3.226 \(\int \sec ^6(e+f x) \sqrt{d \tan (e+f x)} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 (d \tan (e+f x))^{11/2}}{11 d^5 f}+\frac{4 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{3/2}}{3 d f} \]

[Out]

(2*(d*Tan[e + f*x])^(3/2))/(3*d*f) + (4*(d*Tan[e + f*x])^(7/2))/(7*d^3*f) + (2*(d*Tan[e + f*x])^(11/2))/(11*d^
5*f)

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Rubi [A]  time = 0.0538728, antiderivative size = 67, 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 = {2607, 270} \[ \frac{2 (d \tan (e+f x))^{11/2}}{11 d^5 f}+\frac{4 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{3/2}}{3 d f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^6*Sqrt[d*Tan[e + f*x]],x]

[Out]

(2*(d*Tan[e + f*x])^(3/2))/(3*d*f) + (4*(d*Tan[e + f*x])^(7/2))/(7*d^3*f) + (2*(d*Tan[e + f*x])^(11/2))/(11*d^
5*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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sec ^6(e+f x) \sqrt{d \tan (e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{d x} \left (1+x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\sqrt{d x}+\frac{2 (d x)^{5/2}}{d^2}+\frac{(d x)^{9/2}}{d^4}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{4 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{11/2}}{11 d^5 f}\\ \end{align*}

Mathematica [A]  time = 0.187972, size = 52, normalized size = 0.78 \[ \frac{2 (28 \cos (2 (e+f x))+4 \cos (4 (e+f x))+45) \sec ^4(e+f x) (d \tan (e+f x))^{3/2}}{231 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^6*Sqrt[d*Tan[e + f*x]],x]

[Out]

(2*(45 + 28*Cos[2*(e + f*x)] + 4*Cos[4*(e + f*x)])*Sec[e + f*x]^4*(d*Tan[e + f*x])^(3/2))/(231*d*f)

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Maple [A]  time = 0.256, size = 60, normalized size = 0.9 \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+48\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+42 \right ) \sin \left ( fx+e \right ) }{231\,f \left ( \cos \left ( fx+e \right ) \right ) ^{5}}\sqrt{{\frac{d\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^6*(d*tan(f*x+e))^(1/2),x)

[Out]

2/231/f*(32*cos(f*x+e)^4+24*cos(f*x+e)^2+21)*(d*sin(f*x+e)/cos(f*x+e))^(1/2)*sin(f*x+e)/cos(f*x+e)^5

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Maxima [A]  time = 1.11092, size = 69, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (21 \, \left (d \tan \left (f x + e\right )\right )^{\frac{11}{2}} + 66 \, \left (d \tan \left (f x + e\right )\right )^{\frac{7}{2}} d^{2} + 77 \, \left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}} d^{4}\right )}}{231 \, d^{5} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^6*(d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

2/231*(21*(d*tan(f*x + e))^(11/2) + 66*(d*tan(f*x + e))^(7/2)*d^2 + 77*(d*tan(f*x + e))^(3/2)*d^4)/(d^5*f)

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Fricas [A]  time = 1.86404, size = 159, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (32 \, \cos \left (f x + e\right )^{4} + 24 \, \cos \left (f x + e\right )^{2} + 21\right )} \sqrt{\frac{d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{231 \, f \cos \left (f x + e\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^6*(d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/231*(32*cos(f*x + e)^4 + 24*cos(f*x + e)^2 + 21)*sqrt(d*sin(f*x + e)/cos(f*x + e))*sin(f*x + e)/(f*cos(f*x +
e)^5)

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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)**6*(d*tan(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.29671, size = 111, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (21 \, \sqrt{d \tan \left (f x + e\right )} d^{5} \tan \left (f x + e\right )^{5} + 66 \, \sqrt{d \tan \left (f x + e\right )} d^{5} \tan \left (f x + e\right )^{3} + 77 \, \sqrt{d \tan \left (f x + e\right )} d^{5} \tan \left (f x + e\right )\right )}}{231 \, d^{5} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^6*(d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

2/231*(21*sqrt(d*tan(f*x + e))*d^5*tan(f*x + e)^5 + 66*sqrt(d*tan(f*x + e))*d^5*tan(f*x + e)^3 + 77*sqrt(d*tan
(f*x + e))*d^5*tan(f*x + e))/(d^5*f)