3.249 \(\int \sec ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d f} \]

[Out]

(2*(d*Tan[e + f*x])^(7/2))/(7*d*f)

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Rubi [A]  time = 0.0427842, antiderivative size = 22, 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 = {2607, 32} \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^2*(d*Tan[e + f*x])^(5/2),x]

[Out]

(2*(d*Tan[e + f*x])^(7/2))/(7*d*f)

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \sec ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int (d x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 (d \tan (e+f x))^{7/2}}{7 d f}\\ \end{align*}

Mathematica [A]  time = 0.0548, size = 22, normalized size = 1. \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^2*(d*Tan[e + f*x])^(5/2),x]

[Out]

(2*(d*Tan[e + f*x])^(7/2))/(7*d*f)

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Maple [A]  time = 0.02, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{7\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^2*(d*tan(f*x+e))^(5/2),x)

[Out]

2/7*(d*tan(f*x+e))^(7/2)/d/f

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Maxima [A]  time = 0.940318, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \, \left (d \tan \left (f x + e\right )\right )^{\frac{7}{2}}}{7 \, d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

2/7*(d*tan(f*x + e))^(7/2)/(d*f)

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Fricas [B]  time = 1.74948, size = 134, normalized size = 6.09 \begin{align*} -\frac{2 \,{\left (d^{2} \cos \left (f x + e\right )^{2} - d^{2}\right )} \sqrt{\frac{d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{7 \, f \cos \left (f x + e\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-2/7*(d^2*cos(f*x + e)^2 - d^2)*sqrt(d*sin(f*x + e)/cos(f*x + e))*sin(f*x + e)/(f*cos(f*x + e)^3)

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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(sec(f*x+e)**2*(d*tan(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.33562, size = 38, normalized size = 1.73 \begin{align*} \frac{2 \, \sqrt{d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )^{3}}{7 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

2/7*sqrt(d*tan(f*x + e))*d^2*tan(f*x + e)^3/f