3.309 \(\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{4 \sqrt{e \cos (c+d x)}}{5 a d e \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}} \]

[Out]

(-2*Sqrt[e*Cos[c + d*x]])/(5*d*e*(a + a*Sin[c + d*x])^(3/2)) - (4*Sqrt[e*Cos[c + d*x]])/(5*a*d*e*Sqrt[a + a*Si
n[c + d*x]])

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Rubi [A]  time = 0.129558, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac{4 \sqrt{e \cos (c+d x)}}{5 a d e \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)),x]

[Out]

(-2*Sqrt[e*Cos[c + d*x]])/(5*d*e*(a + a*Sin[c + d*x])^(3/2)) - (4*Sqrt[e*Cos[c + d*x]])/(5*a*d*e*Sqrt[a + a*Si
n[c + d*x]])

Rule 2672

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

Rule 2671

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[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{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a+a \sin (c+d x))^{3/2}}+\frac{2 \int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx}{5 a}\\ &=-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a+a \sin (c+d x))^{3/2}}-\frac{4 \sqrt{e \cos (c+d x)}}{5 a d e \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.110112, size = 59, normalized size = 0.78 \[ -\frac{2 (2 \sin (c+d x)+3) \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)}}{5 a^2 d e (\sin (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)),x]

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])]*(3 + 2*Sin[c + d*x]))/(5*a^2*d*e*(1 + Sin[c + d*x])^2)

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Maple [A]  time = 0.113, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 4\,\sin \left ( dx+c \right ) +6 \right ) \cos \left ( dx+c \right ) }{5\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{e\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5/d*(2*sin(d*x+c)+3)*cos(d*x+c)/(a*(1+sin(d*x+c)))^(3/2)/(e*cos(d*x+c))^(1/2)

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Maxima [B]  time = 1.57702, size = 285, normalized size = 3.75 \begin{align*} -\frac{2 \,{\left (3 \, \sqrt{a} \sqrt{e} + \frac{4 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{4 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{3 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{5 \,{\left (a^{2} e + \frac{2 \, a^{2} e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{2} e \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}} \sqrt{-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(3*sqrt(a)*sqrt(e) + 4*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 4*sqrt(a)*sqrt(e)*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 - 3*sqrt(a)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*(sin(d*x + c)^2/(cos(d*x + c) +
1)^2 + 1)^2/((a^2*e + 2*a^2*e*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^2*e*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2)*sqrt(-sin(d*x + c)/(cos(d*x + c) + 1) + 1))

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Fricas [A]  time = 2.81083, size = 181, normalized size = 2.38 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}{\left (2 \, \sin \left (d x + c\right ) + 3\right )}}{5 \,{\left (a^{2} d e \cos \left (d x + c\right )^{2} - 2 \, a^{2} d e \sin \left (d x + c\right ) - 2 \, a^{2} d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*(2*sin(d*x + c) + 3)/(a^2*d*e*cos(d*x + c)^2 - 2*a^2*d*e*sin
(d*x + c) - 2*a^2*d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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