3.297 \(\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx\)

Optimal. Leaf size=150 \[ \frac{32 (a \sin (c+d x)+a)^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}-\frac{16 (a \sin (c+d x)+a)^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{4 (a \sin (c+d x)+a)^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}} \]

[Out]

(-2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(11/2)) + (4*(a + a*Sin[c + d*x])^(7/2))/(a*d*e*(e*Cos[c
+ d*x])^(11/2)) - (16*(a + a*Sin[c + d*x])^(9/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(11/2)) + (32*(a + a*Sin[c + d*
x])^(11/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(11/2))

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Rubi [A]  time = 0.305198, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{32 (a \sin (c+d x)+a)^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}-\frac{16 (a \sin (c+d x)+a)^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{4 (a \sin (c+d x)+a)^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2),x]

[Out]

(-2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(11/2)) + (4*(a + a*Sin[c + d*x])^(7/2))/(a*d*e*(e*Cos[c
+ d*x])^(11/2)) - (16*(a + a*Sin[c + d*x])^(9/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(11/2)) + (32*(a + a*Sin[c + d*
x])^(11/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(11/2))

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))^{13/2}} \, dx &=-\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac{6 \int \frac{(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{13/2}} \, dx}{a}\\ &=-\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac{4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{8 \int \frac{(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{13/2}} \, dx}{a^2}\\ &=-\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac{4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{16 (a+a \sin (c+d x))^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{16 \int \frac{(a+a \sin (c+d x))^{11/2}}{(e \cos (c+d x))^{13/2}} \, dx}{7 a^3}\\ &=-\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac{4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{16 (a+a \sin (c+d x))^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{32 (a+a \sin (c+d x))^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.282116, size = 74, normalized size = 0.49 \[ \frac{2 \left (16 \sin ^3(c+d x)-40 \sin ^2(c+d x)+26 \sin (c+d x)+5\right ) \sec ^6(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{77 d e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2),x]

[Out]

(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^6*(a*(1 + Sin[c + d*x]))^(5/2)*(5 + 26*Sin[c + d*x] - 40*Sin[c + d*x]^2 +
16*Sin[c + d*x]^3))/(77*d*e^7)

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Maple [A]  time = 0.106, size = 70, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 32\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -80\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-84\,\sin \left ( dx+c \right ) +70 \right ) \cos \left ( dx+c \right ) }{77\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{13}{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))^(13/2),x)

[Out]

-2/77/d*(16*cos(d*x+c)^2*sin(d*x+c)-40*cos(d*x+c)^2-42*sin(d*x+c)+35)*(a*(1+sin(d*x+c)))^(5/2)*cos(d*x+c)/(e*c
os(d*x+c))^(13/2)

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Maxima [B]  time = 1.65589, size = 482, normalized size = 3.21 \begin{align*} \frac{2 \,{\left (5 \, a^{\frac{5}{2}} \sqrt{e} + \frac{52 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{150 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{180 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{180 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{150 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{52 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{5 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{77 \,{\left (e^{7} + \frac{4 \, e^{7} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, e^{7} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, e^{7} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{e^{7} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\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{13}{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))^(13/2),x, algorithm="maxima")

[Out]

2/77*(5*a^(5/2)*sqrt(e) + 52*a^(5/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 150*a^(5/2)*sqrt(e)*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 + 180*a^(5/2)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 180*a^(5/2)*sqrt(e)*sin(d
*x + c)^5/(cos(d*x + c) + 1)^5 + 150*a^(5/2)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 52*a^(5/2)*sqrt(e)*
sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 5*a^(5/2)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*(sin(d*x + c)^2/(
cos(d*x + c) + 1)^2 + 1)^4/((e^7 + 4*e^7*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*e^7*sin(d*x + c)^4/(cos(d*x +
c) + 1)^4 + 4*e^7*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + e^7*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin(d*x +
c)/(cos(d*x + c) + 1) + 1)^(3/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2))

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Fricas [A]  time = 2.1243, size = 298, normalized size = 1.99 \begin{align*} -\frac{2 \,{\left (40 \, a^{2} \cos \left (d x + c\right )^{2} - 35 \, a^{2} - 2 \,{\left (8 \, a^{2} \cos \left (d x + c\right )^{2} - 21 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{77 \,{\left (d e^{7} \cos \left (d x + c\right )^{4} + 2 \, d e^{7} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d e^{7} \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))^(5/2)/(e*cos(d*x+c))^(13/2),x, algorithm="fricas")

[Out]

-2/77*(40*a^2*cos(d*x + c)^2 - 35*a^2 - 2*(8*a^2*cos(d*x + c)^2 - 21*a^2)*sin(d*x + c))*sqrt(e*cos(d*x + c))*s
qrt(a*sin(d*x + c) + a)/(d*e^7*cos(d*x + c)^4 + 2*d*e^7*cos(d*x + c)^2*sin(d*x + c) - 2*d*e^7*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))**(5/2)/(e*cos(d*x+c))**(13/2),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))^(13/2),x, algorithm="giac")

[Out]

Timed out