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

Optimal. Leaf size=65 \[ \frac{2 (d \tan (a+b x))^{7/2}}{7 b d^5}+\frac{4 (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac{2}{b d \sqrt{d \tan (a+b x)}} \]

[Out]

-2/(b*d*Sqrt[d*Tan[a + b*x]]) + (4*(d*Tan[a + b*x])^(3/2))/(3*b*d^3) + (2*(d*Tan[a + b*x])^(7/2))/(7*b*d^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0598842, antiderivative size = 65, 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} \[ \frac{2 (d \tan (a+b x))^{7/2}}{7 b d^5}+\frac{4 (d \tan (a+b x))^{3/2}}{3 b d^3}-\frac{2}{b d \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(b*d*Sqrt[d*Tan[a + b*x]]) + (4*(d*Tan[a + b*x])^(3/2))/(3*b*d^3) + (2*(d*Tan[a + b*x])^(7/2))/(7*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 \frac{\sec ^6(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{(d x)^{3/2}} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{(d x)^{3/2}}+\frac{2 \sqrt{d x}}{d^2}+\frac{(d x)^{5/2}}{d^4}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{2}{b d \sqrt{d \tan (a+b x)}}+\frac{4 (d \tan (a+b x))^{3/2}}{3 b d^3}+\frac{2 (d \tan (a+b x))^{7/2}}{7 b d^5}\\ \end{align*}

Mathematica [A]  time = 0.191811, size = 45, normalized size = 0.69 \[ \frac{\tan ^2(a+b x) \left (6 \sec ^2(a+b x)+22\right )-42}{21 b d \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-42 + (22 + 6*Sec[a + b*x]^2)*Tan[a + b*x]^2)/(21*b*d*Sqrt[d*Tan[a + b*x]])

________________________________________________________________________________________

Maple [A]  time = 0.163, size = 60, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-16\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-6 \right ) \sin \left ( bx+a \right ) }{21\,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]

-2/21/b*(32*cos(b*x+a)^4-8*cos(b*x+a)^2-3)*sin(b*x+a)/cos(b*x+a)^5/(d*sin(b*x+a)/cos(b*x+a))^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.963428, size = 73, normalized size = 1.12 \begin{align*} -\frac{2 \,{\left (\frac{21}{\sqrt{d \tan \left (b x + a\right )}} - \frac{3 \, \left (d \tan \left (b x + a\right )\right )^{\frac{7}{2}} + 14 \, \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} d^{2}}{d^{4}}\right )}}{21 \, b d} \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]

-2/21*(21/sqrt(d*tan(b*x + a)) - (3*(d*tan(b*x + a))^(7/2) + 14*(d*tan(b*x + a))^(3/2)*d^2)/d^4)/(b*d)

________________________________________________________________________________________

Fricas [A]  time = 1.87551, size = 162, normalized size = 2.49 \begin{align*} -\frac{2 \,{\left (32 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{21 \, b d^{2} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )} \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]

-2/21*(32*cos(b*x + a)^4 - 8*cos(b*x + a)^2 - 3)*sqrt(d*sin(b*x + a)/cos(b*x + a))/(b*d^2*cos(b*x + a)^3*sin(b
*x + a))

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [A]  time = 1.30494, size = 92, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )^{3} + 14 \, \sqrt{d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right ) - \frac{21 \, d^{4}}{\sqrt{d \tan \left (b x + a\right )}}\right )}}{21 \, 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="giac")

[Out]

2/21*(3*sqrt(d*tan(b*x + a))*d^3*tan(b*x + a)^3 + 14*sqrt(d*tan(b*x + a))*d^3*tan(b*x + a) - 21*d^4/sqrt(d*tan
(b*x + a)))/(b*d^5)