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

Optimal. Leaf size=95 \[ \frac{2 a e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0649123, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2635, 2642, 2641} \[ \frac{2 a e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+a \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} \left (a e^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{\left (a e^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 \sqrt{e \cos (c+d x)}}\\ &=-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sqrt{e \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.363893, size = 75, normalized size = 0.79 \[ -\frac{a (e \cos (c+d x))^{3/2} \left (\sqrt{\cos (c+d x)} (-10 \sin (c+d x)+3 \cos (2 (c+d x))+3)-10 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{15 d \cos ^{\frac{3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(e*Cos[c + d*x])^(3/2)*(-10*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(3 + 3*Cos[2*(c + d*x)] - 10*Si
n[c + d*x])))/(15*d*Cos[c + d*x]^(3/2))

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Maple [A]  time = 0.352, size = 179, normalized size = 1.9 \begin{align*} -{\frac{2\,a{e}^{2}}{15\,d} \left ( -24\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+20\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +36\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+5\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -10\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -18\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+3\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a*e^2*(-24*sin(1/2*d*x+1/2*c)^7+20*sin(1/2*d*x+1/
2*c)^4*cos(1/2*d*x+1/2*c)+36*sin(1/2*d*x+1/2*c)^5+5*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-10*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-18*sin(1/2*d*x+1/2*c)^3+3
*sin(1/2*d*x+1/2*c))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )\right )} \sqrt{e \cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*e*cos(d*x + c)*sin(d*x + c) + a*e*cos(d*x + c))*sqrt(e*cos(d*x + c)), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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