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

Optimal. Leaf size=233 \[ \frac{2 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]

[Out]

(2*(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*(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*(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*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[
(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 = 1.14926, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2852, 2842, 2640, 2639} \[ \frac{2 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]

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]

(2*(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*(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*(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*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[
(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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{2 (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{\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{2 (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 (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{\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{2 (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 (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 (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{\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{2 (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 (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 (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{(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{2 (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 (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 (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 (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{2 (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 (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 (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 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 [A]  time = 2.3048, size = 240, normalized size = 1.03 \[ \frac{(g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sqrt{\cos (e+f x)} \left (-32 \sin \left (\frac{1}{2} (e+f x)\right )+15 \sin \left (\frac{3}{2} (e+f x)\right )+3 \sin \left (\frac{5}{2} (e+f x)\right )-32 \cos \left (\frac{1}{2} (e+f x)\right )-15 \cos \left (\frac{3}{2} (e+f x)\right )+3 \cos \left (\frac{5}{2} (e+f x)\right )\right )+12 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5\right )}{90 c^3 f (\sin (e+f x)-1)^3 \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}} \]

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])^(7/2)),x]

[Out]

((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(12*Elli
pticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + Sqrt[Cos[e + f*x]]*(-32*Cos[(e + f*x)/2] - 15*
Cos[(3*(e + f*x))/2] + 3*Cos[(5*(e + f*x))/2] - 32*Sin[(e + f*x)/2] + 15*Sin[(3*(e + f*x))/2] + 3*Sin[(5*(e +
f*x))/2])))/(90*c^3*f*Cos[e + f*x]^(3/2)*(-1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f
*x]])

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Maple [C]  time = 0.332, size = 955, 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)/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2),x)

[Out]

2/45/f*(g*cos(f*x+e))^(3/2)*(sin(f*x+e)*cos(f*x+e)-sin(f*x+e)-cos(f*x+e)+1)*(-3*I*(1/(cos(f*x+e)+1))^(1/2)*(co
s(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)^4*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-12*I*(1/(cos(
f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^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)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+3*I*(1/(c
os(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)^4*EllipticE(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)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e
),I)*cos(f*x+e)^4+6*I*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)*sin(f*x+e)*cos(f*x+e)^2*EllipticE(
I*(-1+cos(f*x+e))/sin(f*x+e),I)+9*I*sin(f*x+e)*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)-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*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)
+1))^(1/2)*cos(f*x+e)^2*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+6*I*sin(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/s
in(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)*cos(f*x+e)^4*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-3*cos(f*x+e)^4-6*cos(f*x+e)^2*
sin(f*x+e)+5*cos(f*x+e)^3+sin(f*x+e)*cos(f*x+e)+4*cos(f*x+e)^2+5*sin(f*x+e)-11*cos(f*x+e)+5)*(cos(f*x+e)^2+2*c
os(f*x+e)+1)/(a*(1+sin(f*x+e)))^(1/2)/(-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)/(c-c*sin(f*x+e))^(7/2)/(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)^(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}{3 \, a c^{4} \cos \left (f x + e\right )^{3} - 4 \, a c^{4} \cos \left (f x + e\right ) -{\left (a c^{4} \cos \left (f x + e\right )^{3} - 4 \, a c^{4} \cos \left (f x + e\right )\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)/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2),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/(3*a*c^4*cos(f*x + e)^3 -
4*a*c^4*cos(f*x + e) - (a*c^4*cos(f*x + e)^3 - 4*a*c^4*cos(f*x + e))*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)/(c-c*sin(f*x+e))**(7/2)/(a+a*sin(f*x+e))**(1/2),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)/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2),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)