3.6.68 \(\int \frac {(a+b \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx\) [568]

Optimal. Leaf size=210 \[ -\frac {6 a b \left (31 a^2+34 b^2\right ) \sqrt {e \cos (c+d x)}}{35 d e}+\frac {2 \left (7 a^4+28 a^2 b^2+4 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (29 a^2+10 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e} \]

[Out]

2/7*(7*a^4+28*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)*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)-6/35*a*b*(31*a^2+34*b^2)*(e*cos(d*x+c))^(1/2)/d/e-2/35*b*(29*a^2+
10*b^2)*(a+b*sin(d*x+c))*(e*cos(d*x+c))^(1/2)/d/e-26/35*a*b*(a+b*sin(d*x+c))^2*(e*cos(d*x+c))^(1/2)/d/e-2/7*b*
(a+b*sin(d*x+c))^3*(e*cos(d*x+c))^(1/2)/d/e

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Rubi [A]
time = 0.30, 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, 2720} \begin {gather*} -\frac {6 a b \left (31 a^2+34 b^2\right ) \sqrt {e \cos (c+d x)}}{35 d e}-\frac {2 b \left (29 a^2+10 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}+\frac {2 \left (7 a^4+28 a^2 b^2+4 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a*b*(31*a^2 + 34*b^2)*Sqrt[e*Cos[c + d*x]])/(35*d*e) + (2*(7*a^4 + 28*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*
EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) - (2*b*(29*a^2 + 10*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin
[c + d*x]))/(35*d*e) - (26*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(35*d*e) - (2*b*Sqrt[e*Cos[c + d*x
]]*(a + b*Sin[c + d*x])^3)/(7*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))^4}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}+\frac {2}{7} \int \frac {(a+b \sin (c+d x))^2 \left (\frac {7 a^2}{2}+3 b^2+\frac {13}{2} a b \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}+\frac {4}{35} \int \frac {(a+b \sin (c+d x)) \left (\frac {1}{4} a \left (35 a^2+82 b^2\right )+\frac {3}{4} b \left (29 a^2+10 b^2\right ) \sin (c+d x)\right )}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (29 a^2+10 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}+\frac {8}{105} \int \frac {\frac {15}{8} \left (7 a^4+28 a^2 b^2+4 b^4\right )+\frac {9}{8} a b \left (31 a^2+34 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {6 a b \left (31 a^2+34 b^2\right ) \sqrt {e \cos (c+d x)}}{35 d e}-\frac {2 b \left (29 a^2+10 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}+\frac {1}{7} \left (7 a^4+28 a^2 b^2+4 b^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {6 a b \left (31 a^2+34 b^2\right ) \sqrt {e \cos (c+d x)}}{35 d e}-\frac {2 b \left (29 a^2+10 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}+\frac {\left (\left (7 a^4+28 a^2 b^2+4 b^4\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 \sqrt {e \cos (c+d x)}}\\ &=-\frac {6 a b \left (31 a^2+34 b^2\right ) \sqrt {e \cos (c+d x)}}{35 d e}+\frac {2 \left (7 a^4+28 a^2 b^2+4 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (29 a^2+10 b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}-\frac {26 a b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}\\ \end {align*}

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Mathematica [A]
time = 1.18, size = 130, normalized size = 0.62 \begin {gather*} \frac {20 \left (7 a^4+28 a^2 b^2+4 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-b \cos (c+d x) \left (560 a^3+504 a b^2-56 a b^2 \cos (2 (c+d x))+5 b \left (56 a^2+11 b^2\right ) \sin (c+d x)-5 b^3 \sin (3 (c+d x))\right )}{70 d \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*(7*a^4 + 28*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - b*Cos[c + d*x]*(560*a^3 + 504*
a*b^2 - 56*a*b^2*Cos[2*(c + d*x)] + 5*b*(56*a^2 + 11*b^2)*Sin[c + d*x] - 5*b^3*Sin[3*(c + d*x)]))/(70*d*Sqrt[e
*Cos[c + d*x]])

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Maple [A]
time = 6.46, size = 412, normalized size = 1.96

method	result	size
default	\(-\frac {2 \left (80 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+224 a \,b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-280 a^{2} b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 a \,b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{4}+140 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b^{2}+20 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{4}-280 a^{3} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 a^{2} b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-112 a \,b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 a^{3} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+112 a \,b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(412\)

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]

-2/35/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(80*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+2
24*a*b^3*sin(1/2*d*x+1/2*c)^7-120*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-280*a^2*b^2*cos(1/2*d*x+1/2*c)*s
in(1/2*d*x+1/2*c)^4-336*a*b^3*sin(1/2*d*x+1/2*c)^5+35*(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^4+140*(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^2*b^2+20*(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))*b^4-280*a^3*b*sin(1/2*d*x+1/2*c)^3+140*a^2*b^2*cos(1/2*d*x+1/2*c)*s
in(1/2*d*x+1/2*c)^2-112*a*b^3*sin(1/2*d*x+1/2*c)^3+20*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+140*a^3*b*si
n(1/2*d*x+1/2*c)+112*a*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))^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)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 163, normalized size = 0.78 \begin {gather*} -\frac {{\left (5 \, \sqrt {2} {\left (7 i \, a^{4} + 28 i \, a^{2} b^{2} + 4 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, a^{4} - 28 i \, a^{2} b^{2} - 4 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (28 \, a b^{3} \cos \left (d x + c\right )^{2} - 140 \, a^{3} b - 140 \, a b^{3} + 5 \, {\left (b^{4} \cos \left (d x + c\right )^{2} - 14 \, a^{2} b^{2} - 3 \, b^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{35 \, 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]

-1/35*(5*sqrt(2)*(7*I*a^4 + 28*I*a^2*b^2 + 4*I*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
+ 5*sqrt(2)*(-7*I*a^4 - 28*I*a^2*b^2 - 4*I*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*
(28*a*b^3*cos(d*x + c)^2 - 140*a^3*b - 140*a*b^3 + 5*(b^4*cos(d*x + c)^2 - 14*a^2*b^2 - 3*b^4)*sin(d*x + c))*s
qrt(cos(d*x + c)))*e^(-1/2)/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+b*sin(d*x+c))**4/(e*cos(d*x+c))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3065 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+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*e^(-1/2)/sqrt(cos(d*x + c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4}{\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))^4/(e*cos(c + d*x))^(1/2),x)

[Out]

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

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