\(\int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) [380]

Optimal result

Integrand size = 30, antiderivative size = 133 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}} \]

[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)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2822, 2821} \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}} \]

[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*} \text {integral}& = \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*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(335\) vs. \(2(133)=266\).

Time = 9.07 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.52 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {4 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3+3 \sin (e+f x))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{13/2}}-\frac {12 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (3+3 \sin (e+f x))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{13/2}}+\frac {3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (3+3 \sin (e+f x))^{7/2}}{2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{13/2}}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (3+3 \sin (e+f x))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{13/2}} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

(4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3 + 3*Sin[e + f*x])^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^7*(c - c*Sin[e + f*x])^(13/2)) - (12*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(3 + 3*Sin[e + f*x])^(7/2))/(5*
f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + (3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/
2])^5*(3 + 3*Sin[e + f*x])^(7/2))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) -
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(3 + 3*Sin[e + f*x])^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^7*(c - c*Sin[e + f*x])^(13/2))

Maple [A] (verified)

Time = 3.32 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.32

method	result	size
default	\(-\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{3} \left (3 \left (\cos ^{5}\left (f x +e \right )\right )+18 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-54 \left (\cos ^{3}\left (f x +e \right )\right )-106 \sin \left (f x +e \right ) \cos \left (f x +e \right )+129 \cos \left (f x +e \right )+118 \tan \left (f x +e \right )-78 \sec \left (f x +e \right )\right )}{30 f \left (\left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-5 \left (\cos ^{4}\left (f x +e \right )\right )-12 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+20 \left (\cos ^{2}\left (f x +e \right )\right )+16 \sin \left (f x +e \right )-16\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{6}}\)	\(175\)

[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*(a*(sin(f*x+e)+1))^(1/2)*a^3/(cos(f*x+e)^4*sin(f*x+e)-5*cos(f*x+e)^4-12*sin(f*x+e)*cos(f*x+e)^2+20*cos
(f*x+e)^2+16*sin(f*x+e)-16)/(-c*(sin(f*x+e)-1))^(1/2)/c^6*(3*cos(f*x+e)^5+18*cos(f*x+e)^3*sin(f*x+e)-54*cos(f*
x+e)^3-106*sin(f*x+e)*cos(f*x+e)+129*cos(f*x+e)+118*tan(f*x+e)-78*sec(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.34 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {{\left (15 \, a^{3} \cos \left (f x + e\right )^{2} - 18 \, a^{3} + 2 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 11 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} - 32 \, c^{7} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{5} - 16 \, c^{7} f \cos \left (f x + e\right )^{3} + 16 \, c^{7} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

[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))

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.17 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {{\left (20 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 45 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 36 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{480 \, c^{\frac {13}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12}} \]

[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)

Mupad [B] (verification not implemented)

Time = 13.32 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.48 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {224\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}+\frac {416\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}-\frac {32\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c^7\,f}-\frac {32\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^7\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \]

[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))