3.6.67 \(\int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx\) [567]

Optimal. Leaf size=210 \[ -\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}+\frac {2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e} \]

[Out]

-22/315*a*b*(17*a^2+18*b^2)*(e*cos(d*x+c))^(3/2)/d/e-2/105*b*(41*a^2+14*b^2)*(e*cos(d*x+c))^(3/2)*(a+b*sin(d*x
+c))/d/e-10/21*a*b*(e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^2/d/e-2/9*b*(e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^3/d
/e+2/15*(15*a^4+36*a^2*b^2+4*b^4)*(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)

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2771, 2941, 2748, 2721, 2719} \begin {gather*} -\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac {2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4,x]

[Out]

(-22*a*b*(17*a^2 + 18*b^2)*(e*Cos[c + d*x])^(3/2))/(315*d*e) + (2*(15*a^4 + 36*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c +
d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b*(41*a^2 + 14*b^2)*(e*Cos[c + d*x])^(3/2)*(a
+ b*Sin[c + d*x]))/(105*d*e) - (10*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(21*d*e) - (2*b*(e*Cos[
c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3)/(9*d*e)

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 \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx &=-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {2}{9} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (\frac {9 a^2}{2}+3 b^2+\frac {15}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {4}{63} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac {3}{4} a \left (21 a^2+34 b^2\right )+\frac {3}{4} b \left (41 a^2+14 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {8}{315} \int \sqrt {e \cos (c+d x)} \left (\frac {21}{8} \left (15 a^4+36 a^2 b^2+4 b^4\right )+\frac {33}{8} a b \left (17 a^2+18 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {1}{15} \left (15 a^4+36 a^2 b^2+4 b^4\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {\left (\left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}+\frac {2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.12, size = 137, normalized size = 0.65 \begin {gather*} \frac {\sqrt {e \cos (c+d x)} \left (84 \left (15 a^4+36 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )-b \cos ^{\frac {3}{2}}(c+d x) \left (-360 a b^2 \cos (2 (c+d x))+21 b \left (72 a^2+13 b^2\right ) \sin (c+d x)+5 \left (336 a^3+264 a b^2-7 b^3 \sin (3 (c+d x))\right )\right )\right )}{630 d \sqrt {\cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4,x]

[Out]

(Sqrt[e*Cos[c + d*x]]*(84*(15*a^4 + 36*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x)/2, 2] - b*Cos[c + d*x]^(3/2)*(-360
*a*b^2*Cos[2*(c + d*x)] + 21*b*(72*a^2 + 13*b^2)*Sin[c + d*x] + 5*(336*a^3 + 264*a*b^2 - 7*b^3*Sin[3*(c + d*x)
]))))/(630*d*Sqrt[Cos[c + d*x]])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(524\) vs. \(2(214)=428\).
time = 8.57, size = 525, normalized size = 2.50

method	result	size
default	\(\frac {2 e \left (1120 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2880 a \,b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2240 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3024 a^{2} 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 )-5760 a \,b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1064 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 a^{3} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3024 a^{2} 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 )+2640 a \,b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \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^{4}+756 \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^{2} b^{2}+84 \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 ) b^{4}+1680 a^{3} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-756 a^{2} 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 )+240 a \,b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-84 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 a^{3} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-240 a \,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}\)	\(525\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),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*(1120*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)
^10+2880*a*b^3*sin(1/2*d*x+1/2*c)^9-2240*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-3024*a^2*b^2*cos(1/2*d*x+
1/2*c)*sin(1/2*d*x+1/2*c)^6-5760*a*b^3*sin(1/2*d*x+1/2*c)^7+1064*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1
680*a^3*b*sin(1/2*d*x+1/2*c)^5+3024*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+2640*a*b^3*sin(1/2*d*x+1/2
*c)^5+56*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+315*(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^4+756*(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^2*b^2+84*(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))*b^4+1680*a^3*b*sin(1/2*d*x+1/2*c)^3-756*a^2*b^2*cos(1/2*d*x+1/2*
c)*sin(1/2*d*x+1/2*c)^2+240*a*b^3*sin(1/2*d*x+1/2*c)^3-84*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-420*a^3*
b*sin(1/2*d*x+1/2*c)-240*a*b^3*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((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

e^(1/2)*integrate((b*sin(d*x + c) + a)^4*sqrt(cos(d*x + c)), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 196, normalized size = 0.93 \begin {gather*} \frac {21 i \, \sqrt {2} {\left (15 \, a^{4} + 36 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {1}{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} {\left (15 \, a^{4} + 36 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {1}{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 (180 \, a b^{3} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 420 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right ) e^{\frac {1}{2}} + 7 \, {\left (5 \, b^{4} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - {\left (54 \, a^{2} b^{2} + 11 \, b^{4}\right )} \cos \left (d x + c\right ) e^{\frac {1}{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((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/315*(21*I*sqrt(2)*(15*a^4 + 36*a^2*b^2 + 4*b^4)*e^(1/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co
s(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*(15*a^4 + 36*a^2*b^2 + 4*b^4)*e^(1/2)*weierstrassZeta(-4, 0, weie
rstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(180*a*b^3*cos(d*x + c)^3*e^(1/2) - 420*(a^3*b + a*
b^3)*cos(d*x + c)*e^(1/2) + 7*(5*b^4*cos(d*x + c)^3*e^(1/2) - (54*a^2*b^2 + 11*b^4)*cos(d*x + c)*e^(1/2))*sin(
d*x + c))*sqrt(cos(d*x + c)))/d

________________________________________________________________________________________

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((a+b*sin(d*x+c))**4*(e*cos(d*x+c))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3064 deep

________________________________________________________________________________________

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((a+b*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^4*sqrt(cos(d*x + c))*e^(1/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(c + d*x))^(1/2)*(a + b*sin(c + d*x))^4,x)

[Out]

int((e*cos(c + d*x))^(1/2)*(a + b*sin(c + d*x))^4, x)

________________________________________________________________________________________