3.2.100 \(\int \frac {a+a \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx\) [200]

Optimal. Leaf size=61 \[ -\frac {2 a \sqrt {e \cos (c+d x)}}{d e}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2748, 2721, 2720} \begin {gather*} \frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)}}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sqrt[e*Cos[c + d*x]])/(d*e) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]]
)

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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*} \int \frac {a+a \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {2 a \sqrt {e \cos (c+d x)}}{d e}+a \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 a \sqrt {e \cos (c+d x)}}{d e}+\frac {\left (a \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{\sqrt {e \cos (c+d x)}}\\ &=-\frac {2 a \sqrt {e \cos (c+d x)}}{d e}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 14.17, size = 48, normalized size = 0.79 \begin {gather*} -\frac {2 a \left (\cos (c+d x)-\sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{d \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*(Cos[c + d*x] - Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]))/(d*Sqrt[e*Cos[c + d*x]])

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Maple [A]
time = 0.99, size = 103, normalized size = 1.69

method	result	size
default	\(-\frac {2 a \left (\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(103\)
risch	\(-\frac {\left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) a \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}+\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) a \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{i \left (d x +c \right )}}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \sqrt {{\mathrm e}^{3 i \left (d x +c \right )} e +{\mathrm e}^{i \left (d x +c \right )} e}\, \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 66, normalized size = 1.08 \begin {gather*} \frac {{\left (-i \, \sqrt {2} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, a \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*sqrt(2)*a*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + I*sqrt(2)*a*weierstrassPInverse(-4,
0, cos(d*x + c) - I*sin(d*x + c)) - 2*a*sqrt(cos(d*x + c)))*e^(-1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 0.55, size = 45, normalized size = 0.74 \begin {gather*} -\frac {2\,a\,\sqrt {\cos \left (c+d\,x\right )}\,\left (\sqrt {\cos \left (c+d\,x\right )}-\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{d\,\sqrt {e\,\cos \left (c+d\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*cos(c + d*x)^(1/2)*(cos(c + d*x)^(1/2) - ellipticF(c/2 + (d*x)/2, 2)))/(d*(e*cos(c + d*x))^(1/2))

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