3.89 \(\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=290 \[ \frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{a \sin (e+f x)+a}}+\frac{22 a c^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a}} \]

[Out]

(22*a*c^3*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a*c^3*g*Sqr
t[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[
e + f*x]]) + (22*a*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(105*f*g*Sqrt[a + a*Sin[e + f*x]]) + (
2*a*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f
*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 1.42838, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{a \sin (e+f x)+a}}+\frac{22 a c^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(22*a*c^3*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a*c^3*g*Sqr
t[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[
e + f*x]]) + (22*a*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(105*f*g*Sqrt[a + a*Sin[e + f*x]]) + (
2*a*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f
*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]])

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]

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 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 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]

Rubi steps

\begin{align*} \int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx &=-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{3} a \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{21} (11 a c) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{22 a c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{105 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{15} \left (11 a c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{105 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{15} \left (11 a c^3\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{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{105 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}+\frac{\left (11 a c^3 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{15 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{105 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}+\frac{\left (11 a c^3 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{15 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^3 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{105 f g \sqrt{a+a \sin (e+f x)}}+\frac{2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 2.37835, size = 281, normalized size = 0.97 \[ -\frac{c^3 g e^{-4 i (e+f x)} \left (e^{i (e+f x)}-i\right ) \left (\sqrt{1+e^{2 i (e+f x)}} \left (-180 i e^{i (e+f x)}+238 e^{2 i (e+f x)}-540 i e^{3 i (e+f x)}+3696 e^{4 i (e+f x)}-540 i e^{5 i (e+f x)}-238 e^{6 i (e+f x)}-180 i e^{7 i (e+f x)}+35 e^{8 i (e+f x)}-35\right )-2464 e^{6 i (e+f x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right )\right ) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \cos (e+f x)}}{2520 f \left (e^{i (e+f x)}+i\right ) \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

-(c^3*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-35 - (180*I)*E^(I*(e + f*
x)) + 238*E^((2*I)*(e + f*x)) - (540*I)*E^((3*I)*(e + f*x)) + 3696*E^((4*I)*(e + f*x)) - (540*I)*E^((5*I)*(e +
f*x)) - 238*E^((6*I)*(e + f*x)) - (180*I)*E^((7*I)*(e + f*x)) + 35*E^((8*I)*(e + f*x))) - 2464*E^((6*I)*(e +
f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(2520*E^((4*I)*(e +
f*x))*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[c - c*Sin[e + f*x]])

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Maple [C]  time = 0.368, size = 392, normalized size = 1.4 \begin{align*} -{\frac{2}{315\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sin \left ( fx+e \right ) -2 \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( 231\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) -231\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+35\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+90\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+231\,i\sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) -231\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-112\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-154\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+231\,\cos \left ( fx+e \right ) \right ) \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2),x)

[Out]

-2/315/f*(-c*(-1+sin(f*x+e)))^(5/2)*(231*I*sin(f*x+e)*cos(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x
+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-231*I*sin(f*x+e)*cos(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(
cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+35*cos(f*x+e)^6+90*sin(f*x+e)*cos(f
*x+e)^4+231*I*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e)
)/sin(f*x+e),I)-231*I*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+co
s(f*x+e))/sin(f*x+e),I)-112*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
)))^(1/2)/(cos(f*x+e)^2+2*sin(f*x+e)-2)/sin(f*x+e)/cos(f*x+e)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{a \sin \left (f x + e\right ) + a}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (c^{2} g \cos \left (f x + e\right )^{3} + 2 \, c^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, c^{2} g \cos \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(c^2*g*cos(f*x + e)^3 + 2*c^2*g*cos(f*x + e)*sin(f*x + e) - 2*c^2*g*cos(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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(5/2)*(a+a*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError