\(\int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx\) [254]

Optimal result

Integrand size = 23, antiderivative size = 255 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {\cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{\sqrt {a+b} f}-\frac {\cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{\sqrt {a+b} f}-\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}} \]

[Out]

cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-
b*(1+sec(f*x+e))/(a-b))^(1/2)/f/(a+b)^(1/2)-cot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-
b))^(1/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/f/(a+b)^(1/2)-cot(f*x+e)/f/(a+b*sec(
f*x+e))^(1/2)+b^2*tan(f*x+e)/(a^2-b^2)/f/(a+b*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3960, 3918, 21, 3914, 3917, 4089} \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {b^2 \tan (e+f x)}{f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}-\frac {\cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f \sqrt {a+b}}+\frac {\cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f \sqrt {a+b}}-\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}} \]

[In]

Int[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]],x]

[Out]

(Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*
x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(Sqrt[a + b]*f) - (Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a
+ b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e +
f*x]))/(a - b))])/(Sqrt[a + b]*f) - Cot[e + f*x]/(f*Sqrt[a + b*Sec[e + f*x]]) + (b^2*Tan[e + f*x])/((a^2 - b^2
)*f*Sqrt[a + b*Sec[e + f*x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3914

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

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3918

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-b)*Cot[e + f*x]*(
(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a
+ b*Csc[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 2)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 -
b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 3960

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)/cos[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Simp[Tan[e + f*x]*((a
+ b*Csc[e + f*x])^m/f), x] + Dist[b*m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e
, f, m}, x]

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}-\frac {1}{2} b \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx \\ & = -\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}}+\frac {b \int \frac {\sec (e+f x) \left (-\frac {a}{2}-\frac {1}{2} b \sec (e+f x)\right )}{\sqrt {a+b \sec (e+f x)}} \, dx}{a^2-b^2} \\ & = -\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}}-\frac {b \int \sec (e+f x) \sqrt {a+b \sec (e+f x)} \, dx}{2 \left (a^2-b^2\right )} \\ & = -\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}}-\frac {b \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{2 (a+b)}-\frac {b^2 \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx}{2 \left (a^2-b^2\right )} \\ & = \frac {\cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{\sqrt {a+b} f}-\frac {\cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{\sqrt {a+b} f}-\frac {\cot (e+f x)}{f \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.39 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\frac {\sqrt {\sec (e+f x)} \left (\frac {(b+a \cos (e+f x)) (-a+b \cos (e+f x)) \csc (e+f x)}{\left (a^2-b^2\right ) \sqrt {\sec (e+f x)}}+\frac {b \left (-\frac {(a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left (E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}-(b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (-a^2+b^2\right ) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)}}\right )}{f \sqrt {a+b \sec (e+f x)}} \]

[In]

Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]],x]

[Out]

(Sqrt[Sec[e + f*x]]*(((b + a*Cos[e + f*x])*(-a + b*Cos[e + f*x])*Csc[e + f*x])/((a^2 - b^2)*Sqrt[Sec[e + f*x]]
) + (b*(-(((a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*(EllipticE[ArcSin[Tan[(e + f*x)/2]]
, (a - b)/(a + b)] - EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x
])]) - (b + a*Cos[e + f*x])*Tan[(e + f*x)/2]))/((-a^2 + b^2)*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*
Sec[e + f*x]])))/(f*Sqrt[a + b*Sec[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(758\) vs. \(2(233)=466\).

Time = 5.73 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.98

method	result	size
default	\(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (-\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a b \cos \left (f x +e \right )-\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2} \cos \left (f x +e \right )+\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a b \cos \left (f x +e \right )+\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{2} \cos \left (f x +e \right )-\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a b -\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2}+\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a b +\sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2}+a^{2} \cos \left (f x +e \right ) \cot \left (f x +e \right )-a b \cos \left (f x +e \right ) \cot \left (f x +e \right )+a b \cot \left (f x +e \right )-b^{2} \cot \left (f x +e \right )\right )}{f \left (a -b \right ) \left (a +b \right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(759\)

[In]

int(csc(f*x+e)^2/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/f/(a-b)/(a+b)*(a+b*sec(f*x+e))^(1/2)/(b+a*cos(f*x+e))*(-EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2)
)*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*a*b*cos(f*x+e)-EllipticF(c
ot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x
+e)+1))^(1/2)*b^2*cos(f*x+e)+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)
*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*a*b*cos(f*x+e)+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))
^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*b^2*cos(f*x+e)-(
1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*EllipticF(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e
)/(cos(f*x+e)+1))^(1/2)*a*b-(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*EllipticF(cot(f*x+e)-csc(f*x+e),((
a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*b^2+(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*Ellip
ticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*a*b+(1/(a+b)*(b+a*cos(f*x+e)
)/(cos(f*x+e)+1))^(1/2)*EllipticE(cot(f*x+e)-csc(f*x+e),((a-b)/(a+b))^(1/2))*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)
*b^2+a^2*cos(f*x+e)*cot(f*x+e)-a*b*cos(f*x+e)*cot(f*x+e)+a*b*cot(f*x+e)-b^2*cot(f*x+e))

Fricas [F]

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

[In]

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

[Out]

integral(csc(f*x + e)^2/sqrt(b*sec(f*x + e) + a), x)

Sympy [F]

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

[In]

integrate(csc(f*x+e)**2/(a+b*sec(f*x+e))**(1/2),x)

[Out]

Integral(csc(e + f*x)**2/sqrt(a + b*sec(e + f*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

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

[Out]

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