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]

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

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 1.00 \[ \frac {2 (d \tan (e+f x))^{7/2}}{7 d f} \]

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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fricas [B]  time = 0.47, size = 55, normalized size = 2.50 \[ -\frac {2 \, {\left (d^{2} \cos \left (f x + e\right )^{2} - d^{2}\right )} \sqrt {\frac {d \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{7 \, f \cos \left (f x + e\right )^{3}} \]

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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giac [A]  time = 0.74, size = 28, normalized size = 1.27 \[ \frac {2 \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )^{3}}{7 \, f} \]

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

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maple [A]  time = 0.12, size = 19, normalized size = 0.86 \[ \frac {2 \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7 d f} \]

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 = 0.33, size = 18, normalized size = 0.82 \[ \frac {2 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}}}{7 \, d f} \]

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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mupad [B]  time = 5.58, size = 230, normalized size = 10.45 \[ \frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,2{}\mathrm {i}}{7\,f}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,12{}\mathrm {i}}{7\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}+\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,24{}\mathrm {i}}{7\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {d^2\,\sqrt {-\frac {d\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,16{}\mathrm {i}}{7\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^2*(-(d*(exp(e*2i + f*x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*2i)/(7*f) - (d^2*(-(d*(exp(e*2i + f*x*
2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*12i)/(7*f*(exp(e*2i + f*x*2i) + 1)) + (d^2*(-(d*(exp(e*2i + f*x*
2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*24i)/(7*f*(exp(e*2i + f*x*2i) + 1)^2) - (d^2*(-(d*(exp(e*2i + f*
x*2i)*1i - 1i))/(exp(e*2i + f*x*2i) + 1))^(1/2)*16i)/(7*f*(exp(e*2i + f*x*2i) + 1)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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