\(\int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx\) [382]

Optimal result

Integrand size = 30, antiderivative size = 178 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {3 \cos (e+f x) (3+3 \sin (e+f x))^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {27 \cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {27 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac {81 \cos (e+f x)}{280 c^3 f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{11/2}} \]

[Out]

1/8*a*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f/(c-c*sin(f*x+e))^(17/2)-3/56*a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c
/f/(c-c*sin(f*x+e))^(15/2)-1/280*a^4*cos(f*x+e)/c^3/f/(c-c*sin(f*x+e))^(11/2)/(a+a*sin(f*x+e))^(1/2)+1/56*a^3*
cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c^2/f/(c-c*sin(f*x+e))^(13/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2818, 2817} \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=-\frac {a^4 \cos (e+f x)}{280 c^3 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac {3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}} \]

[In]

Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(17/2),x]

[Out]

(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(17/2)) - (3*a^2*Cos[e + f*x]*(a + a*Sin
[e + f*x])^(3/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(56*c^2*f
*(c - c*Sin[e + f*x])^(13/2)) - (a^4*Cos[e + f*x])/(280*c^3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(1
1/2))

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*} \text {integral}& = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {(3 a) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{15/2}} \, dx}{8 c} \\ & = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {\left (3 a^2\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{28 c^2} \\ & = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{11/2}} \, dx}{56 c^3} \\ & = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac {a^4 \cos (e+f x)}{280 c^3 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.15 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.85 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3+3 \sin (e+f x))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {12 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (3+3 \sin (e+f x))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (3+3 \sin (e+f x))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (3+3 \sin (e+f x))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(17/2),x]

[Out]

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3 + 3*Sin[e + f*x])^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*
(c - c*Sin[e + f*x])^(17/2)) - (12*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(3 + 3*Sin[e + f*x])^(7/2))/(7*f*(C
os[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*
(3 + 3*Sin[e + f*x])^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - ((Cos[(e
+ f*x)/2] - Sin[(e + f*x)/2])^7*(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])^(17/2))

Maple [A] (verified)

Time = 3.57 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.28

method	result	size
default	\(-\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{3} \left (3 \left (\cos ^{7}\left (f x +e \right )\right )+24 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-96 \left (\cos ^{5}\left (f x +e \right )\right )-240 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+480 \left (\cos ^{3}\left (f x +e \right )\right )+583 \sin \left (f x +e \right ) \cos \left (f x +e \right )-754 \cos \left (f x +e \right )-402 \tan \left (f x +e \right )+367 \sec \left (f x +e \right )\right )}{35 f \left (\left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )-7 \left (\cos ^{6}\left (f x +e \right )\right )-24 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+56 \left (\cos ^{4}\left (f x +e \right )\right )+80 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-112 \left (\cos ^{2}\left (f x +e \right )\right )-64 \sin \left (f x +e \right )+64\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{8}}\)	\(227\)

[In]

int((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x,method=_RETURNVERBOSE)

[Out]

-1/35/f*(a*(sin(f*x+e)+1))^(1/2)*a^3/(cos(f*x+e)^6*sin(f*x+e)-7*cos(f*x+e)^6-24*cos(f*x+e)^4*sin(f*x+e)+56*cos
(f*x+e)^4+80*sin(f*x+e)*cos(f*x+e)^2-112*cos(f*x+e)^2-64*sin(f*x+e)+64)/(-c*(sin(f*x+e)-1))^(1/2)/c^8*(3*cos(f
*x+e)^7+24*cos(f*x+e)^5*sin(f*x+e)-96*cos(f*x+e)^5-240*cos(f*x+e)^3*sin(f*x+e)+480*cos(f*x+e)^3+583*sin(f*x+e)
*cos(f*x+e)-754*cos(f*x+e)-402*tan(f*x+e)+367*sec(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.15 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=-\frac {{\left (14 \, a^{3} \cos \left (f x + e\right )^{2} - 17 \, a^{3} + {\left (7 \, a^{3} \cos \left (f x + e\right )^{2} - 18 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{35 \, {\left (c^{9} f \cos \left (f x + e\right )^{9} - 32 \, c^{9} f \cos \left (f x + e\right )^{7} + 160 \, c^{9} f \cos \left (f x + e\right )^{5} - 256 \, c^{9} f \cos \left (f x + e\right )^{3} + 128 \, c^{9} f \cos \left (f x + e\right ) + 8 \, {\left (c^{9} f \cos \left (f x + e\right )^{7} - 10 \, c^{9} f \cos \left (f x + e\right )^{5} + 24 \, c^{9} f \cos \left (f x + e\right )^{3} - 16 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm="fricas")

[Out]

-1/35*(14*a^3*cos(f*x + e)^2 - 17*a^3 + (7*a^3*cos(f*x + e)^2 - 18*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)
*sqrt(-c*sin(f*x + e) + c)/(c^9*f*cos(f*x + e)^9 - 32*c^9*f*cos(f*x + e)^7 + 160*c^9*f*cos(f*x + e)^5 - 256*c^
9*f*cos(f*x + e)^3 + 128*c^9*f*cos(f*x + e) + 8*(c^9*f*cos(f*x + e)^7 - 10*c^9*f*cos(f*x + e)^5 + 24*c^9*f*cos
(f*x + e)^3 - 16*c^9*f*cos(f*x + e))*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(17/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {17}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(17/2), x)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.87 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {{\left (56 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 140 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 120 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 35 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{8960 \, c^{\frac {17}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16}} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm="giac")

[Out]

1/8960*(56*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 - 140*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 + 120*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 - 35*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/(c^(17/2)*f*sgn(sin(-1/4*pi + 1/2*
f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^16)

Mupad [B] (verification not implemented)

Time = 15.33 (sec) , antiderivative size = 673, normalized size of antiderivative = 3.78 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Too large to display} \]

[In]

int((a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(17/2),x)

[Out]

-((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*((a^3*exp(e*6i + f*x*6i)*(a + a*((ex
p(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*64i)/(5*c^9*f) + (256*a^3*exp(e*7i + f*x*7i)*(a +
a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*c^9*f) - (a^3*exp(e*8i + f*x*8i)*(a +
a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*832i)/(7*c^9*f) - (1024*a^3*exp(e*9i + f*x*
9i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(7*c^9*f) + (a^3*exp(e*10i + f*x*
10i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*832i)/(7*c^9*f) + (256*a^3*exp(e*
11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*c^9*f) - (a^3*exp(e
*12i + f*x*12i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*64i)/(5*c^9*f)))/(exp(
e*1i + f*x*1i)*16i - 119*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*544i + 1700*exp(e*4i + f*x*4i) + exp(e*5i + f
*x*5i)*3808i - 6188*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*7072i + 4862*exp(e*8i + f*x*8i) + 4862*exp(e*10i +
f*x*10i) + exp(e*11i + f*x*11i)*7072i - 6188*exp(e*12i + f*x*12i) - exp(e*13i + f*x*13i)*3808i + 1700*exp(e*1
4i + f*x*14i) + exp(e*15i + f*x*15i)*544i - 119*exp(e*16i + f*x*16i) - exp(e*17i + f*x*17i)*16i + exp(e*18i +
f*x*18i) + 1)