3.66 \(\int \csc ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx\)

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

[Out]

(2*d*Sqrt[d*Tan[a + b*x]])/b

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Rubi [A]  time = 0.0429873, 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} \[ \frac{2 d \sqrt{d \tan (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]

[Out]

(2*d*Sqrt[d*Tan[a + b*x]])/b

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

Mathematica [A]  time = 0.0496233, size = 18, normalized size = 1. \[ \frac{2 d \sqrt{d \tan (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]

[Out]

(2*d*Sqrt[d*Tan[a + b*x]])/b

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Maple [B]  time = 0.135, size = 58, normalized size = 3.2 \begin{align*} 2\,{\frac{\cos \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) -1 \right ) ^{2} \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}{b \left ( \sin \left ( bx+a \right ) \right ) ^{5}} \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{3/2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2*(d*tan(b*x+a))^(3/2),x)

[Out]

2/b*(d*sin(b*x+a)/cos(b*x+a))^(3/2)*cos(b*x+a)*(cos(b*x+a)-1)^2*(cos(b*x+a)+1)^2/sin(b*x+a)^5

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Maxima [A]  time = 1.50048, size = 31, normalized size = 1.72 \begin{align*} \frac{2 \, \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}}}{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))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.59553, size = 55, normalized size = 3.06 \begin{align*} \frac{2 \, d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**2*(d*tan(b*x+a))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} \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))^(3/2),x, algorithm="giac")

[Out]

integrate((d*tan(b*x + a))^(3/2)*csc(b*x + a)^2, x)