {d}}\right )-\log \left (\frac {2 e^{\frac {1}{2} i (e-2 f x)} \left ((-1)^{3/4} d+\sqrt [4]{-1} c e^{i (e+f x)}+i \sqrt {d} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{\sqrt {d}}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt {d} f}\right )}{2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]

[Out]

(Sqrt[a*(1 + Sin[e + f*x])]*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/f - (I*(c + d)*(L
og[(2*(-1)^(1/4)*c - 2*(-1)^(3/4)*d*E^(I*(e + f*x)) + 2*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*
(e + f*x)))])/(Sqrt[d]*E^(I*e))] - Log[(2*E^((I/2)*(e - 2*f*x))*((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) +
I*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/Sqrt[d]])*(Cos[(e + f*x)/2] - I*Sin[
(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])])/(Sqrt[d]*f)))/(2*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (95) = 190\).
time = 0.64, size = 991, normalized size = 9.44 \begin {gather*} \left [\frac {{\left ({\left (c + d\right )} \cos \left (f x + e\right ) + {\left (c + d\right )} \sin \left (f x + e\right ) + c + d\right )} \sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\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 {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) - 8 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{8 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}}, \frac {{\left ({\left (c + d\right )} \cos \left (f x + e\right ) + {\left (c + d\right )} \sin \left (f x + e\right ) + c + d\right )} \sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\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 {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right ) - 4 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{4 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(((c + d)*cos(f*x + e) + (c + d)*sin(f*x + e) + c + d)*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 +
4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a
*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^
4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 2
6*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)
^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 1
3*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*
a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*
d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a
*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x
+ e) + sin(f*x + e) + 1)) - 8*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos(f*x + e) - sin(f*x + e) +
1))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/4*(((c + d)*cos(f*x + e) + (c + d)*sin(f*x + e) + c + d)*sqrt(a/
d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) +
a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) -
(a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos(f*x + e) -
sin(f*x + e) + 1))/(f*cos(f*x + e) + f*sin(f*x + e) + f)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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