3.105 \(\int \frac{\csc ^2(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]

[Out]

(-2*d)/(7*b*(d*Tan[a + b*x])^(7/2))

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Rubi [A]  time = 0.0428568, 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} \[ -\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2/(d*Tan[a + b*x])^(5/2),x]

[Out]

(-2*d)/(7*b*(d*Tan[a + b*x])^(7/2))

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)}{(d \tan (a+b x))^{5/2}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{1}{x^{9/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.159704, size = 20, normalized size = 1. \[ -\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2/(d*Tan[a + b*x])^(5/2),x]

[Out]

(-2*d)/(7*b*(d*Tan[a + b*x])^(7/2))

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Maple [B]  time = 0.13, size = 38, normalized size = 1.9 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) }{7\,b\sin \left ( bx+a \right ) } \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{5}{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))^(5/2),x)

[Out]

-2/7/b*cos(b*x+a)/sin(b*x+a)/(d*sin(b*x+a)/cos(b*x+a))^(5/2)

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Maxima [A]  time = 1.08351, size = 31, normalized size = 1.55 \begin{align*} -\frac{2}{7 \, \left (d \tan \left (b x + a\right )\right )^{\frac{5}{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))^(5/2),x, algorithm="maxima")

[Out]

-2/7/((d*tan(b*x + a))^(5/2)*b*tan(b*x + a))

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Fricas [B]  time = 2.37278, size = 150, normalized size = 7.5 \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 )^{4}}{7 \,{\left (b d^{3} \cos \left (b x + a\right )^{4} - 2 \, b d^{3} \cos \left (b x + a\right )^{2} + b d^{3}\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))^(5/2),x, algorithm="fricas")

[Out]

-2/7*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)^4/(b*d^3*cos(b*x + a)^4 - 2*b*d^3*cos(b*x + a)^2 + b*d^3)

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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))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.14885, size = 35, normalized size = 1.75 \begin{align*} -\frac{2}{7 \, \sqrt{d \tan \left (b x + a\right )} b d^{2} \tan \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*tan(b*x+a))^(5/2),x, algorithm="giac")

[Out]

-2/7/(sqrt(d*tan(b*x + a))*b*d^2*tan(b*x + a)^3)