\(\int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx\) [20]

Optimal result

Integrand size = 36, antiderivative size = 102 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {\log (\sin (e+f x)) \sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{c f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3021, 2816, 2746, 31, 3027, 3556} \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \log (\sin (e+f x))}{c f}-\frac {a \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

-((a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Log[Sin[e +
f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(c*f)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2816

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

Rule 3021

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x
_)]]), x_Symbol] :> Dist[-d/c, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/c, Int[S
qrt[a + b*Sin[e + f*x]]*(Sqrt[c + d*Sin[e + f*x]]/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] && EqQ[b*c + a*d, 0]

Rule 3027

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(\log (1-\sin (e+f x))-\log (\sin (e+f x))) \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{c f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.98

method	result	size
default	\(\frac {\left (2 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right )}{f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(100\)

[In]

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

[Out]

1/f*(2*ln(csc(f*x+e)-cot(f*x+e)-1)-ln(csc(f*x+e)-cot(f*x+e)))*(a*(1+sin(f*x+e)))^(1/2)*(-cos(f*x+e)+sin(f*x+e)
-1)/(cos(f*x+e)+sin(f*x+e)+1)/(-c*(sin(f*x+e)-1))^(1/2)

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c*cos(f*x + e)^2 + c*sin(f*x + e) - c), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\frac {2 \, \sqrt {a} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{\sqrt {c}} - \frac {\sqrt {a} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{\sqrt {c}}}{f} \]

[In]

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

[Out]

(2*sqrt(a)*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/sqrt(c) - sqrt(a)*log(sin(f*x + e)/(cos(f*x + e) + 1))/sqr
t(c))/f

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx=0 \]

[In]

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

[Out]

0

Mupad [F(-1)]

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

[In]

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

[Out]

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