3.56 \(\int \csc ^2(a+b x) \sqrt{d \tan (a+b x)} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

[Out]

(-2*d)/(b*Sqrt[d*Tan[a + b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0414481, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2591, 30} \[ -\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*d)/(b*Sqrt[d*Tan[a + b*x]])

Rule 2591

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, (b*Tan[e + f*x])/f
f], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \csc ^2(a+b x) \sqrt{d \tan (a+b x)} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d}{b \sqrt{d \tan (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0727914, size = 18, normalized size = 1. \[ -\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*d)/(b*Sqrt[d*Tan[a + b*x]])

________________________________________________________________________________________

Maple [B]  time = 0.136, size = 38, normalized size = 2.1 \begin{align*} -2\,{\frac{\cos \left ( bx+a \right ) }{b\sin \left ( bx+a \right ) }\sqrt{{\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.26284, size = 31, normalized size = 1.72 \begin{align*} -\frac{2 \, \sqrt{d \tan \left (b x + a\right )}}{b \tan \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(d*tan(b*x + a))/(b*tan(b*x + a))

________________________________________________________________________________________

Fricas [B]  time = 1.63919, size = 92, normalized size = 5.11 \begin{align*} -\frac{2 \, \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)/(b*sin(b*x + a))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(d*tan(a + b*x))*csc(a + b*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a)^2, x)