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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0482106, 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} \[ \frac{2 d (d \tan (a+b x))^{3/2}}{3 b}-\frac{2 d^3}{b \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.112681, size = 32, normalized size = 0.78 \[ -\frac{2 d \left (3 \cot ^2(a+b x)-1\right ) (d \tan (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*(-1 + 3*Cot[a + b*x]^2)*(d*Tan[a + b*x])^(3/2))/(3*b)

________________________________________________________________________________________

Maple [A]  time = 0.124, size = 50, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-2 \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{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]

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

________________________________________________________________________________________

Maxima [A]  time = 1.32078, size = 49, normalized size = 1.2 \begin{align*} -\frac{2 \, d^{3}{\left (\frac{3}{\sqrt{d \tan \left (b x + a\right )}} - \frac{\left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}}}{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))^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.20697, size = 134, normalized size = 3.27 \begin{align*} -\frac{2 \,{\left (4 \, d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\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))^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{5}{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))^(5/2),x, algorithm="giac")

[Out]

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