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

Optimal. Leaf size=193 \[ \frac{256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac{128 \sqrt{a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{32}{195 a^2 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}} \]

[Out]

-2/(13*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)) - 16/(117*a*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Si
n[c + d*x])^(3/2)) - 32/(195*a^2*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (128*Sqrt[a + a*Sin[c
+ d*x]])/(195*a^3*d*e*(e*Cos[c + d*x])^(3/2)) + (256*(a + a*Sin[c + d*x])^(3/2))/(585*a^4*d*e*(e*Cos[c + d*x])
^(3/2))

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Rubi [A]  time = 0.375388, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac{128 \sqrt{a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{32}{195 a^2 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(13*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)) - 16/(117*a*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Si
n[c + d*x])^(3/2)) - 32/(195*a^2*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (128*Sqrt[a + a*Sin[c
+ d*x]])/(195*a^3*d*e*(e*Cos[c + d*x])^(3/2)) + (256*(a + a*Sin[c + d*x])^(3/2))/(585*a^4*d*e*(e*Cos[c + d*x])
^(3/2))

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{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}+\frac{8 \int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx}{13 a}\\ &=-\frac{2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac{16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}+\frac{16 \int \frac{1}{(e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}} \, dx}{39 a^2}\\ &=-\frac{2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac{16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac{32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}+\frac{64 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^3}\\ &=-\frac{2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac{16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac{32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}-\frac{128 \sqrt{a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac{128 \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^4}\\ &=-\frac{2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac{16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac{32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}-\frac{128 \sqrt{a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac{256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.330774, size = 76, normalized size = 0.39 \[ -\frac{2 (-40 \sin (c+d x)+80 \sin (3 (c+d x))+136 \cos (2 (c+d x))-16 \cos (4 (c+d x))+77)}{585 d e (a (\sin (c+d x)+1))^{5/2} (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(77 + 136*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] - 40*Sin[c + d*x] + 80*Sin[3*(c + d*x)]))/(585*d*e*(e*Cos
[c + d*x])^(3/2)*(a*(1 + Sin[c + d*x]))^(5/2))

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Maple [A]  time = 0.115, size = 80, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -256\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+640\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +800\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-240\,\sin \left ( dx+c \right ) -150 \right ) \cos \left ( dx+c \right ) }{585\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{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))^(5/2)/(a+a*sin(d*x+c))^(5/2),x)

[Out]

-2/585/d*(-128*cos(d*x+c)^4+320*cos(d*x+c)^2*sin(d*x+c)+400*cos(d*x+c)^2-120*sin(d*x+c)-75)*cos(d*x+c)/(e*cos(
d*x+c))^(5/2)/(a*(1+sin(d*x+c)))^(5/2)

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Maxima [B]  time = 1.69006, size = 609, normalized size = 3.16 \begin{align*} -\frac{2 \,{\left (197 \, \sqrt{a} \sqrt{e} + \frac{400 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{15 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1760 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{2230 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{2230 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1760 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{15 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{400 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{197 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{585 \,{\left (a^{3} e^{3} + \frac{5 \, a^{3} e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{10 \, a^{3} e^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{5 \, a^{3} e^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a^{3} e^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{15}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/585*(197*sqrt(a)*sqrt(e) + 400*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 15*sqrt(a)*sqrt(e)*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 - 1760*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 2230*sqrt(a)*sqrt(e)
*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2230*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1760*sqrt(a)
*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 15*sqrt(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 400*sq
rt(a)*sqrt(e)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 197*sqrt(a)*sqrt(e)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)
*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((a^3*e^3 + 5*a^3*e^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^
3*e^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^3*e^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a^3*e^3*sin(d*x
+ c)^8/(cos(d*x + c) + 1)^8 + a^3*e^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*d*(sin(d*x + c)/(cos(d*x + c) +
1) + 1)^(15/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/2))

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Fricas [A]  time = 2.9877, size = 366, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (128 \, \cos \left (d x + c\right )^{4} - 400 \, \cos \left (d x + c\right )^{2} - 40 \,{\left (8 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 75\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{585 \,{\left (3 \, a^{3} d e^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} d e^{3} \cos \left (d x + c\right )^{2} +{\left (a^{3} d e^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} d e^{3} \cos \left (d x + c\right )^{2}\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))^(5/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2/585*(128*cos(d*x + c)^4 - 400*cos(d*x + c)^2 - 40*(8*cos(d*x + c)^2 - 3)*sin(d*x + c) + 75)*sqrt(e*cos(d*x
+ c))*sqrt(a*sin(d*x + c) + a)/(3*a^3*d*e^3*cos(d*x + c)^4 - 4*a^3*d*e^3*cos(d*x + c)^2 + (a^3*d*e^3*cos(d*x +
c)^4 - 4*a^3*d*e^3*cos(d*x + c)^2)*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))**(5/2)/(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{5}{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))^(5/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

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