3.236 \(\int \sec ^6(a+b x) (d \tan (a+b x))^{3/2} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 (d \tan (a+b x))^{13/2}}{13 b d^5}+\frac{4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac{2 (d \tan (a+b x))^{5/2}}{5 b d} \]

[Out]

(2*(d*Tan[a + b*x])^(5/2))/(5*b*d) + (4*(d*Tan[a + b*x])^(9/2))/(9*b*d^3) + (2*(d*Tan[a + b*x])^(13/2))/(13*b*
d^5)

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Rubi [A]  time = 0.0579711, 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} \[ \frac{2 (d \tan (a+b x))^{13/2}}{13 b d^5}+\frac{4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac{2 (d \tan (a+b x))^{5/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^6*(d*Tan[a + b*x])^(3/2),x]

[Out]

(2*(d*Tan[a + b*x])^(5/2))/(5*b*d) + (4*(d*Tan[a + b*x])^(9/2))/(9*b*d^3) + (2*(d*Tan[a + b*x])^(13/2))/(13*b*
d^5)

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(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (d x)^{3/2} \left (1+x^2\right )^2 \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left ((d x)^{3/2}+\frac{2 (d x)^{7/2}}{d^2}+\frac{(d x)^{11/2}}{d^4}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{2 (d \tan (a+b x))^{5/2}}{5 b d}+\frac{4 (d \tan (a+b x))^{9/2}}{9 b d^3}+\frac{2 (d \tan (a+b x))^{13/2}}{13 b d^5}\\ \end{align*}

Mathematica [A]  time = 0.137308, size = 52, normalized size = 0.78 \[ \frac{2 d \left (45 \sec ^6(a+b x)-5 \sec ^4(a+b x)-8 \sec ^2(a+b x)-32\right ) \sqrt{d \tan (a+b x)}}{585 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^6*(d*Tan[a + b*x])^(3/2),x]

[Out]

(2*d*(-32 - 8*Sec[a + b*x]^2 - 5*Sec[a + b*x]^4 + 45*Sec[a + b*x]^6)*Sqrt[d*Tan[a + b*x]])/(585*b)

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Maple [A]  time = 0.189, size = 60, normalized size = 0.9 \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+80\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+90 \right ) \sin \left ( bx+a \right ) }{585\,b \left ( \cos \left ( bx+a \right ) \right ) ^{5}} \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^6*(d*tan(b*x+a))^(3/2),x)

[Out]

2/585/b*(32*cos(b*x+a)^4+40*cos(b*x+a)^2+45)*(d*sin(b*x+a)/cos(b*x+a))^(3/2)*sin(b*x+a)/cos(b*x+a)^5

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Maxima [A]  time = 0.937855, size = 69, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (45 \, \left (d \tan \left (b x + a\right )\right )^{\frac{13}{2}} + 130 \, \left (d \tan \left (b x + a\right )\right )^{\frac{9}{2}} d^{2} + 117 \, \left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}} d^{4}\right )}}{585 \, b d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^6*(d*tan(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

2/585*(45*(d*tan(b*x + a))^(13/2) + 130*(d*tan(b*x + a))^(9/2)*d^2 + 117*(d*tan(b*x + a))^(5/2)*d^4)/(b*d^5)

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Fricas [A]  time = 2.17808, size = 178, normalized size = 2.66 \begin{align*} -\frac{2 \,{\left (32 \, d \cos \left (b x + a\right )^{6} + 8 \, d \cos \left (b x + a\right )^{4} + 5 \, d \cos \left (b x + a\right )^{2} - 45 \, d\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{585 \, b \cos \left (b x + a\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^6*(d*tan(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

-2/585*(32*d*cos(b*x + a)^6 + 8*d*cos(b*x + a)^4 + 5*d*cos(b*x + a)^2 - 45*d)*sqrt(d*sin(b*x + a)/cos(b*x + a)
)/(b*cos(b*x + a)^6)

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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(b*x+a)**6*(d*tan(b*x+a))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} \sec \left (b x + a\right )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^6*(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((d*tan(b*x + a))^(3/2)*sec(b*x + a)^6, x)