3.227 \(\int \sec ^4(e+f x) \sqrt{d \tan (e+f x)} \, dx\)

Optimal. Leaf size=45 \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{3/2}}{3 d f} \]

[Out]

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

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Rubi [A]  time = 0.0449009, antiderivative size = 45, 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, 14} \[ \frac{2 (d \tan (e+f x))^{7/2}}{7 d^3 f}+\frac{2 (d \tan (e+f x))^{3/2}}{3 d f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^4*Sqrt[d*Tan[e + f*x]],x]

[Out]

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

Mathematica [A]  time = 0.141476, size = 34, normalized size = 0.76 \[ \frac{2 \left (3 \sec ^2(e+f x)+4\right ) (d \tan (e+f x))^{3/2}}{21 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^4*Sqrt[d*Tan[e + f*x]],x]

[Out]

(2*(4 + 3*Sec[e + f*x]^2)*(d*Tan[e + f*x])^(3/2))/(21*d*f)

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Maple [A]  time = 0.163, size = 50, normalized size = 1.1 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+6 \right ) \sin \left ( fx+e \right ) }{21\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\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)^4*(d*tan(f*x+e))^(1/2),x)

[Out]

2/21/f*(4*cos(f*x+e)^2+3)*(d*sin(f*x+e)/cos(f*x+e))^(1/2)*sin(f*x+e)/cos(f*x+e)^3

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Maxima [A]  time = 0.973216, size = 49, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (3 \, \left (d \tan \left (f x + e\right )\right )^{\frac{7}{2}} + 7 \, \left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}} d^{2}\right )}}{21 \, d^{3} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^4*(d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

2/21*(3*(d*tan(f*x + e))^(7/2) + 7*(d*tan(f*x + e))^(3/2)*d^2)/(d^3*f)

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Fricas [A]  time = 1.71276, size = 128, normalized size = 2.84 \begin{align*} \frac{2 \,{\left (4 \, \cos \left (f x + e\right )^{2} + 3\right )} \sqrt{\frac{d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{21 \, 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)^4*(d*tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/21*(4*cos(f*x + e)^2 + 3)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \tan{\left (e + f x \right )}} \sec ^{4}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**4*(d*tan(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(d*tan(e + f*x))*sec(e + f*x)**4, x)

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Giac [A]  time = 1.14318, size = 77, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right )^{3} + 7 \, \sqrt{d \tan \left (f x + e\right )} d^{3} \tan \left (f x + e\right )\right )}}{21 \, d^{3} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^4*(d*tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

2/21*(3*sqrt(d*tan(f*x + e))*d^3*tan(f*x + e)^3 + 7*sqrt(d*tan(f*x + e))*d^3*tan(f*x + e))/(d^3*f)