3.287 \(\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}} \]

[Out]

(-2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2)) + (8*(a + a*Sin[c + d*x])^(5/2))/(3*a*d*e*(e*Cos[
c + d*x])^(7/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(21*a^2*d*e*(e*Cos[c + d*x])^(7/2))

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Rubi [A]  time = 0.229943, 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} \[ -\frac{16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(-2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2)) + (8*(a + a*Sin[c + d*x])^(5/2))/(3*a*d*e*(e*Cos[
c + d*x])^(7/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(21*a^2*d*e*(e*Cos[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))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac{4 \int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx}{a}\\ &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac{8 (a+a \sin (c+d x))^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{8 \int \frac{(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{3 a^2}\\ &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac{8 (a+a \sin (c+d x))^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{16 (a+a \sin (c+d x))^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.196198, size = 105, normalized size = 0.93 \[ \frac{2 a \sqrt{a (\sin (c+d x)+1)} (12 \sin (c+d x)+4 \cos (2 (c+d x))-5)}{21 d e^4 \sqrt{e \cos (c+d x)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(2*a*Sqrt[a*(1 + Sin[c + d*x])]*(-5 + 4*Cos[2*(c + d*x)] + 12*Sin[c + d*x]))/(21*d*e^4*Sqrt[e*Cos[c + d*x]]*(C
os[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 0.102, size = 54, normalized size = 0.5 \begin{align*}{\frac{ \left ( 16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+24\,\sin \left ( dx+c \right ) -18 \right ) \cos \left ( dx+c \right ) }{21\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{3}{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))^(3/2)/(e*cos(d*x+c))^(9/2),x)

[Out]

2/21/d*(8*cos(d*x+c)^2+12*sin(d*x+c)-9)*(a*(1+sin(d*x+c)))^(3/2)*cos(d*x+c)/(e*cos(d*x+c))^(9/2)

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Maxima [B]  time = 1.60803, size = 379, normalized size = 3.35 \begin{align*} -\frac{2 \,{\left (a^{\frac{3}{2}} \sqrt{e} - \frac{24 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{33 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{33 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{24 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{a^{\frac{3}{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}}{21 \,{\left (e^{5} + \frac{3 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{e^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d{\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 )}{\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))^(3/2)/(e*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

-2/21*(a^(3/2)*sqrt(e) - 24*a^(3/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 33*a^(3/2)*sqrt(e)*sin(d*x + c)^
2/(cos(d*x + c) + 1)^2 - 33*a^(3/2)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 24*a^(3/2)*sqrt(e)*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5 - a^(3/2)*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^5 + 3*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*e^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ e^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2)*(-sin(d*x + c)/(cos(d
*x + c) + 1) + 1)^(9/2))

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Fricas [A]  time = 2.2969, size = 215, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (8 \, a \cos \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) - 9 \, a\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} \sin \left (d x + c\right ) - d e^{5} \cos \left (d x + c\right )^{2}\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))^(9/2),x, algorithm="fricas")

[Out]

-2/21*(8*a*cos(d*x + c)^2 + 12*a*sin(d*x + c) - 9*a)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^5*cos(
d*x + c)^2*sin(d*x + c) - d*e^5*cos(d*x + c)^2)

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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))**(3/2)/(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))^(3/2)/(e*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

Timed out