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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.132634, 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} \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{3 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{3 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(3*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (4*Sqrt[a + a*Sin[c + d*x]])/(3*a*d*e*Sqrt[e*Cos[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}{(e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2}{3 d e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}+\frac{2 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{3 a}\\ &=-\frac{2}{3 d e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}+\frac{4 \sqrt{a+a \sin (c+d x)}}{3 a d e \sqrt{e \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.102878, size = 46, normalized size = 0.61 \[ \frac{2 (2 \sin (c+d x)+1)}{3 d e \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.107, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 4\,\sin \left ( dx+c \right ) +2 \right ) \cos \left ( dx+c \right ) }{3\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.6021, size = 284, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (\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{\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}}{3 \,{\left (a e^{2} + \frac{2 \, a e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a e^{2} \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{5}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.62174, size = 177, normalized size = 2.33 \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 ) + 1\right )}}{3 \,{\left (a d e^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d e^{2} \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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