3.118 \(\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\)

Optimal. Leaf size=343 \[ -\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^4 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}} \]

[Out]

-10/77*a^2*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f/g/(c-c*sin(f*x+e))^(1/2)-2/33*a*c*(g*cos(f*x+e))^(5
/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(1/2)+2/11*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(7/2)/f/g/(
c-c*sin(f*x+e))^(1/2)-2/3*a^4*c*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+2*a^4*c
*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*c
os(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-2/7*a^3*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e
))^(1/2)/f/g/(c-c*sin(f*x+e))^(1/2)

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Rubi [A]  time = 1.72, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 a^4 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

(-2*a^4*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^4*c*g*Sqrt[
Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f
*x]]) - (2*a^3*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c
*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x]
)^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin
[e + f*x])^(7/2))/(11*f*g*Sqrt[c - c*Sin[e + f*x]])

Rule 2639

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

Rule 2640

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

Rule 2842

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(g*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), In
t[(g*Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2
, 0]

Rule 2851

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

Rubi steps

\begin {align*} \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx &=\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{11} (3 c) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{11} (5 a c) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{7} \left (5 a^2 c\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {2 a^3 c (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\left (a^3 c\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\left (a^4 c\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (a^4 c g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (a^4 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^4 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [C]  time = 4.32, size = 360, normalized size = 1.05 \[ \frac {\sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (-\frac {a^3 \sqrt {\cos (e+f x)} \sqrt {a (\sin (e+f x)+1)} (1374 \cos (e+f x)+423 \cos (3 (e+f x))-7 (44 \sin (2 (e+f x))-22 \sin (4 (e+f x))+3 \cos (5 (e+f x))-528 \cot (e)))}{1848 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {(2+2 i) a^4 e^{\frac {1}{2} i (e+f x)} \left (e^{i (e+f x)}+i\right ) \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )\right )^{3/2} \left (\left (-1+e^{2 i e}\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (e+f x)}\right )+\sqrt {1+e^{2 i (e+f x)}}\right )}{\left (-1+e^{2 i e}\right ) f \left (1+e^{2 i (e+f x)}\right )^{3/2} \sqrt {-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2}}\right )}{\cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(((2 + 2*I)*a^4*E^((I/2)*(e + f*x))*(I + E^(I*(e + f*x)))*((1
+ E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))] + (-1 + E^((2*I)*e))*Hypergeomet
ric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(e + f*x))]))/((-1 + E^((2*I)*e))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*
(e + f*x))]*(1 + E^((2*I)*(e + f*x)))^(3/2)*f) - (a^3*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])]*(1374*Cos[
e + f*x] + 423*Cos[3*(e + f*x)] - 7*(3*Cos[5*(e + f*x)] - 528*Cot[e] + 44*Sin[2*(e + f*x)] - 22*Sin[4*(e + f*x
)])))/(1848*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2
]))

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right ) + {\left (a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(3*a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e) + (a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e))*sin(f
*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)*sqrt(-c*sin(f*x + e) + c), x)

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maple [C]  time = 0.72, size = 425, normalized size = 1.24 \[ -\frac {2 \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (21 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+77 \left (\cos ^{6}\left (f x +e \right )\right )-132 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-154 \left (\cos ^{4}\left (f x +e \right )\right )-154 \left (\cos ^{2}\left (f x +e \right )\right )+231 \cos \left (f x +e \right )\right ) \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}}{231 f \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )-4\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x)

[Out]

-2/231/f*(-c*(sin(f*x+e)-1))^(1/2)*(21*cos(f*x+e)^6*sin(f*x+e)+231*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos
(f*x+e)+1))^(1/2)*cos(f*x+e)*sin(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-231*I*(1/(cos(f*x+e)+1))^(1/
2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*sin(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+77*cos(f*
x+e)^6-132*sin(f*x+e)*cos(f*x+e)^4+231*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)
*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-231*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*si
n(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-154*cos(f*x+e)^4-154*cos(f*x+e)^2+231*cos(f*x+e))*(g*cos(f*
x+e))^(3/2)*(a*(1+sin(f*x+e)))^(7/2)/(cos(f*x+e)^2*sin(f*x+e)+3*cos(f*x+e)^2-4*sin(f*x+e)-4)/sin(f*x+e)/cos(f*
x+e)^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)*sqrt(-c*sin(f*x + e) + c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2),x)

[Out]

int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2), 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((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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