3.6.56 \(\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx\) [556]

Optimal. Leaf size=197 \[ -\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e} \]

[Out]

-2/231*b*(43*a^2+12*b^2)*(e*cos(d*x+c))^(7/2)/d/e+2/15*a*(3*a^2+2*b^2)*e*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/d-10/
33*a*b*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))/d/e-2/11*b*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^2/d/e+2/5*a*(3*a
^2+2*b^2)*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 = 0.19, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2771, 2941, 2748, 2715, 2721, 2719} \begin {gather*} \frac {2 a e^2 \left (3 a^2+2 b^2\right ) 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 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a e \left (3 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3,x]

[Out]

(-2*b*(43*a^2 + 12*b^2)*(e*Cos[c + d*x])^(7/2))/(231*d*e) + (2*a*(3*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*Elli
pticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(3*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/
(15*d) - (10*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(33*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Si
n[c + d*x])^2)/(11*d*e)

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])

Rule 2771

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 - 1)/(f*g*(m + p))), x] + Dist[1/(m + p), Int[(g*Cos[e + f*x]
)^p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ
[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ
[m])

Rule 2941

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*
d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] &&
GtQ[m, 0] &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin {align*} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (\frac {11 a^2}{2}+2 b^2+\frac {15}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {4}{99} \int (e \cos (c+d x))^{5/2} \left (\frac {33}{4} a \left (3 a^2+2 b^2\right )+\frac {3}{4} b \left (43 a^2+12 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {1}{3} \left (a \left (3 a^2+2 b^2\right )\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {1}{5} \left (a \left (3 a^2+2 b^2\right ) e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {\left (a \left (3 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) 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 \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\\ \end {align*}

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Mathematica [A]
time = 1.42, size = 150, normalized size = 0.76 \begin {gather*} \frac {(e \cos (c+d x))^{5/2} \left (1848 \left (3 a^3+2 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (-1980 a^2 b-345 b^3-60 \left (33 a^2 b+4 b^3\right ) \cos (2 (c+d x))+105 b^3 \cos (4 (c+d x))+1848 a^3 \sin (c+d x)+462 a b^2 \sin (c+d x)-770 a b^2 \sin (3 (c+d x))\right )\right )}{4620 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3,x]

[Out]

((e*Cos[c + d*x])^(5/2)*(1848*(3*a^3 + 2*a*b^2)*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-1980*a^2*b -
345*b^3 - 60*(33*a^2*b + 4*b^3)*Cos[2*(c + d*x)] + 105*b^3*Cos[4*(c + d*x)] + 1848*a^3*Sin[c + d*x] + 462*a*b^
2*Sin[c + d*x] - 770*a*b^2*Sin[3*(c + d*x)])))/(4620*d*Cos[c + d*x]^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(533\) vs. \(2(201)=402\).
time = 9.69, size = 534, normalized size = 2.71

method	result	size
default	\(\frac {2 e^{3} \left (6720 b^{3} \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12320 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20160 b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7920 a^{2} b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24640 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22560 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1848 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15840 a^{2} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-17248 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11520 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1848 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11880 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4928 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2340 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+693 \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 ) a^{3}+462 \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 ) a \,b^{2}+462 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3960 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-462 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+60 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-495 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-60 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1155 \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}\)	\(534\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

2/1155/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^3*(6720*b^3*sin(1/2*d*x+1/2*c)^13-12320*a*b^2*
cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10-20160*b^3*sin(1/2*d*x+1/2*c)^11-7920*a^2*b*sin(1/2*d*x+1/2*c)^9+24640
*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+22560*b^3*sin(1/2*d*x+1/2*c)^9+1848*a^3*cos(1/2*d*x+1/2*c)*sin(
1/2*d*x+1/2*c)^6+15840*a^2*b*sin(1/2*d*x+1/2*c)^7-17248*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-11520*b^
3*sin(1/2*d*x+1/2*c)^7-1848*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-11880*a^2*b*sin(1/2*d*x+1/2*c)^5+4928*
a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+2340*b^3*sin(1/2*d*x+1/2*c)^5+693*(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))*a^3+462*(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))*a*b^2+462*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d
*x+1/2*c)^2+3960*a^2*b*sin(1/2*d*x+1/2*c)^3-462*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+60*b^3*sin(1/2*d
*x+1/2*c)^3-495*a^2*b*sin(1/2*d*x+1/2*c)-60*b^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 \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+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

e^(5/2)*integrate((b*sin(d*x + c) + a)^3*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 = 0.14, size = 181, normalized size = 0.92 \begin {gather*} \frac {231 i \, \sqrt {2} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} 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 ) - 231 i \, \sqrt {2} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} 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 (105 \, b^{3} \cos \left (d x + c\right )^{5} e^{\frac {5}{2}} - 165 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - 77 \, {\left (5 \, a b^{2} \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {5}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{1155 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/1155*(231*I*sqrt(2)*(3*a^3 + 2*a*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
+ I*sin(d*x + c))) - 231*I*sqrt(2)*(3*a^3 + 2*a*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0
, cos(d*x + c) - I*sin(d*x + c))) + 2*(105*b^3*cos(d*x + c)^5*e^(5/2) - 165*(3*a^2*b + b^3)*cos(d*x + c)^3*e^(
5/2) - 77*(5*a*b^2*cos(d*x + c)^3*e^(5/2) - (3*a^3 + 2*a*b^2)*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+b*sin(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 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+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^3*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+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3,x)

[Out]

int((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3, x)

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