3.3.76 \(\int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx\) [276]

Optimal. Leaf size=34 \[ \frac {2 \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2750} \begin {gather*} \frac {2 \sqrt {a \sin (c+d x)+a}}{d e \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2),x]

[Out]

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

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {2 \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 34, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a (1+\sin (c+d x))}}{d e \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2),x]

[Out]

(2*Sqrt[a*(1 + Sin[c + d*x])])/(d*e*Sqrt[e*Cos[c + d*x]])

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Maple [A]
time = 0.18, size = 34, normalized size = 1.00

method	result	size
default	\(\frac {2 \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (27) = 54\).
time = 0.55, size = 77, normalized size = 2.26 \begin {gather*} \frac {2 \, {\left (\sqrt {a} - \frac {\sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} e^{\left (-\frac {3}{2}\right )}}{d \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 27, normalized size = 0.79 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} e^{\left (-\frac {3}{2}\right )}}{d \sqrt {\cos \left (d x + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sin(c + d*x) + 1))/(e*cos(c + d*x))**(3/2), 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))^(1/2)/(e*cos(d*x+c))^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.31, size = 30, normalized size = 0.88 \begin {gather*} \frac {2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{d\,e\,\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))^(1/2)/(e*cos(c + d*x))^(3/2),x)

[Out]

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

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