3.6.41 \(\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \, dx\) [541]

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

[Out]

-2/5*b*(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

________________________________________________________________________________________

Rubi [A]
time = 0.05, 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 = {2748, 2715, 2721, 2720} \begin {gather*} \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 \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(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 2715

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] && Integ
erQ[2*n]

Rule 2720

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

Rule 2721

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

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[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+b \sin (c+d x)) \, dx &=-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e}+a \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 b (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 b (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 b (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 = 0.49, size = 79, normalized size = 0.83 \begin {gather*} \frac {(e \cos (c+d x))^{3/2} \left (10 a F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sqrt {\cos (c+d x)} (-3 b-3 b \cos (2 (c+d x))+10 a \sin (c+d x))\right )}{15 d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 3.95, size = 185, normalized size = 1.95

method	result	size
default	\(-\frac {2 e^{2} \left (-24 b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+36 b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a -10 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18 b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 93, normalized size = 0.98 \begin {gather*} \frac {-5 i \, \sqrt {2} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (3 \, b \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - 5 \, a e^{\frac {3}{2}} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{15 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(-5*I*sqrt(2)*a*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*I*sqrt(2)*a*e^(3/2)
*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(3*b*cos(d*x + c)^2*e^(3/2) - 5*a*e^(3/2)*sin(d
*x + c))*sqrt(cos(d*x + c)))/d

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________