3.2.97 \(\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx\) [197]

Optimal. Leaf size=95 \[ -\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {6 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]

[Out]

-2/7*a*(e*cos(d*x+c))^(7/2)/d/e+2/5*a*e*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/d+6/5*a*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/
2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)

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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, 2719} \begin {gather*} \frac {6 a e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*(e*Cos[c + d*x])^(7/2))/(7*d*e) + (6*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos
[c + d*x]]) + (2*a*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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))^{5/2} (a+a \sin (c+d x)) \, dx &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+a \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} \left (3 a e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\left (3 a e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {6 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 1.79, size = 264, normalized size = 2.78 \begin {gather*} \frac {a e^3 \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \left (-154 \cos (d x)-182 \cos (2 c+d x)+14 \cos (2 c+3 d x)-14 \cos (4 c+3 d x)-30 \sin (c)+168 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right ) (\cos (d x)-i \sin (d x)) \sqrt {1+\cos (2 (c+d x))+i \sin (2 (c+d x))}+56 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right ) (\cos (d x)+i \sin (d x)) \sqrt {1+\cos (2 (c+d x))+i \sin (2 (c+d x))}+20 \sin (c+2 d x)-20 \sin (3 c+2 d x)+5 \sin (3 c+4 d x)-5 \sin (5 c+4 d x)\right )}{560 d \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e^3*Csc[c/2]*Sec[c/2]*(-154*Cos[d*x] - 182*Cos[2*c + d*x] + 14*Cos[2*c + 3*d*x] - 14*Cos[4*c + 3*d*x] - 30*
Sin[c] + 168*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] - I*Sin[d*x])
*Sqrt[1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]] + 56*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c]
+ I*Sin[c])^2)]*(Cos[d*x] + I*Sin[d*x])*Sqrt[1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]] + 20*Sin[c + 2*d*x] -
20*Sin[3*c + 2*d*x] + 5*Sin[3*c + 4*d*x] - 5*Sin[5*c + 4*d*x]))/(560*d*Sqrt[e*Cos[c + d*x]])

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Maple [A]
time = 1.77, size = 214, normalized size = 2.25

method	result	size
default	\(\frac {2 a \,e^{3} \left (-80 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+160 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-120 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+40 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(214\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/35/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a*e^3*(-80*sin(1/2*d*x+1/2*c)^9+56*sin(1/2*d*x+1/2
*c)^6*cos(1/2*d*x+1/2*c)+160*sin(1/2*d*x+1/2*c)^7-56*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-120*sin(1/2*d*x+1
/2*c)^5+21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))
+14*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+40*sin(1/2*d*x+1/2*c)^3-5*sin(1/2*d*x+1/2*c))/d

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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))^(5/2)*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 105, normalized size = 1.11 \begin {gather*} \frac {21 i \, \sqrt {2} a e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} a e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (5 \, a \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - 7 \, a \cos \left (d x + c\right ) e^{\frac {5}{2}} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{35 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/35*(21*I*sqrt(2)*a*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)))
- 21*I*sqrt(2)*a*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) -
2*(5*a*cos(d*x + c)^3*e^(5/2) - 7*a*cos(d*x + c)*e^(5/2)*sin(d*x + c))*sqrt(cos(d*x + c)))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4849 deep

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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))^(5/2)*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (a+a\,\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))^(5/2)*(a + a*sin(c + d*x)),x)

[Out]

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

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