\(\int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx\) [542]

Optimal result

Integrand size = 23, antiderivative size = 63 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2748, 2721, 2719} \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e} \]

[In]

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

[Out]

(-2*b*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d
*x]])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+a \int \sqrt {e \cos (c+d x)} \, dx \\ & = -\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac {\left (a \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{\sqrt {\cos (c+d x)}} \\ & = -\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=-\frac {2 \sqrt {e \cos (c+d x)} \left (b \cos ^{\frac {3}{2}}(c+d x)-3 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{3 d \sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*(b*Cos[c + d*x]^(3/2) - 3*a*EllipticE[(c + d*x)/2, 2]))/(3*d*Sqrt[Cos[c + d*x]])

Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.95

method	result	size
default	\(\frac {2 e \left (-4 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +4 b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(123\)
parts	\(\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 b \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e}\)	\(163\)

[In]

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

[Out]

2/3/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e*(-4*sin(1/2*d*x+1/2*c)^5*b+3*(sin(1/2*d*x+1/2*c)^
2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a+4*b*sin(1/2*d*x+1/2*c)^3-b*s
in(1/2*d*x+1/2*c))/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.35 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=\frac {3 i \, \sqrt {2} a \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} a \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, \sqrt {e \cos \left (d x + c\right )} b \cos \left (d x + c\right )}{3 \, d} \]

[In]

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

[Out]

1/3*(3*I*sqrt(2)*a*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) -
3*I*sqrt(2)*a*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*s
qrt(e*cos(d*x + c))*b*cos(d*x + c))/d

Sympy [F]

\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=\int \sqrt {e \cos {\left (c + d x \right )}} \left (a + b \sin {\left (c + d x \right )}\right )\, dx \]

[In]

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

[Out]

Integral(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)), x)

Maxima [F]

\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,\left (a+b\,\sin \left (c+d\,x\right )\right ) \,d x \]

[In]

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

[Out]

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