\(\int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx\) [1371]

Optimal result

Integrand size = 33, antiderivative size = 433 \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=-\frac {b \sqrt {g} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}+\frac {b^{3/2} \sqrt {g} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \sqrt [4]{-a^2+b^2} f}+\frac {b \sqrt {g} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}-\frac {b^{3/2} \sqrt {g} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^2 \sqrt [4]{-a^2+b^2} f}-\frac {(g \cos (e+f x))^{3/2} \csc (e+f x)}{a f g}-\frac {\sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {\cos (e+f x)}}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}} \] Output:

-b*g^(1/2)*arctan((g*cos(f*x+e))^(1/2)/g^(1/2))/a^2/f+b^(3/2)*g^(1/2)*arct 
an(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2+b^2)^(1/4)/g^(1/2))/a^2/(-a^2+b^2)^( 
1/4)/f+b*g^(1/2)*arctanh((g*cos(f*x+e))^(1/2)/g^(1/2))/a^2/f-b^(3/2)*g^(1/ 
2)*arctanh(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2+b^2)^(1/4)/g^(1/2))/a^2/(-a^ 
2+b^2)^(1/4)/f-(g*cos(f*x+e))^(3/2)*csc(f*x+e)/a/f/g-(g*cos(f*x+e))^(1/2)* 
EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/a/f/cos(f*x+e)^(1/2)+b*g*cos(f*x+e)^ 
(1/2)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))/a/(b 
-(-a^2+b^2)^(1/2))/f/(g*cos(f*x+e))^(1/2)+b*g*cos(f*x+e)^(1/2)*EllipticPi( 
sin(1/2*f*x+1/2*e),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))/a/(b+(-a^2+b^2)^(1/2) 
)/f/(g*cos(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 38.06 (sec) , antiderivative size = 1550, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

Integrate[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]
 

Output:

-((Sqrt[g*Cos[e + f*x]]*Cot[e + f*x])/(a*f)) + (Sqrt[g*Cos[e + f*x]]*((4*a 
*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f 
*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^ 
2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^ 
2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 
+ b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)* 
Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sq 
rt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b] 
*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) + ( 
5*b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6 
*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e 
+ f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 
 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2 
)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f* 
x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 
- Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + 
Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b 
]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4 
)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/ 
4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e +...
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3042, 3377, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\csc ^2(e+f x) \sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {g \cos (e+f x)}}{\sin (e+f x)^2 (a+b \sin (e+f x))}dx\)

\(\Big \downarrow \) 3377

\(\displaystyle \int \left (\frac {b^2 \sqrt {g \cos (e+f x)}}{a^2 (a+b \sin (e+f x))}-\frac {b \csc (e+f x) \sqrt {g \cos (e+f x)}}{a^2}+\frac {\csc ^2(e+f x) \sqrt {g \cos (e+f x)}}{a}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^{3/2} \sqrt {g} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^2 f \sqrt [4]{b^2-a^2}}-\frac {b \sqrt {g} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}-\frac {b^{3/2} \sqrt {g} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^2 f \sqrt [4]{b^2-a^2}}+\frac {b \sqrt {g} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^2 f}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a f \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}+\frac {b g \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a f \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}-\frac {\csc (e+f x) (g \cos (e+f x))^{3/2}}{a f g}-\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f \sqrt {\cos (e+f x)}}\)

Input:

Int[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]
 

Output:

-((b*Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f)) + (b^(3/2)*Sqr 
t[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/ 
(a^2*(-a^2 + b^2)^(1/4)*f) + (b*Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[ 
g]])/(a^2*f) - (b^(3/2)*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((- 
a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(1/4)*f) - ((g*Cos[e + f*x]) 
^(3/2)*Csc[e + f*x])/(a*f*g) - (Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2 
, 2])/(a*f*Sqrt[Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/ 
(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b - Sqrt[-a^2 + b^2])*f*Sqrt[ 
g*Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 
 + b^2]), (e + f*x)/2, 2])/(a*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x] 
])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3377

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^(n_))/((a 
_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(g*cos[e + 
 f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, 
 g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/ 
2, 0])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.18 (sec) , antiderivative size = 1230, normalized size of antiderivative = 2.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1230\)

Input:

int((g*cos(f*x+e))^(1/2)*csc(f*x+e)^2/(a+b*sin(f*x+e)),x,method=_RETURNVER 
BOSE)
 

Output:

(4*g*b*(1/4/a^2/(-g)^(1/2)*ln((-2*g+2*(-g)^(1/2)*(2*g*cos(1/2*f*x+1/2*e)^2 
-g)^(1/2))/cos(1/2*f*x+1/2*e))+1/8/a^2/g^(1/2)*ln((4*cos(1/2*f*x+1/2*e)*g+ 
2*g^(1/2)*(-2*g*sin(1/2*f*x+1/2*e)^2+g)^(1/2)-2*g)/(cos(1/2*f*x+1/2*e)-1)) 
+1/8/a^2/g^(1/2)*ln((-4*cos(1/2*f*x+1/2*e)*g+2*g^(1/2)*(-2*g*sin(1/2*f*x+1 
/2*e)^2+g)^(1/2)-2*g)/(cos(1/2*f*x+1/2*e)+1))+1/16/a^2/(g^2*(a^2-b^2)/b^2) 
^(1/4)*2^(1/2)*(ln((2*g*cos(1/2*f*x+1/2*e)^2-g-(g^2*(a^2-b^2)/b^2)^(1/4)*( 
2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)*2^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/2))/(2*g* 
cos(1/2*f*x+1/2*e)^2-g+(g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2*e)^2 
-g)^(1/2)*2^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan(2^(1/2)/(g^2*(a^2-b 
^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)+1)-2*arctan(-2^(1/2)/(g^ 
2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)+1)))+1/8*(g*(-1+ 
2*cos(1/2*f*x+1/2*e)^2)*sin(1/2*f*x+1/2*e)^2)^(1/2)/a*g/cos(1/2*f*x+1/2*e) 
/(-2*g*sin(1/2*f*x+1/2*e)^4+g*sin(1/2*f*x+1/2*e)^2)^(3/2)*(8*cos(1/2*f*x+1 
/2*e)*(sin(1/2*f*x+1/2*e)^2)^(1/2)*EllipticE(cos(1/2*f*x+1/2*e),2^(1/2))*( 
-1+2*sin(1/2*f*x+1/2*e)^2)^(3/2)*sin(1/2*f*x+1/2*e)^2*g+32*sin(1/2*f*x+1/2 
*e)^8*g-48*sin(1/2*f*x+1/2*e)^6*g-cos(1/2*f*x+1/2*e)/a^2*sum(1/_alpha*(8*( 
sin(1/2*f*x+1/2*e)^2)^(1/2)*(-1+2*sin(1/2*f*x+1/2*e)^2)^(1/2)*EllipticPi(c 
os(1/2*f*x+1/2*e),(-4*_alpha^2*b^2+4*b^2)/a^2,2^(1/2))*(g*(2*_alpha^2*b^2+ 
a^2-2*b^2)/b^2)^(1/2)*_alpha^3*b^2-8*b^2*_alpha*(sin(1/2*f*x+1/2*e)^2)^(1/ 
2)*(-1+2*sin(1/2*f*x+1/2*e)^2)^(1/2)*EllipticPi(cos(1/2*f*x+1/2*e),(-4*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))^(1/2)*csc(f*x+e)^2/(a+b*sin(f*x+e)),x, algorithm= 
"fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g \cos {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}}{a + b \sin {\left (e + f x \right )}}\, dx \] Input:

integrate((g*cos(f*x+e))**(1/2)*csc(f*x+e)**2/(a+b*sin(f*x+e)),x)
 

Output:

Integral(sqrt(g*cos(e + f*x))*csc(e + f*x)**2/(a + b*sin(e + f*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*cos(f*x+e))^(1/2)*csc(f*x+e)^2/(a+b*sin(f*x+e)),x, algorithm= 
"maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )} \csc \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a} \,d x } \] Input:

integrate((g*cos(f*x+e))^(1/2)*csc(f*x+e)^2/(a+b*sin(f*x+e)),x, algorithm= 
"giac")
 

Output:

integrate(sqrt(g*cos(f*x + e))*csc(f*x + e)^2/(b*sin(f*x + e) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}}{{\sin \left (e+f\,x\right )}^2\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:

int((g*cos(e + f*x))^(1/2)/(sin(e + f*x)^2*(a + b*sin(e + f*x))),x)
 

Output:

int((g*cos(e + f*x))^(1/2)/(sin(e + f*x)^2*(a + b*sin(e + f*x))), x)
 

Reduce [F]

\[ \int \frac {\sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx=\sqrt {g}\, \left (\int \frac {\sqrt {\cos \left (f x +e \right )}\, \csc \left (f x +e \right )^{2}}{\sin \left (f x +e \right ) b +a}d x \right ) \] Input:

int((g*cos(f*x+e))^(1/2)*csc(f*x+e)^2/(a+b*sin(f*x+e)),x)
 

Output:

sqrt(g)*int((sqrt(cos(e + f*x))*csc(e + f*x)**2)/(sin(e + f*x)*b + a),x)