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

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

[Out]

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

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Rubi [A]  time = 0.0478271, 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} \[ \frac{2 d \sqrt{d \tan (a+b x)}}{b}-\frac{2 d^3}{3 b (d \tan (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0794664, size = 30, normalized size = 0.73 \[ -\frac{2 d \left (\csc ^2(a+b x)-4\right ) \sqrt{d \tan (a+b x)}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.155, size = 50, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-6 \right ) \cos \left ( bx+a \right ) }{3\,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]

-2/3/b*(4*cos(b*x+a)^2-3)*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 = 1.69516, size = 46, normalized size = 1.12 \begin{align*} -\frac{2 \, d^{3}{\left (\frac{1}{\left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}}} - \frac{3 \, \sqrt{d \tan \left (b x + a\right )}}{d^{2}}\right )}}{3 \, 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]

-2/3*d^3*(1/(d*tan(b*x + a))^(3/2) - 3*sqrt(d*tan(b*x + a))/d^2)/b

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Fricas [A]  time = 1.45203, size = 120, normalized size = 2.93 \begin{align*} \frac{2 \,{\left (4 \, d \cos \left (b x + a\right )^{2} - 3 \, d\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\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]

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

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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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} \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))^(3/2),x, algorithm="giac")

[Out]

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