3.4.80 \(\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) [380]

Optimal. Leaf size=133 \[ \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}} \]

[Out]

1/12*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(13/2)+1/60*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c
-c*sin(f*x+e))^(11/2)+1/480*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+e))^(9/2)

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Rubi [A]
time = 0.19, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2822, 2821} \begin {gather*} \frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(12*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*
x])^(7/2))/(60*c*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(480*c^2*f*(c - c*
Sin[e + f*x])^(9/2))

Rule 2821

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

Rule 2822

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{6 c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{60 c^2}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 6.18, size = 128, normalized size = 0.96 \begin {gather*} \frac {a^3 \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))} (21-15 \cos (2 (e+f x))+39 \sin (e+f x)-5 \sin (3 (e+f x)))}{60 c^6 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))^6 \sqrt {c-c \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(21 - 15*Cos[2*(e + f*x)] + 39*Sin[e + f
*x] - 5*Sin[3*(e + f*x)]))/(60*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^6*Sqrt[c - c*Si
n[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(275\) vs. \(2(115)=230\).
time = 17.08, size = 276, normalized size = 2.08

method	result	size
default	\(\frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \left (3 \left (\cos ^{6}\left (f x +e \right )\right )-3 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+18 \left (\cos ^{5}\left (f x +e \right )\right )+21 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-72 \left (\cos ^{4}\left (f x +e \right )\right )+51 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-106 \left (\cos ^{3}\left (f x +e \right )\right )-157 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+235 \left (\cos ^{2}\left (f x +e \right )\right )-78 \cos \left (f x +e \right ) \sin \left (f x +e \right )+118 \cos \left (f x +e \right )+196 \sin \left (f x +e \right )-196\right )}{30 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {13}{2}} \left (\cos ^{4}\left (f x +e \right )+\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-8 \left (\cos ^{2}\left (f x +e \right )\right )-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+8 \sin \left (f x +e \right )+8\right )}\)	\(276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13/2),x,method=_RETURNVERBOSE)

[Out]

1/30/f*sin(f*x+e)*(a*(1+sin(f*x+e)))^(7/2)*(3*cos(f*x+e)^6-3*cos(f*x+e)^5*sin(f*x+e)+18*cos(f*x+e)^5+21*sin(f*
x+e)*cos(f*x+e)^4-72*cos(f*x+e)^4+51*cos(f*x+e)^3*sin(f*x+e)-106*cos(f*x+e)^3-157*sin(f*x+e)*cos(f*x+e)^2+235*
cos(f*x+e)^2-78*cos(f*x+e)*sin(f*x+e)+118*cos(f*x+e)+196*sin(f*x+e)-196)/(-c*(sin(f*x+e)-1))^(13/2)/(cos(f*x+e
)^4+cos(f*x+e)^3*sin(f*x+e)+3*cos(f*x+e)^3-4*sin(f*x+e)*cos(f*x+e)^2-8*cos(f*x+e)^2-4*cos(f*x+e)*sin(f*x+e)-4*
cos(f*x+e)+8*sin(f*x+e)+8)

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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))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(13/2), x)

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Fricas [A]
time = 0.35, size = 191, normalized size = 1.44 \begin {gather*} \frac {{\left (15 \, a^{3} \cos \left (f x + e\right )^{2} - 18 \, a^{3} + 2 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 11 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} - 32 \, c^{7} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{5} - 16 \, c^{7} f \cos \left (f x + e\right )^{3} + 16 \, c^{7} 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))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="fricas")

[Out]

1/30*(15*a^3*cos(f*x + e)^2 - 18*a^3 + 2*(5*a^3*cos(f*x + e)^2 - 11*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a
)*sqrt(-c*sin(f*x + e) + c)/(c^7*f*cos(f*x + e)^7 - 18*c^7*f*cos(f*x + e)^5 + 48*c^7*f*cos(f*x + e)^3 - 32*c^7
*f*cos(f*x + e) + 2*(3*c^7*f*cos(f*x + e)^5 - 16*c^7*f*cos(f*x + e)^3 + 16*c^7*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))**(7/2)/(c-c*sin(f*x+e))**(13/2),x)

[Out]

Timed out

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Giac [A]
time = 0.52, size = 164, normalized size = 1.23 \begin {gather*} \frac {{\left (20 \, a^{3} \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 )^{6} - 45 \, a^{3} \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 )^{4} + 36 \, a^{3} \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} - 10 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{480 \, c^{\frac {13}{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 )^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="giac")

[Out]

1/480*(20*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^6 - 45*a^3*sgn(cos(-1/4*pi +
1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4 + 36*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi +
1/2*f*x + 1/2*e)^2 - 10*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/(c^(13/2)*f*sgn(sin(-1/4*pi + 1/2*f*x
+ 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^12)

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Mupad [B]
time = 12.06, size = 330, normalized size = 2.48 \begin {gather*} -\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {224\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}+\frac {416\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}-\frac {32\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c^7\,f}-\frac {32\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^7\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(13/2),x)

[Out]

-((c - c*sin(e + f*x))^(1/2)*((224*a^3*exp(e*7i + f*x*7i)*(a + a*sin(e + f*x))^(1/2))/(5*c^7*f) + (416*a^3*exp
(e*7i + f*x*7i)*sin(e + f*x)*(a + a*sin(e + f*x))^(1/2))/(5*c^7*f) - (32*a^3*exp(e*7i + f*x*7i)*cos(2*e + 2*f*
x)*(a + a*sin(e + f*x))^(1/2))/(c^7*f) - (32*a^3*exp(e*7i + f*x*7i)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2
))/(3*c^7*f)))/(858*exp(e*7i + f*x*7i)*cos(3*e + 3*f*x) - 858*cos(e + f*x)*exp(e*7i + f*x*7i) - 130*exp(e*7i +
f*x*7i)*cos(5*e + 5*f*x) + 2*exp(e*7i + f*x*7i)*cos(7*e + 7*f*x) + 1144*exp(e*7i + f*x*7i)*sin(2*e + 2*f*x) -
416*exp(e*7i + f*x*7i)*sin(4*e + 4*f*x) + 24*exp(e*7i + f*x*7i)*sin(6*e + 6*f*x))

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