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

Optimal. Leaf size=20 \[ -\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]

[Out]

(-2*d)/(3*b*(d*Tan[a + b*x])^(3/2))

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Rubi [A]  time = 0.0363285, antiderivative size = 20, 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} \[ -\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(3*b*(d*Tan[a + b*x])^(3/2))

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 \frac{\csc ^2(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0997169, size = 20, normalized size = 1. \[ -\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d)/(3*b*(d*Tan[a + b*x])^(3/2))

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Maple [B]  time = 0.145, size = 38, normalized size = 1.9 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) }{3\,b\sin \left ( bx+a \right ) }{\frac{1}{\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]

-2/3/b*cos(b*x+a)/sin(b*x+a)/(d*sin(b*x+a)/cos(b*x+a))^(1/2)

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Maxima [A]  time = 1.09345, size = 31, normalized size = 1.55 \begin{align*} -\frac{2}{3 \, \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]

-2/3/(sqrt(d*tan(b*x + a))*b*tan(b*x + a))

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Fricas [B]  time = 2.14189, size = 109, normalized size = 5.45 \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 )^{2}}{3 \,{\left (b d \cos \left (b x + a\right )^{2} - b d\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]

2/3*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)^2/(b*d*cos(b*x + a)^2 - b*d)

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

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Giac [A]  time = 1.11735, size = 31, normalized size = 1.55 \begin{align*} -\frac{2}{3 \, \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="giac")

[Out]

-2/3/(sqrt(d*tan(b*x + a))*b*tan(b*x + a))