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

Optimal. Leaf size=237 \[ -\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x
])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(15*c^
2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*El
lipticE[(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 1.16495, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2850, 2852, 2842, 2640, 2639} \[ -\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]

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])^(7/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x
])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(15*c^
2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*El
lipticE[(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2850

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

Rule 2852

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*(c + d*Sin[e + f*x])^
n)/(a*f*g*(2*m + p + 1)), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), 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] && E
qQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] &&  !LtQ[m, n, -1] && 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 \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{a \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{3 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{a \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{15 c^2}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{a \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx}{15 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{(a g \cos (e+f x)) \int \sqrt{g \cos (e+f x)} \, dx}{15 c^3 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (a g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{15 c^3 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{2 a 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 c^3 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 2.27547, size = 256, normalized size = 1.08 \[ \frac{4 e^{3 i (e+f x)} \left (g e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )\right )^{3/2} \left (\left (e^{i (e+f x)}-i\right )^5 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right )+\sqrt{1+e^{2 i (e+f x)}} \left (e^{i (e+f x)}+15 i e^{2 i (e+f x)}-3 e^{3 i (e+f x)}-29 i\right )\right ) \sqrt{a (\sin (e+f x)+1)}}{45 c^3 f \left (e^{i (e+f x)}-i\right )^4 \left (e^{i (e+f x)}+i\right ) \left (1+e^{2 i (e+f x)}\right )^{3/2} \sqrt{i c e^{-i (e+f x)} \left (e^{i (e+f x)}-i\right )^2}} \]

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])^(7/2),x]

[Out]

(4*E^((3*I)*(e + f*x))*(((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-
29*I + E^(I*(e + f*x)) + (15*I)*E^((2*I)*(e + f*x)) - 3*E^((3*I)*(e + f*x))) + (-I + E^(I*(e + f*x)))^5*Hyperg
eometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(45*c^3*(-I + E^(I*(e + f*x)))^4
*Sqrt[(I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(I + E^(I*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(3/2)*f)

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Maple [C]  time = 0.361, size = 966, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

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))^(1/2)/(c-c*sin(f*x+e))^(7/2),x)

[Out]

2/45/f*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(1/2)*(sin(f*x+e)*cos(f*x+e)-sin(f*x+e)-cos(f*x+e)+1)*(6*I*(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)*sin(f*x+e)-12
*I*cos(f*x+e)^2*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))
^(1/2)-6*I*(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
)*sin(f*x+e)-6*I*(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)-6*I*(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)*cos(f*x+e)^4+12*I*cos(f*x+e)^2*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+
e)/(cos(f*x+e)+1))^(1/2)+9*I*(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)*sin(f*x+e)*cos(f*x+e)^2+6*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ell
ipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4+6*I*(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)-3*I*(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)*sin(f*x+e)*cos(f*x+e)^4+3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(co
s(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^4-9*I*(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)*sin(f*x+e)*cos(f*x+e)^2+3*cos(
f*x+e)^4+6*cos(f*x+e)^2*sin(f*x+e)+10*cos(f*x+e)^3-16*sin(f*x+e)*cos(f*x+e)-19*cos(f*x+e)^2+10*sin(f*x+e)-4*co
s(f*x+e)+10)*(cos(f*x+e)^2+2*cos(f*x+e)+1)/(1+sin(f*x+e))/(-c*(-1+sin(f*x+e)))^(7/2)/sin(f*x+e)^5/cos(f*x+e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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{7}{2}}}\,{d x} \end{align*}

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))^(1/2)/(c-c*sin(f*x+e))^(7/2),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)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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} g \cos \left (f x + e\right )}{c^{4} \cos \left (f x + e\right )^{4} - 8 \, c^{4} \cos \left (f x + e\right )^{2} + 8 \, c^{4} + 4 \,{\left (c^{4} \cos \left (f x + e\right )^{2} - 2 \, c^{4}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

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

[Out]

integral(sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*g*cos(f*x + e)/(c^4*cos(f*x +
e)^4 - 8*c^4*cos(f*x + e)^2 + 8*c^4 + 4*(c^4*cos(f*x + e)^2 - 2*c^4)*sin(f*x + e)), 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)*(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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{7}{2}}}\,{d x} \end{align*}

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

[Out]

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