\(\int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx\) [381]

Optimal result

Integrand size = 30, antiderivative size = 178 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}} \]

[Out]

1/14*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(15/2)+1/56*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c
-c*sin(f*x+e))^(13/2)+1/280*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+e))^(11/2)+1/2240*cos(f*x+e)*
(a+a*sin(f*x+e))^(7/2)/c^3/f/(c-c*sin(f*x+e))^(9/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2822, 2821} \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]

[In]

Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2),x]

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*
x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(280*c^2*f*(c - c*
Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2240*c^3*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}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {3 \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{14 c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c 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}{28 c^2} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 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}{280 c^3} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.33 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.87 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {8 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3+3 \sin (e+f x))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (3+3 \sin (e+f x))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {6 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (3+3 \sin (e+f x))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (3+3 \sin (e+f x))^{7/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2),x]

[Out]

(8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3 + 3*Sin[e + f*x])^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^7*(c - c*Sin[e + f*x])^(15/2)) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(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])^(15/2)) + (6*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])
^5*(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])^(15/2)) - ((C
os[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(3 + 3*Sin[e + f*x])^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7
*(c - c*Sin[e + f*x])^(15/2))

Maple [A] (verified)

Time = 3.45 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{3} \left (13 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-91 \left (\cos ^{5}\left (f x +e \right )\right )-312 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+728 \left (\cos ^{3}\left (f x +e \right )\right )+1075 \sin \left (f x +e \right ) \cos \left (f x +e \right )-1393 \cos \left (f x +e \right )-916 \tan \left (f x +e \right )+756 \sec \left (f x +e \right )\right )}{140 f \left (\cos ^{6}\left (f x +e \right )+6 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-18 \left (\cos ^{4}\left (f x +e \right )\right )-32 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+48 \left (\cos ^{2}\left (f x +e \right )\right )+32 \sin \left (f x +e \right )-32\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{7}}\)	\(200\)

[In]

int((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x,method=_RETURNVERBOSE)

[Out]

1/140/f*(a*(sin(f*x+e)+1))^(1/2)*a^3/(cos(f*x+e)^6+6*cos(f*x+e)^4*sin(f*x+e)-18*cos(f*x+e)^4-32*sin(f*x+e)*cos
(f*x+e)^2+48*cos(f*x+e)^2+32*sin(f*x+e)-32)/(-c*(sin(f*x+e)-1))^(1/2)/c^7*(13*cos(f*x+e)^5*sin(f*x+e)-91*cos(f
*x+e)^5-312*cos(f*x+e)^3*sin(f*x+e)+728*cos(f*x+e)^3+1075*sin(f*x+e)*cos(f*x+e)-1393*cos(f*x+e)-916*tan(f*x+e)
+756*sec(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {{\left (63 \, a^{3} \cos \left (f x + e\right )^{2} - 76 \, a^{3} + 7 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{140 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) - {\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="fricas")

[Out]

1/140*(63*a^3*cos(f*x + e)^2 - 76*a^3 + 7*(5*a^3*cos(f*x + e)^2 - 12*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) +
a)*sqrt(-c*sin(f*x + e) + c)/(7*c^8*f*cos(f*x + e)^7 - 56*c^8*f*cos(f*x + e)^5 + 112*c^8*f*cos(f*x + e)^3 - 64
*c^8*f*cos(f*x + e) - (c^8*f*cos(f*x + e)^7 - 24*c^8*f*cos(f*x + e)^5 + 80*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos
(f*x + e))*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(15/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(15/2), x)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {{\left (35 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 84 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 70 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 20 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{2240 \, c^{8} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14}} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="giac")

[Out]

1/2240*(35*a^3*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)^6 - 84*a^3*sqrt(c)*s
gn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4 + 70*a^3*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 - 20*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/(c^8
*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^14)

Mupad [B] (verification not implemented)

Time = 14.65 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.63 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Too large to display} \]

[In]

int((a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(15/2),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*((exp
(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*144i)/(5*c^8*f) - (8*a^3*exp(e*5i + f*x*5i)*(a + a
*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^8*f) + (344*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^8*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)*2848i)/(35*c^8*f) - (344*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))/(5*c^8*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)*144i)/(5*c^8*f) + (8*a^3*exp(e*11
i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^8*f)))/(exp(e*1i + f*
x*1i)*14i - 90*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*350i + 910*exp(e*4i + f*x*4i) + exp(e*5i + f*x*5i)*1638
i - 2002*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*1430i - exp(e*9i + f*x*9i)*1430i + 2002*exp(e*10i + f*x*10i)
+ exp(e*11i + f*x*11i)*1638i - 910*exp(e*12i + f*x*12i) - exp(e*13i + f*x*13i)*350i + 90*exp(e*14i + f*x*14i)
+ exp(e*15i + f*x*15i)*14i - exp(e*16i + f*x*16i) + 1)