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

Optimal. Leaf size=115 \[ \frac{16 \sqrt{a \sin (c+d x)+a}}{21 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{8}{21 a d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}} \]

[Out]

-2/(7*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) - 8/(21*a*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c
+ d*x]]) + (16*Sqrt[a + a*Sin[c + d*x]])/(21*a^2*d*e*Sqrt[e*Cos[c + d*x]])

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Rubi [A]  time = 0.209761, 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} \[ \frac{16 \sqrt{a \sin (c+d x)+a}}{21 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{8}{21 a d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(7*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) - 8/(21*a*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c
+ d*x]]) + (16*Sqrt[a + a*Sin[c + d*x]])/(21*a^2*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))^{3/2}} \, dx &=-\frac{2}{7 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}+\frac{4 \int \frac{1}{(e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}} \, dx}{7 a}\\ &=-\frac{2}{7 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac{8}{21 a d e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}+\frac{8 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{21 a^2}\\ &=-\frac{2}{7 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac{8}{21 a d e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}+\frac{16 \sqrt{a+a \sin (c+d x)}}{21 a^2 d e \sqrt{e \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.103559, size = 56, normalized size = 0.49 \[ \frac{16 \sin ^2(c+d x)+24 \sin (c+d x)+2}{21 d e (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2 + 24*Sin[c + d*x] + 16*Sin[c + d*x]^2)/(21*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d*x]))^(3/2))

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Maple [A]  time = 0.093, size = 54, normalized size = 0.5 \begin{align*}{\frac{ \left ( -16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+24\,\sin \left ( dx+c \right ) +18 \right ) \cos \left ( dx+c \right ) }{21\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{3}{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))^(3/2),x)

[Out]

2/21/d*(-8*cos(d*x+c)^2+12*sin(d*x+c)+9)*cos(d*x+c)/(e*cos(d*x+c))^(3/2)/(a*(1+sin(d*x+c)))^(3/2)

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Maxima [B]  time = 1.6065, size = 397, normalized size = 3.45 \begin{align*} \frac{2 \,{\left (\sqrt{a} \sqrt{e} + \frac{24 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{33 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{33 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{24 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{\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}}{21 \,{\left (a^{2} e^{2} + \frac{3 \, a^{2} e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{2} e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{2} e^{2} \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{9}{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))^(3/2),x, algorithm="maxima")

[Out]

2/21*(sqrt(a)*sqrt(e) + 24*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 33*sqrt(a)*sqrt(e)*sin(d*x + c)^2
/(cos(d*x + c) + 1)^2 - 33*sqrt(a)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 24*sqrt(a)*sqrt(e)*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5 - 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^2*e^2 + 3*a^2*e^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*e^2*sin(d*x + c)^4/(cos(d*x +
c) + 1)^4 + a^2*e^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(-sin(
d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2))

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Fricas [A]  time = 2.82499, size = 252, normalized size = 2.19 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (8 \, \cos \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) - 9\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{21 \,{\left (a^{2} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} d e^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d e^{2} \cos \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))^(3/2),x, algorithm="fricas")

[Out]

2/21*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 12*sin(d*x + c) - 9)*sqrt(a*sin(d*x + c) + a)/(a^2*d*e^2*cos(d*x
+ c)^3 - 2*a^2*d*e^2*cos(d*x + c)*sin(d*x + c) - 2*a^2*d*e^2*cos(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))**(3/2),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{3}{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))^(3/2),x, algorithm="giac")

[Out]

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