3.3.70 \(\int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx\) [270]

Optimal. Leaf size=191 \[ -\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )} \]

[Out]

-2/13*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))^4-10/117*(e*cos(d*x+c))^(3/2)/a/d/e/(a+a*sin(d*x+c))^3-2/39*(e
*cos(d*x+c))^(3/2)/d/e/(a^2+a^2*sin(d*x+c))^2-2/39*(e*cos(d*x+c))^(3/2)/d/e/(a^4+a^4*sin(d*x+c))-2/39*(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)/a^4/d/cos(d
*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 191, 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 = {2760, 2762, 2721, 2719} \begin {gather*} -\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4 \sin (c+d x)+a^4\right )}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(39*a^4*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))
/(13*d*e*(a + a*Sin[c + d*x])^4) - (10*(e*Cos[c + d*x])^(3/2))/(117*a*d*e*(a + a*Sin[c + d*x])^3) - (2*(e*Cos[
c + d*x])^(3/2))/(39*d*e*(a^2 + a^2*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(39*d*e*(a^4 + a^4*Sin[c + d
*x]))

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 2760

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1))), x] + Dist[(m + p + 1)/(a*(2*m + p + 1)),
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] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rule 2762

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*((g*Cos[e
+ f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*Sin[e + f*x]))), x] + Dist[p/(a*(p - 1)), Int[(g*Cos[e + f*x])^p, x], x
] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] &&  !GeQ[p, 1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}+\frac {5 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx}{13 a}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}+\frac {5 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx}{39 a^2}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {\int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{39 a^3}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\int \sqrt {e \cos (c+d x)} \, dx}{39 a^4}\\ &=-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{39 a^4 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.05, size = 66, normalized size = 0.35 \begin {gather*} -\frac {(e \cos (c+d x))^{3/2} \, _2F_1\left (\frac {3}{4},\frac {17}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{12 \sqrt [4]{2} a^4 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]

-1/12*((e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 17/4, 7/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^4*d*e*(1 + S
in[c + d*x])^(3/4))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(693\) vs. \(2(195)=390\).
time = 14.62, size = 694, normalized size = 3.63

method	result	size
default	\(-\frac {2 \left (192 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-384 \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-576 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1152 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+720 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1472 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-480 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1024 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+180 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-280 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-36 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-208 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \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 )-120 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+208 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}{117 \left (64 \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-192 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-160 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+60 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a^{4} \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}\)	\(694\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/117/(64*sin(1/2*d*x+1/2*c)^12-192*sin(1/2*d*x+1/2*c)^10+240*sin(1/2*d*x+1/2*c)^8-160*sin(1/2*d*x+1/2*c)^6+6
0*sin(1/2*d*x+1/2*c)^4-12*sin(1/2*d*x+1/2*c)^2+1)/a^4/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(
192*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/
2*d*x+1/2*c)^12-384*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-576*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(co
s(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^10+1152*cos(1/2*d*x+1/2*c)*sin(1/2*d
*x+1/2*c)^12+720*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*sin(1/2*d*x+1/2*c)^8-1472*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-480*EllipticE(cos(1/2*d*x+1/2*c),2^(
1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^6+1024*cos(1/2*d*x+1/2*
c)*sin(1/2*d*x+1/2*c)^8+180*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+
1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-280*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-36*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2-40*sin(1/2*d*
x+1/2*c)^4*cos(1/2*d*x+1/2*c)-208*sin(1/2*d*x+1/2*c)^5+3*(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))-120*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+208*sin(1/2*d*x+1/2
*c)^3+20*sin(1/2*d*x+1/2*c))*e/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))^(1/2)/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 520, normalized size = 2.72 \begin {gather*} \frac {3 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} e^{\frac {1}{2}} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} + 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} + 4 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} - 8 i \, \sqrt {2} e^{\frac {1}{2}}\right )} \sin \left (d x + c\right ) - 8 i \, \sqrt {2} e^{\frac {1}{2}}\right )} {\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 ) + 3 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} e^{\frac {1}{2}} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 4 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + 8 i \, \sqrt {2} e^{\frac {1}{2}}\right )} \sin \left (d x + c\right ) + 8 i \, \sqrt {2} e^{\frac {1}{2}}\right )} {\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 (3 \, \cos \left (d x + c\right )^{4} e^{\frac {1}{2}} + 12 \, \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 14 \, \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 32 \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + {\left (3 \, \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 9 \, \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 23 \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + 9 \, e^{\frac {1}{2}}\right )} \sin \left (d x + c\right ) - 9 \, e^{\frac {1}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{117 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 3 \, a^{4} d \cos \left (d x + c\right )^{3} - 8 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + 8 \, a^{4} d - {\left (a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d \cos \left (d x + c\right ) - 8 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/117*(3*(-I*sqrt(2)*cos(d*x + c)^4*e^(1/2) + 3*I*sqrt(2)*cos(d*x + c)^3*e^(1/2) + 8*I*sqrt(2)*cos(d*x + c)^2*
e^(1/2) - 4*I*sqrt(2)*cos(d*x + c)*e^(1/2) + (I*sqrt(2)*cos(d*x + c)^3*e^(1/2) + 4*I*sqrt(2)*cos(d*x + c)^2*e^
(1/2) - 4*I*sqrt(2)*cos(d*x + c)*e^(1/2) - 8*I*sqrt(2)*e^(1/2))*sin(d*x + c) - 8*I*sqrt(2)*e^(1/2))*weierstras
sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(I*sqrt(2)*cos(d*x + c)^4*e^(1/2)
- 3*I*sqrt(2)*cos(d*x + c)^3*e^(1/2) - 8*I*sqrt(2)*cos(d*x + c)^2*e^(1/2) + 4*I*sqrt(2)*cos(d*x + c)*e^(1/2) +
(-I*sqrt(2)*cos(d*x + c)^3*e^(1/2) - 4*I*sqrt(2)*cos(d*x + c)^2*e^(1/2) + 4*I*sqrt(2)*cos(d*x + c)*e^(1/2) +
8*I*sqrt(2)*e^(1/2))*sin(d*x + c) + 8*I*sqrt(2)*e^(1/2))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos
(d*x + c) - I*sin(d*x + c))) + 2*(3*cos(d*x + c)^4*e^(1/2) + 12*cos(d*x + c)^3*e^(1/2) - 14*cos(d*x + c)^2*e^(
1/2) - 32*cos(d*x + c)*e^(1/2) + (3*cos(d*x + c)^3*e^(1/2) - 9*cos(d*x + c)^2*e^(1/2) - 23*cos(d*x + c)*e^(1/2
) + 9*e^(1/2))*sin(d*x + c) - 9*e^(1/2))*sqrt(cos(d*x + c)))/(a^4*d*cos(d*x + c)^4 - 3*a^4*d*cos(d*x + c)^3 -
8*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + 8*a^4*d - (a^4*d*cos(d*x + c)^3 + 4*a^4*d*cos(d*x + c)^2 - 4*a
^4*d*cos(d*x + c) - 8*a^4*d)*sin(d*x + c))

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\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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