3.57 \(\int \csc ^4(a+b x) \sqrt{d \tan (a+b x)} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0447967, antiderivative size = 41, 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} \[ -\frac{2 d^3}{5 b (d \tan (a+b x))^{5/2}}-\frac{2 d}{b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^4*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*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 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 \csc ^4(a+b x) \sqrt{d \tan (a+b x)} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{d^2+x^2}{x^{7/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{d \operatorname{Subst}\left (\int \left (\frac{d^2}{x^{7/2}}+\frac{1}{x^{3/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 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 = 0.116051, size = 30, normalized size = 0.73 \[ -\frac{2 d \left (\csc ^2(a+b x)+4\right )}{5 b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^4*Sqrt[d*Tan[a + b*x]],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.181, size = 50, normalized size = 1.2 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-10 \right ) \cos \left ( bx+a \right ) }{5\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}}\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)^4*(d*tan(b*x+a))^(1/2),x)

[Out]

2/5/b*(4*cos(b*x+a)^2-5)*cos(b*x+a)*(d*sin(b*x+a)/cos(b*x+a))^(1/2)/sin(b*x+a)^3

________________________________________________________________________________________

Maxima [A]  time = 2.99058, size = 45, normalized size = 1.1 \begin{align*} -\frac{2 \,{\left (5 \, d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )} d}{5 \, \left (d \tan \left (b x + a\right )\right )^{\frac{5}{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))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.68853, size = 154, normalized size = 3.76 \begin{align*} -\frac{2 \,{\left (4 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{5 \,{\left (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)^4*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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 )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^4*(d*tan(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a)^4, x)