3.3.3 \(\int \sin (e+f x) \sqrt {a+b \sin (e+f x)} \, dx\) [203]

Optimal. Leaf size=172 \[ -\frac {2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 f}+\frac {2 a E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{3 b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{3 b f \sqrt {a+b \sin (e+f x)}} \]

[Out]

-2/3*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/f-2/3*a*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*E
llipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(f*x+e))^(1/2)/b/f/((a+b*sin(f*x+e))/(a+b)
)^(1/2)+2/3*(a^2-b^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi
+1/2*f*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^(1/2)/b/f/(a+b*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 b f \sqrt {a+b \sin (e+f x)}}-\frac {2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 f}+\frac {2 a \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*f) + (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a +
b*Sin[e + f*x]])/(3*b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*(a^2 - b^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*
b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(3*b*f*Sqrt[a + b*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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

Rule 2832

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rubi steps

\begin {align*} \int \sin (e+f x) \sqrt {a+b \sin (e+f x)} \, dx &=-\frac {2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 f}+\frac {2}{3} \int \frac {\frac {b}{2}+\frac {1}{2} a \sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx\\ &=-\frac {2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 f}+\frac {a \int \sqrt {a+b \sin (e+f x)} \, dx}{3 b}-\frac {\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx}{3 b}\\ &=-\frac {2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 f}+\frac {\left (a \sqrt {a+b \sin (e+f x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}} \, dx}{3 b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}} \, dx}{3 b \sqrt {a+b \sin (e+f x)}}\\ &=-\frac {2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 f}+\frac {2 a E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{3 b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{3 b f \sqrt {a+b \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 2.67, size = 143, normalized size = 0.83 \begin {gather*} -\frac {2 \left (b \cos (e+f x) (a+b \sin (e+f x))+a (a+b) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}-\left (a^2-b^2\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}\right )}{3 b f \sqrt {a+b \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*Cos[e + f*x]*(a + b*Sin[e + f*x]) + a*(a + b)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a +
b*Sin[e + f*x])/(a + b)] - (a^2 - b^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*
x])/(a + b)]))/(3*b*f*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. \(459\) vs. \(2(222)=444\).
time = 3.42, size = 460, normalized size = 2.67

method	result	size
default	\(\frac {\frac {2 \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b}{3}-\frac {2 \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{3}}{3}-\frac {2 \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3}}{3}+\frac {2 \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{2}}{3}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right ) b^{3}}{3}+\frac {2 \left (\sin ^{2}\left (f x +e \right )\right ) a \,b^{2}}{3}-\frac {2 b^{3} \sin \left (f x +e \right )}{3}-\frac {2 a \,b^{2}}{3}}{b^{2} \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(460\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a+b))^(1/2)*(-(1+sin(f*x+e))*b/(a-b))^(1/2)*EllipticF(
((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b-((a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a
+b))^(1/2)*(-(1+sin(f*x+e))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^3-(
(a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a+b))^(1/2)*(-(1+sin(f*x+e))*b/(a-b))^(1/2)*EllipticE(((a+b*
sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3+((a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a+b))^(1/
2)*(-(1+sin(f*x+e))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^2+sin(f*x
+e)^3*b^3+sin(f*x+e)^2*a*b^2-b^3*sin(f*x+e)-a*b^2)/b^2/cos(f*x+e)/(a+b*sin(f*x+e))^(1/2)/f

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 420, normalized size = 2.44 \begin {gather*} \frac {-3 i \, \sqrt {2} a \sqrt {i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 i \, \sqrt {2} a \sqrt {-i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, \sqrt {b \sin \left (f x + e\right ) + a} b^{2} \cos \left (f x + e\right ) - \sqrt {2} {\left (2 \, a^{2} - 3 \, b^{2}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right ) - \sqrt {2} {\left (2 \, a^{2} - 3 \, b^{2}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )}{9 \, b^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(-3*I*sqrt(2)*a*sqrt(I*b)*b*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, wei
erstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) - 3*I*b*sin(
f*x + e) - 2*I*a)/b)) + 3*I*sqrt(2)*a*sqrt(-I*b)*b*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 +
9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(
f*x + e) + 3*I*b*sin(f*x + e) + 2*I*a)/b)) - 6*sqrt(b*sin(f*x + e) + a)*b^2*cos(f*x + e) - sqrt(2)*(2*a^2 - 3*
b^2)*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x
+ e) - 3*I*b*sin(f*x + e) - 2*I*a)/b) - sqrt(2)*(2*a^2 - 3*b^2)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 -
3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) + 3*I*b*sin(f*x + e) + 2*I*a)/b))/(b^2*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sin \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(sin(e + f*x)*(a + b*sin(e + f*x))^(1/2),x)

[Out]

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

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