3.6.59 \(\int \frac {(a+b \sin (c+d x))^3}{\sqrt {e \cos (c+d x)}} \, dx\) [559]

Optimal. Leaf size=152 \[ -\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{5 d e}+\frac {2 a \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}-\frac {6 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e} \]

[Out]

2*a*(a^2+2*b^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)-2/5*b*(11*a^2+4*b^2)*(e*cos(d*x+c))^(1/2)/d/e-6/5*a*b*(a+b*sin(d*x+c))*(e*cos(
d*x+c))^(1/2)/d/e-2/5*b*(a+b*sin(d*x+c))^2*(e*cos(d*x+c))^(1/2)/d/e

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Rubi [A]
time = 0.16, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2771, 2941, 2748, 2721, 2720} \begin {gather*} -\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{5 d e}+\frac {2 a \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}-\frac {6 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(11*a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]])/(5*d*e) + (2*a*(a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*
x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]]) - (6*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(5*d*e) - (2*b*Sqrt[e*C
os[c + d*x]]*(a + b*Sin[c + d*x])^2)/(5*d*e)

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 \frac {(a+b \sin (c+d x))^3}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}+\frac {2}{5} \int \frac {(a+b \sin (c+d x)) \left (\frac {5 a^2}{2}+2 b^2+\frac {9}{2} a b \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {6 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}+\frac {4}{15} \int \frac {\frac {15}{4} a \left (a^2+2 b^2\right )+\frac {3}{4} b \left (11 a^2+4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{5 d e}-\frac {6 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}+\left (a \left (a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{5 d e}-\frac {6 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}+\frac {\left (a \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{\sqrt {e \cos (c+d x)}}\\ &=-\frac {2 b \left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}}{5 d e}+\frac {2 a \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}-\frac {6 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}\\ \end {align*}

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Mathematica [A]
time = 0.82, size = 94, normalized size = 0.62 \begin {gather*} \frac {10 a \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \cos (c+d x) \left (-30 a^2-9 b^2+b^2 \cos (2 (c+d x))-10 a b \sin (c+d x)\right )}{5 d \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*a*(a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*Cos[c + d*x]*(-30*a^2 - 9*b^2 + b^2*Cos[2
*(c + d*x)] - 10*a*b*Sin[c + d*x]))/(5*d*Sqrt[e*Cos[c + d*x]])

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Maple [A]
time = 6.06, size = 279, normalized size = 1.84

method	result	size
default	\(-\frac {2 \left (8 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}-30 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+4 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(279\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(8*b^3*sin(1/2*d*x+1/2*c)^7-20*a*b^2*cos(1/2*d*x+1
/2*c)*sin(1/2*d*x+1/2*c)^4-12*b^3*sin(1/2*d*x+1/2*c)^5+5*(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+10*(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^2-30*a^2*b*sin(1/2*d*x+1/2*c)^3+10*a*b^2*cos(1/2*d*x+1/2*c)*si
n(1/2*d*x+1/2*c)^2-4*b^3*sin(1/2*d*x+1/2*c)^3+15*a^2*b*sin(1/2*d*x+1/2*c)+4*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((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

e^(-1/2)*integrate((b*sin(d*x + c) + a)^3/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 = 0.10, size = 124, normalized size = 0.82 \begin {gather*} -\frac {{\left (5 \, \sqrt {2} {\left (i \, a^{3} + 2 i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, a^{3} - 2 i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} - 5 \, a b^{2} \sin \left (d x + c\right ) - 15 \, a^{2} b - 5 \, b^{3}\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{5 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-1/5*(5*sqrt(2)*(I*a^3 + 2*I*a*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-I*
a^3 - 2*I*a*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(b^3*cos(d*x + c)^2 - 5*a*b^2*s
in(d*x + c) - 15*a^2*b - 5*b^3)*sqrt(cos(d*x + c)))*e^(-1/2)/d

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c))**3/(e*cos(d*x+c))**(1/2),x)

[Out]

Timed out

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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+b*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^3*e^(-1/2)/sqrt(cos(d*x + c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(1/2),x)

[Out]

int((a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(1/2), x)

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