3.4.56 \(\int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) [356]

Optimal. Leaf size=92 \[ \frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 \cos (e+f x)}{12 c f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \]

[Out]

-1/12*a^2*cos(f*x+e)/c/f/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2)+1/4*a*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)
/f/(c-c*sin(f*x+e))^(9/2)

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Rubi [A]
time = 0.12, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2818, 2817} \begin {gather*} \frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 \cos (e+f x)}{12 c f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*Cos[e + f*x])/(12*c*f*Sqrt[a
+ a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))

Rule 2817

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rule 2818

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Dist[b*((2*m - 1)
/(d*(2*n + 1))), Int[(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}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] &&  !(ILtQ[m + n, 0] && G
tQ[2*m + n + 1, 0])

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx &=\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac {a \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx}{4 c}\\ &=\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 \cos (e+f x)}{12 c f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.66, size = 106, normalized size = 1.15 \begin {gather*} \frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (1+2 \sin (e+f x))}{6 c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^4 \sqrt {c-c \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(1 + 2*Sin[e + f*x]))/(6*c^4*f*(Cos[(e + f
*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(80)=160\).
time = 9.30, size = 168, normalized size = 1.83

method	result	size
default	\(\frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (\cos ^{4}\left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+4 \left (\cos ^{3}\left (f x +e \right )\right )+5 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-12 \left (\cos ^{2}\left (f x +e \right )\right )+7 \cos \left (f x +e \right ) \sin \left (f x +e \right )-10 \cos \left (f x +e \right )-17 \sin \left (f x +e \right )+17\right )}{6 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {9}{2}} \left (\cos ^{2}\left (f x +e \right )+\cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-2 \sin \left (f x +e \right )-2\right )}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/6/f*sin(f*x+e)*(a*(1+sin(f*x+e)))^(3/2)*(cos(f*x+e)^4-cos(f*x+e)^3*sin(f*x+e)+4*cos(f*x+e)^3+5*sin(f*x+e)*co
s(f*x+e)^2-12*cos(f*x+e)^2+7*cos(f*x+e)*sin(f*x+e)-10*cos(f*x+e)-17*sin(f*x+e)+17)/(-c*(sin(f*x+e)-1))^(9/2)/(
cos(f*x+e)^2+cos(f*x+e)*sin(f*x+e)+cos(f*x+e)-2*sin(f*x+e)-2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)/(-c*sin(f*x + e) + c)^(9/2), x)

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Fricas [A]
time = 0.34, size = 123, normalized size = 1.34 \begin {gather*} \frac {{\left (2 \, a \sin \left (f x + e\right ) + a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{6 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} + 8 \, c^{5} f \cos \left (f x + e\right ) + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{3} - 2 \, c^{5} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

1/6*(2*a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^5*f*cos(f*x + e)^5 - 8*c^5*f*
cos(f*x + e)^3 + 8*c^5*f*cos(f*x + e) + 4*(c^5*f*cos(f*x + e)^3 - 2*c^5*f*cos(f*x + e))*sin(f*x + e))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(3/2)/(c-c*sin(f*x+e))**(9/2),x)

[Out]

Timed out

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Giac [A]
time = 0.47, size = 98, normalized size = 1.07 \begin {gather*} \frac {{\left (4 \, a \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{96 \, c^{5} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

1/96*(4*a*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 3*a*sqrt(c)*sgn(cos(-
1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/(c^5*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^
8)

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Mupad [B]
time = 11.81, size = 195, normalized size = 2.12 \begin {gather*} \frac {\left (\frac {16\,a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^5\,f}+\frac {32\,a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^5\,f}\right )\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{84\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}-54\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )+2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )-96\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )+16\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(9/2),x)

[Out]

(((16*a*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2))/(3*c^5*f) + (32*a*exp(e*5i + f*x*5i)*sin(e + f*x)*(a +
a*sin(e + f*x))^(1/2))/(3*c^5*f))*(c - c*sin(e + f*x))^(1/2))/(84*cos(e + f*x)*exp(e*5i + f*x*5i) - 54*exp(e*5
i + f*x*5i)*cos(3*e + 3*f*x) + 2*exp(e*5i + f*x*5i)*cos(5*e + 5*f*x) - 96*exp(e*5i + f*x*5i)*sin(2*e + 2*f*x)
+ 16*exp(e*5i + f*x*5i)*sin(4*e + 4*f*x))

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