3.58 \(\int \csc ^6(a+b x) \sqrt{d \tan (a+b x)} \, dx\)

Optimal. Leaf size=63 \[ -\frac{2 d^5}{9 b (d \tan (a+b x))^{9/2}}-\frac{4 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0515714, antiderivative size = 63, 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} \[ -\frac{2 d^5}{9 b (d \tan (a+b x))^{9/2}}-\frac{4 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]

[Out]

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

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

Mathematica [A]  time = 0.158643, size = 50, normalized size = 0.79 \[ \frac{2 d (20 \cos (2 (a+b x))-4 \cos (4 (a+b x))-21) \csc ^4(a+b x)}{45 b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^6*Sqrt[d*Tan[a + b*x]],x]

[Out]

(2*d*(-21 + 20*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4)/(45*b*Sqrt[d*Tan[a + b*x]])

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Maple [A]  time = 0.187, size = 60, normalized size = 1. \begin{align*} -{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-144\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+90 \right ) \cos \left ( bx+a \right ) }{45\,b \left ( \sin \left ( bx+a \right ) \right ) ^{5}}\sqrt{{\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x)

[Out]

-2/45/b*(32*cos(b*x+a)^4-72*cos(b*x+a)^2+45)*cos(b*x+a)*(d*sin(b*x+a)/cos(b*x+a))^(1/2)/sin(b*x+a)^5

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Maxima [A]  time = 1.21788, size = 65, normalized size = 1.03 \begin{align*} -\frac{2 \,{\left (45 \, d^{4} \tan \left (b x + a\right )^{4} + 18 \, d^{4} \tan \left (b x + a\right )^{2} + 5 \, d^{4}\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)^6*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

-2/45*(45*d^4*tan(b*x + a)^4 + 18*d^4*tan(b*x + a)^2 + 5*d^4)*d/((d*tan(b*x + a))^(9/2)*b)

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Fricas [A]  time = 1.79873, size = 213, normalized size = 3.38 \begin{align*} -\frac{2 \,{\left (32 \, \cos \left (b x + a\right )^{5} - 72 \, \cos \left (b x + a\right )^{3} + 45 \, \cos \left (b x + a\right )\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{45 \,{\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\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))^(1/2),x, algorithm="fricas")

[Out]

-2/45*(32*cos(b*x + a)^5 - 72*cos(b*x + a)^3 + 45*cos(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a))/((b*cos(b*x
+ a)^4 - 2*b*cos(b*x + a)^2 + b)*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))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^6*(d*tan(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a)^6, x)