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

Optimal. Leaf size=115 \[ -\frac{16 \sqrt{e \cos (c+d x)}}{45 a^2 d e \sqrt{a \sin (c+d x)+a}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}} \]

[Out]

(-2*Sqrt[e*Cos[c + d*x]])/(9*d*e*(a + a*Sin[c + d*x])^(5/2)) - (8*Sqrt[e*Cos[c + d*x]])/(45*a*d*e*(a + a*Sin[c
+ d*x])^(3/2)) - (16*Sqrt[e*Cos[c + d*x]])/(45*a^2*d*e*Sqrt[a + a*Sin[c + d*x]])

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Rubi [A]  time = 0.20212, antiderivative size = 115, 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} \[ -\frac{16 \sqrt{e \cos (c+d x)}}{45 a^2 d e \sqrt{a \sin (c+d x)+a}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]])/(9*d*e*(a + a*Sin[c + d*x])^(5/2)) - (8*Sqrt[e*Cos[c + d*x]])/(45*a*d*e*(a + a*Sin[c
+ d*x])^(3/2)) - (16*Sqrt[e*Cos[c + d*x]])/(45*a^2*d*e*Sqrt[a + a*Sin[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}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}+\frac{4 \int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx}{9 a}\\ &=-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}+\frac{8 \int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx}{45 a^2}\\ &=-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}-\frac{16 \sqrt{e \cos (c+d x)}}{45 a^2 d e \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.142075, size = 69, normalized size = 0.6 \[ -\frac{2 \left (8 \sin ^2(c+d x)+20 \sin (c+d x)+17\right ) \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)}}{45 a^3 d e (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])]*(17 + 20*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(45*a^3*d*e*(1
+ Sin[c + d*x])^3)

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Maple [A]  time = 0.115, 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}}}{\frac{1}{\sqrt{e\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.61961, size = 387, normalized size = 3.37 \begin{align*} -\frac{2 \,{\left (17 \, \sqrt{a} \sqrt{e} + \frac{40 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{49 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{49 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{40 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{17 \, \sqrt{a} \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 (a^{3} e + \frac{3 \, a^{3} e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{3} e \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{3} e \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{11}{2}} \sqrt{-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.10313, size = 251, normalized size = 2.18 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (8 \, \cos \left (d x + c\right )^{2} - 20 \, \sin \left (d x + c\right ) - 25\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{45 \,{\left (3 \, a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e +{\left (a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 20*sin(d*x + c) - 25)*sqrt(a*sin(d*x + c) + a)/(3*a^3*d*e*cos(d
*x + c)^2 - 4*a^3*d*e + (a^3*d*e*cos(d*x + c)^2 - 4*a^3*d*e)*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/(a+a*sin(d*x+c))**(5/2)/(e*cos(d*x+c))**(1/2),x)

[Out]

Timed out

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

[Out]

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