3.98 \(\int \frac{\csc ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac{4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]

[Out]

(-2*d^5)/(13*b*(d*Tan[a + b*x])^(13/2)) - (4*d^3)/(9*b*(d*Tan[a + b*x])^(9/2)) - (2*d)/(5*b*(d*Tan[a + b*x])^(
5/2))

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Rubi [A]  time = 0.0541373, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2591, 270} \[ -\frac{2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac{4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^6/(d*Tan[a + b*x])^(3/2),x]

[Out]

(-2*d^5)/(13*b*(d*Tan[a + b*x])^(13/2)) - (4*d^3)/(9*b*(d*Tan[a + b*x])^(9/2)) - (2*d)/(5*b*(d*Tan[a + b*x])^(
5/2))

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\csc ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{\left (d^2+x^2\right )^2}{x^{15/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{d \operatorname{Subst}\left (\int \left (\frac{d^4}{x^{15/2}}+\frac{2 d^2}{x^{11/2}}+\frac{1}{x^{7/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d^5}{13 b (d \tan (a+b x))^{13/2}}-\frac{4 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.121051, size = 54, normalized size = 0.83 \[ \frac{-90 \csc ^6(a+b x)+10 \csc ^4(a+b x)+16 \csc ^2(a+b x)+64}{585 b d \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^6/(d*Tan[a + b*x])^(3/2),x]

[Out]

(64 + 16*Csc[a + b*x]^2 + 10*Csc[a + b*x]^4 - 90*Csc[a + b*x]^6)/(585*b*d*Sqrt[d*Tan[a + b*x]])

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Maple [A]  time = 0.2, size = 60, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-208\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+234 \right ) \cos \left ( bx+a \right ) }{585\,b \left ( \sin \left ( bx+a \right ) \right ) ^{5}} \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x)

[Out]

-2/585/b*(32*cos(b*x+a)^4-104*cos(b*x+a)^2+117)*cos(b*x+a)/sin(b*x+a)^5/(d*sin(b*x+a)/cos(b*x+a))^(3/2)

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Maxima [A]  time = 1.16132, size = 65, normalized size = 1. \begin{align*} -\frac{2 \,{\left (117 \, d^{4} \tan \left (b x + a\right )^{4} + 130 \, d^{4} \tan \left (b x + a\right )^{2} + 45 \, d^{4}\right )} d}{585 \, \left (d \tan \left (b x + a\right )\right )^{\frac{13}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

-2/585*(117*d^4*tan(b*x + a)^4 + 130*d^4*tan(b*x + a)^2 + 45*d^4)*d/((d*tan(b*x + a))^(13/2)*b)

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Fricas [B]  time = 2.14254, size = 269, normalized size = 4.14 \begin{align*} \frac{2 \,{\left (32 \, \cos \left (b x + a\right )^{7} - 104 \, \cos \left (b x + a\right )^{5} + 117 \, \cos \left (b x + a\right )^{3}\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{585 \,{\left (b d^{2} \cos \left (b x + a\right )^{6} - 3 \, b d^{2} \cos \left (b x + a\right )^{4} + 3 \, b d^{2} \cos \left (b x + a\right )^{2} - b d^{2}\right )} \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

2/585*(32*cos(b*x + a)^7 - 104*cos(b*x + a)^5 + 117*cos(b*x + a)^3)*sqrt(d*sin(b*x + a)/cos(b*x + a))/((b*d^2*
cos(b*x + a)^6 - 3*b*d^2*cos(b*x + a)^4 + 3*b*d^2*cos(b*x + a)^2 - b*d^2)*sin(b*x + a))

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

[Out]

Timed out

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Giac [A]  time = 1.15975, size = 78, normalized size = 1.2 \begin{align*} -\frac{2 \,{\left (117 \, d^{6} \tan \left (b x + a\right )^{4} + 130 \, d^{6} \tan \left (b x + a\right )^{2} + 45 \, d^{6}\right )}}{585 \, \sqrt{d \tan \left (b x + a\right )} b d^{7} \tan \left (b x + a\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6/(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

-2/585*(117*d^6*tan(b*x + a)^4 + 130*d^6*tan(b*x + a)^2 + 45*d^6)/(sqrt(d*tan(b*x + a))*b*d^7*tan(b*x + a)^6)