3.285 \(\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0725385, antiderivative size = 36, normalized size of antiderivative = 1., 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 = {2671} \[ \frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2),x]

[Out]

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

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{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx &=\frac{2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.106918, size = 36, normalized size = 1. \[ \frac{2 (a (\sin (c+d x)+1))^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 34, normalized size = 0.9 \begin{align*}{\frac{2\,\cos \left ( dx+c \right ) }{3\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.57786, size = 177, normalized size = 4.92 \begin{align*} \frac{2 \,{\left (a^{\frac{3}{2}} \sqrt{e} - \frac{a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{3 \,{\left (e^{3} + \frac{e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.78553, size = 112, normalized size = 3.11 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a} a}{3 \,{\left (d e^{3} \sin \left (d x + c\right ) - d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**(3/2)/(e*cos(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out