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

Optimal. Leaf size=115 \[ \frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}} \]

[Out]

-2/(5*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (8*Sqrt[a + a*Sin[c + d*x]])/(5*a*d*e*(e*Cos[c +
d*x])^(3/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(15*a^2*d*e*(e*Cos[c + d*x])^(3/2))

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Rubi [A]  time = 0.208582, 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} \[ \frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(5*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (8*Sqrt[a + a*Sin[c + d*x]])/(5*a*d*e*(e*Cos[c +
d*x])^(3/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(15*a^2*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} \sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2}{5 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}+\frac{4 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{5 a}\\ &=-\frac{2}{5 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}-\frac{8 \sqrt{a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac{8 \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{5 a^2}\\ &=-\frac{2}{5 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}-\frac{8 \sqrt{a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac{16 (a+a \sin (c+d x))^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.09899, size = 56, normalized size = 0.49 \[ \frac{2 \left (8 \sin ^2(c+d x)+4 \sin (c+d x)-7\right )}{15 d e \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-7 + 4*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(15*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 0.112, size = 54, normalized size = 0.5 \begin{align*}{\frac{ \left ( -16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,\sin \left ( dx+c \right ) +2 \right ) \cos \left ( dx+c \right ) }{15\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}}} \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))^(1/2),x)

[Out]

2/15/d*(-8*cos(d*x+c)^2+4*sin(d*x+c)+1)*cos(d*x+c)/(e*cos(d*x+c))^(5/2)/(a*(1+sin(d*x+c)))^(1/2)

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Maxima [B]  time = 1.62497, size = 387, normalized size = 3.37 \begin{align*} -\frac{2 \,{\left (7 \, \sqrt{a} \sqrt{e} - \frac{8 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{25 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{25 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{8 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{7 \, \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}}{15 \,{\left (a e^{3} + \frac{3 \, a e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a e^{3} \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{7}{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))^(1/2),x, algorithm="maxima")

[Out]

-2/15*(7*sqrt(a)*sqrt(e) - 8*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 25*sqrt(a)*sqrt(e)*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 + 25*sqrt(a)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 8*sqrt(a)*sqrt(e)*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5 - 7*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*e^3 + 3*a*e^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a*e^3*sin(d*x + c)^4/(cos(d*x + c)
+ 1)^4 + a*e^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2)*(-sin(d*x +
c)/(cos(d*x + c) + 1) + 1)^(5/2))

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Fricas [A]  time = 2.26388, size = 211, normalized size = 1.83 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (8 \, \cos \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) - 1\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \,{\left (a d e^{3} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d e^{3} \cos \left (d x + c\right )^{2}\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))^(1/2),x, algorithm="fricas")

[Out]

-2/15*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 4*sin(d*x + c) - 1)*sqrt(a*sin(d*x + c) + a)/(a*d*e^3*cos(d*x +
c)^2*sin(d*x + c) + a*d*e^3*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(1/(e*cos(d*x+c))**(5/2)/(a+a*sin(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}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}} \sqrt{a \sin \left (d x + c\right ) + a}}\,{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))^(1/2),x, algorithm="giac")

[Out]

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