3.106 \(\int \frac{\csc ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]

[Out]

(-2*d^3)/(11*b*(d*Tan[a + b*x])^(11/2)) - (2*d)/(7*b*(d*Tan[a + b*x])^(7/2))

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Rubi [A]  time = 0.0496892, 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} \[ -\frac{2 d^3}{11 b (d \tan (a+b x))^{11/2}}-\frac{2 d}{7 b (d \tan (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.172509, size = 50, normalized size = 1.16 \[ \frac{2 (2 \cos (2 (a+b x))-9) \cot ^4(a+b x) \csc ^2(a+b x) \sqrt{d \tan (a+b x)}}{77 b d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]

[Out]

(2*(-9 + 2*Cos[2*(a + b*x)])*Cot[a + b*x]^4*Csc[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(77*b*d^3)

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Maple [A]  time = 0.148, size = 50, normalized size = 1.2 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-22 \right ) \cos \left ( bx+a \right ) }{77\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}} \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)^4/(d*tan(b*x+a))^(5/2),x)

[Out]

2/77/b*(4*cos(b*x+a)^2-11)*cos(b*x+a)/sin(b*x+a)^3/(d*sin(b*x+a)/cos(b*x+a))^(5/2)

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Maxima [A]  time = 1.12031, size = 47, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (11 \, d^{2} \tan \left (b x + a\right )^{2} + 7 \, d^{2}\right )} d}{77 \, \left (d \tan \left (b x + a\right )\right )^{\frac{11}{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))^(5/2),x, algorithm="maxima")

[Out]

-2/77*(11*d^2*tan(b*x + a)^2 + 7*d^2)*d/((d*tan(b*x + a))^(11/2)*b)

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Fricas [B]  time = 2.80336, size = 217, normalized size = 5.05 \begin{align*} -\frac{2 \,{\left (4 \, \cos \left (b x + a\right )^{6} - 11 \, \cos \left (b x + a\right )^{4}\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{77 \,{\left (b d^{3} \cos \left (b x + a\right )^{6} - 3 \, b d^{3} \cos \left (b x + a\right )^{4} + 3 \, 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)^4/(d*tan(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

-2/77*(4*cos(b*x + a)^6 - 11*cos(b*x + a)^4)*sqrt(d*sin(b*x + a)/cos(b*x + a))/(b*d^3*cos(b*x + a)^6 - 3*b*d^3
*cos(b*x + a)^4 + 3*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)**4/(d*tan(b*x+a))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.14505, size = 61, normalized size = 1.42 \begin{align*} -\frac{2 \,{\left (11 \, d^{3} \tan \left (b x + a\right )^{2} + 7 \, d^{3}\right )}}{77 \, \sqrt{d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="giac")

[Out]

-2/77*(11*d^3*tan(b*x + a)^2 + 7*d^3)/(sqrt(d*tan(b*x + a))*b*d^5*tan(b*x + a)^5)