\(\int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx\) [13]

Optimal result

Integrand size = 34, antiderivative size = 69 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{c f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3013, 2852, 212, 2815, 2752} \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a}}{c f}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{c f} \]

[In]

Int[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]),x]

[Out]

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

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 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

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]))

Rule 2852

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

Rule 3013

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.70 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\sin (e+f x)\right ) \sec (e+f x) \sqrt {a (1+\sin (e+f x))}}{c f} \]

[In]

Integrate[(Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]),x]

[Out]

(2*Hypergeometric2F1[-1/2, 1, 1/2, 1 - Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])])/(c*f)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) \left (a^{\frac {3}{2}}-\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}}{\sqrt {a}}\right ) a \sqrt {a -a \sin \left (f x +e \right )}\right )}{\sqrt {a}\, c \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(78\)

[In]

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

[Out]

2*(1+sin(f*x+e))*(a^(3/2)-arctanh((a-a*sin(f*x+e))^(1/2)/a^(1/2))*a*(a-a*sin(f*x+e))^(1/2))/a^(1/2)/c/cos(f*x+
e)/(a+a*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. 202 vs. \(2 (61) = 122\).

Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.93 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {\sqrt {a} \cos \left (f x + e\right ) \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, c f \cos \left (f x + e\right )} \]

[In]

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

[Out]

1/2*(sqrt(a)*cos(f*x + e)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*
sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8
*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) -
cos(f*x + e) - 1)) + 4*sqrt(a*sin(f*x + e) + a))/(c*f*cos(f*x + e))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\int { -\frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) - c\right )} \sin \left (f x + e\right )} \,d x } \]

[In]

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

[Out]

-integrate(sqrt(a*sin(f*x + e) + a)/((c*sin(f*x + e) - c)*sin(f*x + e)), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=-\frac {\sqrt {2} {\left (\frac {\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c} + \frac {2 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}\right )} \sqrt {a}}{2 \, f} \]

[In]

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

[Out]

-1/2*sqrt(2)*(sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1
/2*f*x + 1/2*e)))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/c + 2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/(c*sin(-1/4*pi
+ 1/2*f*x + 1/2*e)))*sqrt(a)/f

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (c-c\,\sin \left (e+f\,x\right )\right )} \,d x \]

[In]

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

[Out]

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