\(\int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx\) [322]

Optimal result

Integrand size = 28, antiderivative size = 110 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{3 c f \sqrt {c-c \sin (e+f x)}} \]

[Out]

3/4*cos(f*x+e)/a/f/(c-c*sin(f*x+e))^(3/2)+3/8*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))/a
/c^(3/2)/f*2^(1/2)-sec(f*x+e)/a/c/f/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2815, 2766, 2729, 2728, 212} \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \]

[In]

Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2)),x]

[Out]

(3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*a*c^(3/2)*f) + (3*Cos[e + f*
x])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - Sec[e + f*x]/(a*c*f*Sqrt[c - c*Sin[e + f*x]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

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

Rule 2766

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(-b)*((
g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[a*((2*p + 1)/(2*g^2*(p + 1))), In
t[(g*Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0
] && LtQ[p, -1] && IntegerQ[2*p]

Rule 2815

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c} \\ & = -\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{2 a} \\ & = \frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{8 a c} \\ & = \frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}-\frac {3 \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{4 a c f} \\ & = \frac {3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.60 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\sec (e+f x) \left (1+(3+3 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-3 \sin (e+f x)\right )}{12 c f \sqrt {c-c \sin (e+f x)}} \]

[In]

Integrate[1/((3 + 3*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2)),x]

[Out]

-1/12*(Sec[e + f*x]*(1 + (3 + 3*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 3*Sin[e + f*x]))/(c*f*Sqrt[c - c*Sin[e +
f*x]])

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.22

method	result	size
default	\(-\frac {3 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \sqrt {c \left (\sin \left (f x +e \right )+1\right )}-6 c^{\frac {3}{2}} \sin \left (f x +e \right )+2 c^{\frac {3}{2}}}{8 c^{\frac {5}{2}} a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(134\)

[In]

int(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/8/c^(5/2)/a*(3*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*sin(f
*x+e)*c-3*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*c*(c*(sin(f*x+e)+1))^(1/2)-6*c^(3/2)*s
in(f*x+e)+2*c^(3/2))/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (100) = 200\).

Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \, \sqrt {2} {\left (\cos \left (f x + e\right ) \sin \left (f x + e\right ) - \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (3 \, \sin \left (f x + e\right ) - 1\right )}}{16 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]

[In]

integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/16*(3*sqrt(2)*(cos(f*x + e)*sin(f*x + e) - cos(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*
sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*
x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*sqrt(-c*sin(f*x + e)
+ c)*(3*sin(f*x + e) - 1))/(a*c^2*f*cos(f*x + e)*sin(f*x + e) - a*c^2*f*cos(f*x + e))

Sympy [F]

\[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\int \frac {1}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a} \]

[In]

integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))**(3/2),x)

[Out]

Integral(1/(-c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2 + c*sqrt(-c*sin(e + f*x) + c)), x)/a

Maxima [F]

\[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(3/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (100) = 200\).

Time = 0.33 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.63 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\frac {6 \, \sqrt {2} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{a c^{\frac {3}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (\sqrt {c} + \frac {14 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {3 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )}}{a c^{2} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{32 \, f} \]

[In]

integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

1/32*(6*sqrt(2)*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1))/(a*c^(3/2)*sgn
(sin(-1/4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(a*c^(3/2)*(cos(-1/4*pi + 1/2
*f*x + 1/2*e) + 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + sqrt(2)*(sqrt(c) + 14*sqrt(c)*(cos(-1/4*pi + 1/2*f*x
+ 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1
/4*pi + 1/2*f*x + 1/2*e) + 1)^2)/(a*c^2*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e)
+ 1) + (cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2)*sgn(sin(-1/4*pi + 1/2*f*
x + 1/2*e))))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

int(1/((a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)),x)

[Out]

int(1/((a + a*sin(e + f*x))*(c - c*sin(e + f*x))^(3/2)), x)