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

Optimal. Leaf size=20 \[ \frac{2 d (d \tan (a+b x))^{3/2}}{3 b} \]

[Out]

(2*d*(d*Tan[a + b*x])^(3/2))/(3*b)

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Rubi [A]  time = 0.0420846, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2591, 30} \[ \frac{2 d (d \tan (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0751839, size = 20, normalized size = 1. \[ \frac{2 d (d \tan (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(d*Tan[a + b*x])^(3/2))/(3*b)

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Maple [B]  time = 0.123, size = 38, normalized size = 1.9 \begin{align*}{\frac{2\,\cos \left ( bx+a \right ) }{3\,b\sin \left ( bx+a \right ) } \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)^2*(d*tan(b*x+a))^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.39602, size = 31, normalized size = 1.55 \begin{align*} \frac{2 \, \left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}}}{3 \, b \tan \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(d*tan(b*x + a))^(5/2)/(b*tan(b*x + a))

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Fricas [B]  time = 2.14551, size = 99, normalized size = 4.95 \begin{align*} \frac{2 \, d^{2} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.11302, size = 32, normalized size = 1.6 \begin{align*} \frac{2 \, \sqrt{d \tan \left (b x + a\right )} d^{2} \tan \left (b x + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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