3.294 \(\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 d e (e \cos (c+d x))^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0746067, 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} \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 d e (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2),x]

[Out]

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

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))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx &=\frac{2 (a+a \sin (c+d x))^{5/2}}{5 d e (e \cos (c+d x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.173051, size = 36, normalized size = 1. \[ \frac{2 (a (\sin (c+d x)+1))^{5/2}}{5 d e (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.092, size = 34, normalized size = 0.9 \begin{align*}{\frac{2\,\cos \left ( dx+c \right ) }{5\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(7/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.5902, size = 177, normalized size = 4.92 \begin{align*} \frac{2 \,{\left (a^{\frac{5}{2}} \sqrt{e} - \frac{a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{5 \,{\left (e^{4} + \frac{e^{4} \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{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.66705, size = 265, normalized size = 7.36 \begin{align*} -\frac{2 \,{\left (a^{2} \cos \left (d x + c\right ) + a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{5 \,{\left (d e^{4} \cos \left (d x + c\right )^{2} - d e^{4} \cos \left (d x + c\right ) - 2 \, d e^{4} +{\left (d e^{4} \cos \left (d x + c\right ) + 2 \, d e^{4}\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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))**(5/2)/(e*cos(d*x+c))**(7/2),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))^(5/2)/(e*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out