3.267 \(\int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=120 \[ \frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^4 d \sqrt {e \cos (c+d x)}}+\frac {20 e^3 \sqrt {e \cos (c+d x)}}{21 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {4 e (e \cos (c+d x))^{5/2}}{7 a d (a \sin (c+d x)+a)^3} \]

[Out]

-4/7*e*(e*cos(d*x+c))^(5/2)/a/d/(a+a*sin(d*x+c))^3+10/21*e^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*E
llipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/a^4/d/(e*cos(d*x+c))^(1/2)+20/21*e^3*(e*cos(d*x+c))^(1/2
)/d/(a^4+a^4*sin(d*x+c))

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Rubi [A]  time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2680, 2642, 2641} \[ \frac {20 e^3 \sqrt {e \cos (c+d x)}}{21 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^4 d \sqrt {e \cos (c+d x)}}-\frac {4 e (e \cos (c+d x))^{5/2}}{7 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^4,x]

[Out]

(10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^4*d*Sqrt[e*Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^
(5/2))/(7*a*d*(a + a*Sin[c + d*x])^3) + (20*e^3*Sqrt[e*Cos[c + d*x]])/(21*d*(a^4 + a^4*Sin[c + d*x]))

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rubi steps

\begin {align*} \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^4} \, dx &=-\frac {4 e (e \cos (c+d x))^{5/2}}{7 a d (a+a \sin (c+d x))^3}-\frac {\left (5 e^2\right ) \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^2} \, dx}{7 a^2}\\ &=-\frac {4 e (e \cos (c+d x))^{5/2}}{7 a d (a+a \sin (c+d x))^3}+\frac {20 e^3 \sqrt {e \cos (c+d x)}}{21 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac {\left (5 e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 a^4}\\ &=-\frac {4 e (e \cos (c+d x))^{5/2}}{7 a d (a+a \sin (c+d x))^3}+\frac {20 e^3 \sqrt {e \cos (c+d x)}}{21 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac {\left (5 e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^4 \sqrt {e \cos (c+d x)}}\\ &=\frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^4 d \sqrt {e \cos (c+d x)}}-\frac {4 e (e \cos (c+d x))^{5/2}}{7 a d (a+a \sin (c+d x))^3}+\frac {20 e^3 \sqrt {e \cos (c+d x)}}{21 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 66, normalized size = 0.55 \[ -\frac {(e \cos (c+d x))^{9/2} \, _2F_1\left (\frac {9}{4},\frac {11}{4};\frac {13}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{9\ 2^{3/4} a^4 d e (\sin (c+d x)+1)^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^4,x]

[Out]

-1/9*((e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[9/4, 11/4, 13/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^4*d*e*(1 + S
in[c + d*x])^(9/4))

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} e^{3} \cos \left (d x + c\right )^{3}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*cos(d*x + c))*e^3*cos(d*x + c)^3/(a^4*cos(d*x + c)^4 - 8*a^4*cos(d*x + c)^2 + 8*a^4 - 4*(a^4*c
os(d*x + c)^2 - 2*a^4)*sin(d*x + c)), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 2.69, size = 401, normalized size = 3.34 \[ -\frac {2 \left (40 \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 ) \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 )-60 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-128 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+30 \EllipticF \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 ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+112 \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}}\, \EllipticF \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}+16 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-112 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{4}}{21 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^4,x)

[Out]

-2/21/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*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)*(40*(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))*(sin(1
/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^6-60*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-128*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+30*Ellip
ticF(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)^2+128*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+112*sin(1/2*d*x+1/2*c)^5-5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Elli
pticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)+16*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-
112*sin(1/2*d*x+1/2*c)^3+4*sin(1/2*d*x+1/2*c))*e^4/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(7/2)/(a*sin(d*x + c) + a)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^4,x)

[Out]

int((e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**(7/2)/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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