3.296 \(\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}} \]

[Out]

(2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(9/2)) - (8*(a + a*Sin[c + d*x])^(7/2))/(5*a*d*e*(e*Cos[c
+ d*x])^(9/2)) + (16*(a + a*Sin[c + d*x])^(9/2))/(45*a^2*d*e*(e*Cos[c + d*x])^(9/2))

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Rubi [A]  time = 0.223381, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2),x]

[Out]

(2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(9/2)) - (8*(a + a*Sin[c + d*x])^(7/2))/(5*a*d*e*(e*Cos[c
+ d*x])^(9/2)) + (16*(a + a*Sin[c + d*x])^(9/2))/(45*a^2*d*e*(e*Cos[c + d*x])^(9/2))

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))^{11/2}} \, dx &=\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac{4 \int \frac{(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a}\\ &=\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac{8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{8 \int \frac{(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{11/2}} \, dx}{5 a^2}\\ &=\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac{8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{16 (a+a \sin (c+d x))^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.217391, size = 64, normalized size = 0.57 \[ \frac{2 \left (8 \sin ^2(c+d x)-20 \sin (c+d x)+17\right ) \sec ^5(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{45 d e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2),x]

[Out]

(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^5*(a*(1 + Sin[c + d*x]))^(5/2)*(17 - 20*Sin[c + d*x] + 8*Sin[c + d*x]^2))
/(45*d*e^6)

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Maple [A]  time = 0.105, size = 54, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+40\,\sin \left ( dx+c \right ) -50 \right ) \cos \left ( dx+c \right ) }{45\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{11}{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))^(11/2),x)

[Out]

-2/45/d*(8*cos(d*x+c)^2+20*sin(d*x+c)-25)*(a*(1+sin(d*x+c)))^(5/2)*cos(d*x+c)/(e*cos(d*x+c))^(11/2)

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Maxima [B]  time = 1.59627, size = 381, normalized size = 3.37 \begin{align*} \frac{2 \,{\left (17 \, a^{\frac{5}{2}} \sqrt{e} - \frac{40 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{49 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{49 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{40 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{17 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{45 \,{\left (e^{6} + \frac{3 \, e^{6} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, e^{6} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{e^{6} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{11}{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))^(11/2),x, algorithm="maxima")

[Out]

2/45*(17*a^(5/2)*sqrt(e) - 40*a^(5/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 49*a^(5/2)*sqrt(e)*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 - 49*a^(5/2)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 40*a^(5/2)*sqrt(e)*sin(d*x
+ c)^5/(cos(d*x + c) + 1)^5 - 17*a^(5/2)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*(sin(d*x + c)^2/(cos(d*
x + c) + 1)^2 + 1)^3/((e^6 + 3*e^6*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*e^6*sin(d*x + c)^4/(cos(d*x + c) +
1)^4 + e^6*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*sqrt(sin(d*x + c)/(cos(d*x + c) + 1) + 1)*(-sin(d*x + c)/(co
s(d*x + c) + 1) + 1)^(11/2))

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Fricas [A]  time = 2.49685, size = 254, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (8 \, a^{2} \cos \left (d x + c\right )^{2} + 20 \, a^{2} \sin \left (d x + c\right ) - 25 \, a^{2}\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{45 \,{\left (d e^{6} \cos \left (d x + c\right )^{3} + 2 \, d e^{6} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, d e^{6} \cos \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))^(11/2),x, algorithm="fricas")

[Out]

2/45*(8*a^2*cos(d*x + c)^2 + 20*a^2*sin(d*x + c) - 25*a^2)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^
6*cos(d*x + c)^3 + 2*d*e^6*cos(d*x + c)*sin(d*x + c) - 2*d*e^6*cos(d*x + c))

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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))**(11/2),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))^(11/2),x, algorithm="giac")

[Out]

Timed out