3.295 \(\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=76 \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}} \]

[Out]

(2*(a + a*Sin[c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^(7/2))/(21*a*d*e*(e*Co
s[c + d*x])^(7/2))

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Rubi [A]  time = 0.148437, 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} \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(2*(a + a*Sin[c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^(7/2))/(21*a*d*e*(e*Co
s[c + d*x])^(7/2))

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{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx &=\frac{2 (a+a \sin (c+d x))^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{2 \int \frac{(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{3 a}\\ &=\frac{2 (a+a \sin (c+d x))^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{4 (a+a \sin (c+d x))^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.181336, size = 54, normalized size = 0.71 \[ -\frac{2 (2 \sin (c+d x)-5) \sec ^4(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{21 d e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(a*(1 + Sin[c + d*x]))^(5/2)*(-5 + 2*Sin[c + d*x]))/(21*d*e^5)

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Maple [A]  time = 0.095, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 4\,\sin \left ( dx+c \right ) -10 \right ) \cos \left ( dx+c \right ) }{21\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{9}{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))^(9/2),x)

[Out]

-2/21/d*(2*sin(d*x+c)-5)*(a*(1+sin(d*x+c)))^(5/2)*cos(d*x+c)/(e*cos(d*x+c))^(9/2)

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Maxima [B]  time = 1.58881, size = 279, normalized size = 3.67 \begin{align*} \frac{2 \,{\left (5 \, a^{\frac{5}{2}} \sqrt{e} - \frac{4 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{4 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{5 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\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 )}^{2}}{21 \,{\left (e^{5} + \frac{2 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{e^{5} \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{9}{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))^(9/2),x, algorithm="maxima")

[Out]

2/21*(5*a^(5/2)*sqrt(e) - 4*a^(5/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 4*a^(5/2)*sqrt(e)*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 - 5*a^(5/2)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*sqrt(sin(d*x + c)/(cos(d*x + c)
+ 1) + 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^2/((e^5 + 2*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + e^5
*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*d*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2))

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Fricas [A]  time = 2.59937, size = 185, normalized size = 2.43 \begin{align*} \frac{2 \,{\left (2 \, a^{2} \sin \left (d x + c\right ) - 5 \, a^{2}\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{21 \,{\left (d e^{5} \cos \left (d x + c\right )^{2} + 2 \, d e^{5} \sin \left (d x + c\right ) - 2 \, d e^{5}\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))^(9/2),x, algorithm="fricas")

[Out]

2/21*(2*a^2*sin(d*x + c) - 5*a^2)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^5*cos(d*x + c)^2 + 2*d*e^
5*sin(d*x + c) - 2*d*e^5)

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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))**(9/2),x)

[Out]

Timed out

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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))^(9/2),x, algorithm="giac")

[Out]

Timed out