3.97 \(\int \frac{\csc ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]

[Out]

(-2*d^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 = 0.0496197, antiderivative size = 43, 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, 14} \[ -\frac{2 d^3}{9 b (d \tan (a+b x))^{9/2}}-\frac{2 d}{5 b (d \tan (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4/(d*Tan[a + b*x])^(3/2),x]

[Out]

(-2*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\csc ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{d^2+x^2}{x^{11/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{d \operatorname{Subst}\left (\int \left (\frac{d^2}{x^{11/2}}+\frac{1}{x^{7/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 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 = 0.0945412, size = 42, normalized size = 0.98 \[ \frac{2 \left (-5 \csc ^4(a+b x)+\csc ^2(a+b x)+4\right )}{45 b d \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^4/(d*Tan[a + b*x])^(3/2),x]

[Out]

(2*(4 + Csc[a + b*x]^2 - 5*Csc[a + b*x]^4))/(45*b*d*Sqrt[d*Tan[a + b*x]])

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Maple [A]  time = 0.149, size = 50, normalized size = 1.2 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-18 \right ) \cos \left ( bx+a \right ) }{45\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}} \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)^4/(d*tan(b*x+a))^(3/2),x)

[Out]

2/45/b*(4*cos(b*x+a)^2-9)*cos(b*x+a)/(d*sin(b*x+a)/cos(b*x+a))^(3/2)/sin(b*x+a)^3

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Maxima [A]  time = 1.12163, size = 47, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (9 \, d^{2} \tan \left (b x + a\right )^{2} + 5 \, d^{2}\right )} d}{45 \, \left (d \tan \left (b x + a\right )\right )^{\frac{9}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4/(d*tan(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

-2/45*(9*d^2*tan(b*x + a)^2 + 5*d^2)*d/((d*tan(b*x + a))^(9/2)*b)

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Fricas [B]  time = 2.13236, size = 201, normalized size = 4.67 \begin{align*} \frac{2 \,{\left (4 \, \cos \left (b x + a\right )^{5} - 9 \, \cos \left (b x + a\right )^{3}\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{45 \,{\left (b d^{2} \cos \left (b x + a\right )^{4} - 2 \, 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)^4/(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

2/45*(4*cos(b*x + a)^5 - 9*cos(b*x + a)^3)*sqrt(d*sin(b*x + a)/cos(b*x + a))/((b*d^2*cos(b*x + a)^4 - 2*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)**4/(d*tan(b*x+a))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16506, size = 61, normalized size = 1.42 \begin{align*} -\frac{2 \,{\left (9 \, d^{4} \tan \left (b x + a\right )^{2} + 5 \, d^{4}\right )}}{45 \, \sqrt{d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4/(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

-2/45*(9*d^4*tan(b*x + a)^2 + 5*d^4)/(sqrt(d*tan(b*x + a))*b*d^5*tan(b*x + a)^4)