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]

-2/5*a*(e*cos(d*x+c))^(5/2)/d/e+2/3*a*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*
x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/(e*cos(d*x+c))^(1/2)+2/3*a*e*sin(d*x+c)*(e*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 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 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]

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 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])

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*}

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Mathematica [A]  time = 0.38, 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]

-1/15*(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*Sin[c + d*x])))/(d*Cos[c + d*x]^(3/2))

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}\,{d x} \]

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)

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maple [A]  time = 0.62, size = 179, normalized size = 1.88 \[ -\frac {2 a \,e^{2} \left (-24 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+36 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-18 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]

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*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),
2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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