3.256 \(\int \frac{\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=318 \[ -\frac{(3 a-b) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{f (a-b) (a+b)^{3/2}}+\frac{4 a b^2 \tan (e+f x)}{f \left (a^2-b^2\right )^2 \sqrt{a+b \sec (e+f x)}}+\frac{b^2 \tan (e+f x)}{f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac{4 a \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f (a-b) (a+b)^{3/2}} \]

[Out]

(4*a*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))])/((a - b)*(a + b)^(3/2)*f) - ((3*a - b)*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))])/((a - b)*(a + b)^(3/2)*f) - Cot[e + f*x]/(f*(a + b*Sec[e + f*x])^(3/
2)) + (b^2*Tan[e + f*x])/((a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (4*a*b^2*Tan[e + f*x])/((a^2 - b^2)^2*f*
Sqrt[a + b*Sec[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.52453, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3875, 3833, 4003, 4005, 3832, 4004} \[ \frac{4 a b^2 \tan (e+f x)}{f \left (a^2-b^2\right )^2 \sqrt{a+b \sec (e+f x)}}+\frac{b^2 \tan (e+f x)}{f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}-\frac{(3 a-b) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f (a-b) (a+b)^{3/2}}+\frac{4 a \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{f (a-b) (a+b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2),x]

[Out]

(4*a*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))])/((a - b)*(a + b)^(3/2)*f) - ((3*a - b)*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))])/((a - b)*(a + b)^(3/2)*f) - Cot[e + f*x]/(f*(a + b*Sec[e + f*x])^(3/
2)) + (b^2*Tan[e + f*x])/((a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (4*a*b^2*Tan[e + f*x])/((a^2 - b^2)^2*f*
Sqrt[a + b*Sec[e + f*x]])

Rule 3875

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 3833

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 4003

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

Rule 4005

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/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, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A
^2 - B^2, 0]

Rule 3832

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

Rule 4004

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 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), 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*} \int \frac{\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx &=-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}-\frac{1}{2} (3 b) \int \frac{\sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac{b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac{b \int \frac{\sec (e+f x) \left (-\frac{3 a}{2}+\frac{1}{2} b \sec (e+f x)\right )}{(a+b \sec (e+f x))^{3/2}} \, dx}{a^2-b^2}\\ &=-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac{b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac{4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt{a+b \sec (e+f x)}}-\frac{(2 b) \int \frac{\sec (e+f x) \left (\frac{1}{4} \left (3 a^2+b^2\right )+a b \sec (e+f x)\right )}{\sqrt{a+b \sec (e+f x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac{b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac{4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt{a+b \sec (e+f x)}}-\frac{((3 a-b) b) \int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx}{2 (a-b) (a+b)^2}-\frac{\left (2 a b^2\right ) \int \frac{\sec (e+f x) (1+\sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{4 a \cot (e+f x) E\left (\sin ^{-1}\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}}}{(a-b) (a+b)^{3/2} f}-\frac{(3 a-b) \cot (e+f x) F\left (\sin ^{-1}\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}}}{(a-b) (a+b)^{3/2} f}-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac{b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac{4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt{a+b \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 7.7989, size = 259, normalized size = 0.81 \[ \frac{-2 b \left (3 a^2+4 a b+b^2\right ) \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{a-b}{a+b}\right )-(a-b) \csc (e+f x) (a (a-3 b) \cos (e+f x)+b (3 a-b))+8 a b (a+b) \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a-b}{a+b}\right )}{f \left (a^2-b^2\right )^2 \sqrt{a+b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2),x]

[Out]

(-((a - b)*((3*a - b)*b + a*(a - 3*b)*Cos[e + f*x])*Csc[e + f*x]) + 8*a*b*(a + b)*Cos[(e + f*x)/2]^2*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)]*Sec[e + f*
x]*Sqrt[(1 + Sec[e + f*x])^(-1)] - 2*b*(3*a^2 + 4*a*b + b^2)*Cos[(e + f*x)/2]^2*Sqrt[(b + a*Cos[e + f*x])/((a
+ b)*(1 + Cos[e + f*x]))]*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sec[e + f*x]*Sqrt[(1 + Sec[e +
f*x])^(-1)])/((a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])

________________________________________________________________________________________

Maple [B]  time = 0.258, size = 1065, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x)

[Out]

-1/2/f/(a-b)^2/(a+b)^2*(4*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*sin(f*x+e)*(cos
(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a^2*b+4*EllipticE((-1+cos(f*x+e)
)/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+
e)+b)/(1+cos(f*x+e)))^(1/2)*a*b^2-3*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*sin(f
*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a^2*b-4*EllipticF((-1+
cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+e)*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(
a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*b^2-EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*cos(f*x+
e)*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*b^3+4*Elliptic
E((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos
(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a^2*b+4*EllipticE((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*
(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*b^2-3*EllipticF((-1+cos(f*
x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(
1+cos(f*x+e)))^(1/2)*a^2*b-4*EllipticF((-1+cos(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*x+e)/
(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*b^2-EllipticF((-1+cos(f*x+e))/sin(f*x+
e),((a-b)/(a+b))^(1/2))*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))
^(1/2)*b^3+cos(f*x+e)^2*a^3-4*a^2*b*cos(f*x+e)^2+3*cos(f*x+e)^2*a*b^2+3*cos(f*x+e)*a^2*b-4*a*b^2*cos(f*x+e)+co
s(f*x+e)*b^3)*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*4^(1/2)/(a*cos(f*x+e)+b)/sin(f*x+e)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{2}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)^2/(b*sec(f*x + e) + a)^(3/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2}}{b^{2} \sec \left (f x + e\right )^{2} + 2 \, a b \sec \left (f x + e\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sec{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**2/(a+b*sec(f*x+e))**(3/2),x)

[Out]

Integral(csc(e + f*x)**2/(a + b*sec(e + f*x))**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{2}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2/(a+b*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(csc(f*x + e)^2/(b*sec(f*x + e) + a)^(3/2), x)