3.6.57 \(\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx\) [557]

Optimal. Leaf size=197 \[ -\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e} \]

[Out]

-2/315*b*(89*a^2+28*b^2)*(e*cos(d*x+c))^(5/2)/d/e-26/63*a*b*(e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))/d/e-2/9*b*(e
*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^2/d/e+2/21*a*(7*a^2+6*b^2)*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)+2/21*a*(7*a^2+6*b^2)*e*sin(
d*x+c)*(e*cos(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.18, 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, 2720} \begin {gather*} \frac {2 a e^2 \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a e \left (7 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(89*a^2 + 28*b^2)*(e*Cos[c + d*x])^(5/2))/(315*d*e) + (2*a*(7*a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*Ellipt
icF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (2*a*(7*a^2 + 6*b^2)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(
21*d) - (26*a*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(63*d*e) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin
[c + d*x])^2)/(9*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 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])

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))^{3/2} (a+b \sin (c+d x))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {2}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (\frac {9 a^2}{2}+2 b^2+\frac {13}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {4}{63} \int (e \cos (c+d x))^{3/2} \left (\frac {9}{4} a \left (7 a^2+6 b^2\right )+\frac {1}{4} b \left (89 a^2+28 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {1}{7} \left (a \left (7 a^2+6 b^2\right )\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {1}{21} \left (a \left (7 a^2+6 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {\left (a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {e \cos (c+d x)}}\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\\ \end {align*}

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Mathematica [A]
time = 1.42, size = 153, normalized size = 0.78 \begin {gather*} \frac {(e \cos (c+d x))^{3/2} \left (80 \left (7 a^3+6 a b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {2}{3} \sqrt {\cos (c+d x)} \left (-756 a^2 b-147 b^3-28 \left (27 a^2 b+4 b^3\right ) \cos (2 (c+d x))+35 b^3 \cos (4 (c+d x))+840 a^3 \sin (c+d x)+450 a b^2 \sin (c+d x)-270 a b^2 \sin (3 (c+d x))\right )\right )}{840 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])^3,x]

[Out]

((e*Cos[c + d*x])^(3/2)*(80*(7*a^3 + 6*a*b^2)*EllipticF[(c + d*x)/2, 2] + (2*Sqrt[Cos[c + d*x]]*(-756*a^2*b -
147*b^3 - 28*(27*a^2*b + 4*b^3)*Cos[2*(c + d*x)] + 35*b^3*Cos[4*(c + d*x)] + 840*a^3*Sin[c + d*x] + 450*a*b^2*
Sin[c + d*x] - 270*a*b^2*Sin[3*(c + d*x)]))/3))/(840*d*Cos[c + d*x]^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(449\) vs. \(2(201)=402\).
time = 8.31, size = 450, normalized size = 2.28

method	result	size
default	\(-\frac {2 e^{2} \left (1120 b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2160 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 )-2800 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1512 a^{2} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3240 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 )+2296 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+420 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 )+2268 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1260 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 )-644 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \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^{3}+90 \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 \,b^{2}-210 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 )-1134 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+90 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 )-28 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+189 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+28 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \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}\)	\(450\)

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))^3,x,method=_RETURNVERBOSE)

[Out]

-2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^2*(1120*b^3*sin(1/2*d*x+1/2*c)^11-2160*a*b^2*c
os(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-2800*b^3*sin(1/2*d*x+1/2*c)^9-1512*a^2*b*sin(1/2*d*x+1/2*c)^7+3240*a*b^
2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+2296*b^3*sin(1/2*d*x+1/2*c)^7+420*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^4+2268*a^2*b*sin(1/2*d*x+1/2*c)^5-1260*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-644*b^3*sin(1/2*d
*x+1/2*c)^5+105*(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^3+90*(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*b^2-210*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-1134*a^2*b*sin(1/2*d*x+1/2*c)^3+90*a*b^2*cos(1/2*d*x+
1/2*c)*sin(1/2*d*x+1/2*c)^2-28*b^3*sin(1/2*d*x+1/2*c)^3+189*a^2*b*sin(1/2*d*x+1/2*c)+28*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))^(3/2)*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 169, normalized size = 0.86 \begin {gather*} \frac {-15 i \, \sqrt {2} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} 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 (35 \, b^{3} \cos \left (d x + c\right )^{4} e^{\frac {3}{2}} - 63 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - 15 \, {\left (9 \, a b^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{315 \, 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))^3,x, algorithm="fricas")

[Out]

1/315*(-15*I*sqrt(2)*(7*a^3 + 6*a*b^2)*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*
I*sqrt(2)*(7*a^3 + 6*a*b^2)*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(35*b^3*cos(
d*x + c)^4*e^(3/2) - 63*(3*a^2*b + b^3)*cos(d*x + c)^2*e^(3/2) - 15*(9*a*b^2*cos(d*x + c)^2*e^(3/2) - (7*a^3 +
6*a*b^2)*e^(3/2))*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))**(3/2)*(a+b*sin(d*x+c))**3,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))^(3/2)*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

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

[Out]

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

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