3.12 \(\int \frac{1}{\sqrt{a+b \csc ^2(c+d x)}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \csc ^2(c+d x)}}\right )}{\sqrt{a} d} \]

[Out]

-(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b*Csc[c + d*x]^2]]/(Sqrt[a]*d))

________________________________________________________________________________________

Rubi [A]  time = 0.0360174, antiderivative size = 40, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4128, 377, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Csc[c + d*x]^2],x]

[Out]

-(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(Sqrt[a]*d))

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b \csc ^2(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{\cot (c+d x)}{\sqrt{a+b+b \cot ^2(c+d x)}}\right )}{d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b+b \cot ^2(c+d x)}}\right )}{\sqrt{a} d}\\ \end{align*}

Mathematica [B]  time = 0.144836, size = 98, normalized size = 2.51 \[ -\frac{\csc (c+d x) \sqrt{a \cos (2 (c+d x))-a-2 b} \log \left (\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right )}{\sqrt{2} \sqrt{a} d \sqrt{a+b \csc ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Csc[c + d*x]^2],x]

[Out]

-((Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]*Csc[c + d*x]*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2
*(c + d*x)]]])/(Sqrt[2]*Sqrt[a]*d*Sqrt[a + b*Csc[c + d*x]^2]))

________________________________________________________________________________________

Maple [B]  time = 0.371, size = 182, normalized size = 4.7 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}\ln \left ( 4\,\cos \left ( dx+c \right ) \sqrt{-a}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}}-4\,a\cos \left ( dx+c \right ) +4\,\sqrt{-a}\sqrt{-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}} \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a-b}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*csc(d*x+c)^2)^(1/2),x)

[Out]

1/d/(-a)^(1/2)*sin(d*x+c)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)*ln(4*cos(d*x+c)*(-a)^(1/2)*(-(a*cos(d
*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2)-4*a*cos(d*x+c)+4*(-a)^(1/2)*(-(a*cos(d*x+c)^2-a-b)/(cos(d*x+c)+1)^2)^(1/2
))/((a*cos(d*x+c)^2-a-b)/(cos(d*x+c)^2-1))^(1/2)/(-1+cos(d*x+c))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 0.685864, size = 995, normalized size = 25.51 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (128 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \,{\left (a^{4} + a^{3} b\right )} \cos \left (d x + c\right )^{6} + 160 \,{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \,{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 24 \,{\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{3} -{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )\right )}{8 \, a d}, \frac{\arctan \left (\frac{{\left (8 \, a^{2} \cos \left (d x + c\right )^{4} - 8 \,{\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )}{4 \,{\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 3 \,{\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{3} +{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )\right )}}\right )}{4 \, \sqrt{a} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/8*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*c
os(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c
)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 -
(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*
sin(d*x + c))/(a*d), 1/4*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*
sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^
2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)))/(sqrt(a)*d)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \csc ^{2}{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c)**2)**(1/2),x)

[Out]

Integral(1/sqrt(a + b*csc(c + d*x)**2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \csc \left (d x + c\right )^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csc(d*x+c)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(b*csc(d*x + c)^2 + a), x)