3.319 \(\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=154 \[ \frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{16}{77 a^2 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt{e \cos (c+d x)}} \]

[Out]

-2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)) - 12/(77*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c +
d*x])^(3/2)) - 16/(77*a^2*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (32*Sqrt[a + a*Sin[c + d*x]])/
(77*a^3*d*e*Sqrt[e*Cos[c + d*x]])

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Rubi [A]  time = 0.296674, antiderivative size = 154, 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} \[ \frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{16}{77 a^2 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]

[Out]

-2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)) - 12/(77*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c +
d*x])^(3/2)) - 16/(77*a^2*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (32*Sqrt[a + a*Sin[c + d*x]])/
(77*a^3*d*e*Sqrt[e*Cos[c + d*x]])

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{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{2}{11 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}+\frac{6 \int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx}{11 a}\\ &=-\frac{2}{11 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac{12}{77 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}+\frac{24 \int \frac{1}{(e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}} \, dx}{77 a^2}\\ &=-\frac{2}{11 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac{12}{77 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac{16}{77 a^2 d e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}+\frac{16 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{77 a^3}\\ &=-\frac{2}{11 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac{12}{77 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac{16}{77 a^2 d e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}+\frac{32 \sqrt{a+a \sin (c+d x)}}{77 a^3 d e \sqrt{e \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.137724, size = 66, normalized size = 0.43 \[ \frac{32 \sin ^3(c+d x)+80 \sin ^2(c+d x)+52 \sin (c+d x)-10}{77 d e (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]

[Out]

(-10 + 52*Sin[c + d*x] + 80*Sin[c + d*x]^2 + 32*Sin[c + d*x]^3)/(77*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d
*x]))^(5/2))

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Maple [A]  time = 0.112, 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 ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),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)*cos(d*x+c)/(e*cos(d*x+c))^(3/2)/(a*(1+si
n(d*x+c)))^(5/2)

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Maxima [B]  time = 1.66377, size = 504, normalized size = 3.27 \begin{align*} -\frac{2 \,{\left (5 \, \sqrt{a} \sqrt{e} - \frac{52 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{150 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{180 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{180 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{150 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{52 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{5 \, \sqrt{a} \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 (a^{3} e^{2} + \frac{4 \, a^{3} e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{3} e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a^{3} e^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{3} e^{2} \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{13}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.89052, size = 328, normalized size = 2.13 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (40 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, \cos \left (d x + c\right )^{2} - 21\right )} \sin \left (d x + c\right ) - 35\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{77 \,{\left (3 \, a^{3} d e^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{3} d e^{2} \cos \left (d x + c\right ) +{\left (a^{3} d e^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{3} d e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/77*sqrt(e*cos(d*x + c))*(40*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 - 21)*sin(d*x + c) - 35)*sqrt(a*sin(d*x + c
) + a)/(3*a^3*d*e^2*cos(d*x + c)^3 - 4*a^3*d*e^2*cos(d*x + c) + (a^3*d*e^2*cos(d*x + c)^3 - 4*a^3*d*e^2*cos(d*
x + c))*sin(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(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^(5/2)), x)