3.4.57 \(\int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\) [357]

Optimal. Leaf size=92 \[ \frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{20 c f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}} \]

[Out]

-1/20*a^2*cos(f*x+e)/c/f/(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(1/2)+1/5*a*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)
/f/(c-c*sin(f*x+e))^(11/2)

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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}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{20 c f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*f*(c - c*Sin[e + f*x])^(11/2)) - (a^2*Cos[e + f*x])/(20*c*f*Sqrt[
a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))

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))^{11/2}} \, dx &=\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{5 c}\\ &=\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{20 c f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.95, 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))} (3+5 \sin (e+f x))}{20 c^5 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))^5 \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])^(11/2),x]

[Out]

-1/20*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(3 + 5*Sin[e + f*x]))/(c^5*f*(Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^5*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. \(195\) vs. \(2(80)=160\).
time = 17.52, size = 196, normalized size = 2.13

method	result	size
default	\(-\frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (3 \left (\cos ^{5}\left (f x +e \right )\right )+3 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-18 \left (\cos ^{4}\left (f x +e \right )\right )+15 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-36 \left (\cos ^{3}\left (f x +e \right )\right )-51 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+96 \left (\cos ^{2}\left (f x +e \right )\right )-45 \cos \left (f x +e \right ) \sin \left (f x +e \right )+53 \cos \left (f x +e \right )+98 \sin \left (f x +e \right )-98\right )}{20 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {11}{2}} \left (\cos ^{2}\left (f x +e \right )+\cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-2 \sin \left (f x +e \right )-2\right )}\)	\(196\)

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))^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/20/f*sin(f*x+e)*(a*(1+sin(f*x+e)))^(3/2)*(3*cos(f*x+e)^5+3*sin(f*x+e)*cos(f*x+e)^4-18*cos(f*x+e)^4+15*cos(f
*x+e)^3*sin(f*x+e)-36*cos(f*x+e)^3-51*sin(f*x+e)*cos(f*x+e)^2+96*cos(f*x+e)^2-45*cos(f*x+e)*sin(f*x+e)+53*cos(
f*x+e)+98*sin(f*x+e)-98)/(-c*(sin(f*x+e)-1))^(11/2)/(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))^(11/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)/(-c*sin(f*x + e) + c)^(11/2), x)

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Fricas [A]
time = 0.34, size = 141, normalized size = 1.53 \begin {gather*} \frac {{\left (5 \, a \sin \left (f x + e\right ) + 3 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{20 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{5} - 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right ) - {\left (c^{6} f \cos \left (f x + e\right )^{5} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} 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))^(11/2),x, algorithm="fricas")

[Out]

1/20*(5*a*sin(f*x + e) + 3*a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^6*f*cos(f*x + e)^5 - 20*
c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e) - (c^6*f*cos(f*x + e)^5 - 12*c^6*f*cos(f*x + e)^3 + 16*c^6*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))**(11/2),x)

[Out]

Timed out

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Giac [A]
time = 0.49, size = 92, normalized size = 1.00 \begin {gather*} \frac {{\left (5 \, a \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} - 4 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{320 \, c^{\frac {11}{2}} 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 )^{10}} \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))^(11/2),x, algorithm="giac")

[Out]

1/320*(5*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 4*a*sgn(cos(-1/4*pi + 1/2*f*
x + 1/2*e)))*sqrt(a)/(c^(11/2)*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^10)

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Mupad [B]
time = 12.59, size = 225, normalized size = 2.45 \begin {gather*} \frac {\left (\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,48{}\mathrm {i}}{5\,c^6\,f}+\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,16{}\mathrm {i}}{c^6\,f}\right )\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,264{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,220{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )\,20{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,330{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,88{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )\,2{}\mathrm {i}} \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))^(11/2),x)

[Out]

(((a*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*48i)/(5*c^6*f) + (a*exp(e*6i + f*x*6i)*sin(e + f*x)*(a + a*
sin(e + f*x))^(1/2)*16i)/(c^6*f))*(c - c*sin(e + f*x))^(1/2))/(cos(e + f*x)*exp(e*6i + f*x*6i)*264i - exp(e*6i
+ f*x*6i)*cos(3*e + 3*f*x)*220i + exp(e*6i + f*x*6i)*cos(5*e + 5*f*x)*20i - exp(e*6i + f*x*6i)*sin(2*e + 2*f*
x)*330i + exp(e*6i + f*x*6i)*sin(4*e + 4*f*x)*88i - exp(e*6i + f*x*6i)*sin(6*e + 6*f*x)*2i)

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