3.3.25 \(\int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx\) [225]

Optimal. Leaf size=178 \[ -\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}+\frac {22 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e} \]

[Out]

-22/9*a^4*(e*cos(d*x+c))^(3/2)/d/e-2/9*a*(e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^3/d/e-10/21*(e*cos(d*x+c))^(3/2
)*(a^2+a^2*sin(d*x+c))^2/d/e-22/21*(e*cos(d*x+c))^(3/2)*(a^4+a^4*sin(d*x+c))/d/e+22/3*a^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)

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Rubi [A]
time = 0.14, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2757, 2748, 2721, 2719} \begin {gather*} -\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {22 \left (a^4 \sin (c+d x)+a^4\right ) (e \cos (c+d x))^{3/2}}{21 d e}+\frac {22 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}-\frac {10 \left (a^2 \sin (c+d x)+a^2\right )^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-22*a^4*(e*Cos[c + d*x])^(3/2))/(9*d*e) + (22*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*d*Sqrt[C
os[c + d*x]]) - (2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3)/(9*d*e) - (10*(e*Cos[c + d*x])^(3/2)*(a^2
+ a^2*Sin[c + d*x])^2)/(21*d*e) - (22*(e*Cos[c + d*x])^(3/2)*(a^4 + a^4*Sin[c + d*x]))/(21*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 2757

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

Rubi steps

\begin {align*} \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}+\frac {1}{3} (5 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}+\frac {1}{21} \left (55 a^2\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac {1}{3} \left (11 a^3\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x)) \, dx\\ &=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac {1}{3} \left (11 a^4\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac {\left (11 a^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{3 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}+\frac {22 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.06, size = 66, normalized size = 0.37 \begin {gather*} -\frac {32\ 2^{3/4} a^4 (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {15}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (1+\sin (c+d x))^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*2^(3/4)*a^4*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-15/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 +
Sin[c + d*x])^(3/4))

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Maple [A]
time = 2.10, size = 258, normalized size = 1.45

method	result	size
default	\(\frac {2 a^{4} e \left (224 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+576 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-392 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1152 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+616 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+192 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-168 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+384 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-132 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(258\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^4*e*(224*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c
)-448*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+576*sin(1/2*d*x+1/2*c)^9-392*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/
2*c)-1152*sin(1/2*d*x+1/2*c)^7+616*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+192*sin(1/2*d*x+1/2*c)^5+231*(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))-168*sin(1/2*d*x+
1/2*c)^2*cos(1/2*d*x+1/2*c)+384*sin(1/2*d*x+1/2*c)^3-132*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((a+a*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 143, normalized size = 0.80 \begin {gather*} \frac {231 i \, \sqrt {2} a^{4} 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 ) - 231 i \, \sqrt {2} a^{4} 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 (36 \, a^{4} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 168 \, a^{4} \cos \left (d x + c\right ) e^{\frac {1}{2}} + 7 \, {\left (a^{4} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 13 \, a^{4} \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 )}}{63 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/63*(231*I*sqrt(2)*a^4*e^(1/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c
))) - 231*I*sqrt(2)*a^4*e^(1/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c
))) + 2*(36*a^4*cos(d*x + c)^3*e^(1/2) - 168*a^4*cos(d*x + c)*e^(1/2) + 7*(a^4*cos(d*x + c)^3*e^(1/2) - 13*a^4
*cos(d*x + c)*e^(1/2))*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((a+a*sin(d*x+c))**4*(e*cos(d*x+c))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3064 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((a+a*sin(d*x+c))^4*(e*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\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 + a*sin(c + d*x))^4,x)

[Out]

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

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