3.4 \(\int \sec ^4(a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{\tan (a+b x)}{b} \]

[Out]

Tan[a + b*x]/b + Tan[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0112607, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{\tan (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^4,x]

[Out]

Tan[a + b*x]/b + Tan[a + b*x]^3/(3*b)

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \sec ^4(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (a+b x)\right )}{b}\\ &=\frac{\tan (a+b x)}{b}+\frac{\tan ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0409817, size = 23, normalized size = 0.88 \[ \frac{\frac{1}{3} \tan ^3(a+b x)+\tan (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^4,x]

[Out]

(Tan[a + b*x] + Tan[a + b*x]^3/3)/b

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Maple [A]  time = 0.043, size = 24, normalized size = 0.9 \begin{align*} -{\frac{\tan \left ( bx+a \right ) }{b} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^4,x)

[Out]

-1/b*(-2/3-1/3*sec(b*x+a)^2)*tan(b*x+a)

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Maxima [A]  time = 1.16608, size = 30, normalized size = 1.15 \begin{align*} \frac{\tan \left (b x + a\right )^{3} + 3 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^4,x, algorithm="maxima")

[Out]

1/3*(tan(b*x + a)^3 + 3*tan(b*x + a))/b

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Fricas [A]  time = 1.34596, size = 81, normalized size = 3.12 \begin{align*} \frac{{\left (2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*(2*cos(b*x + a)^2 + 1)*sin(b*x + a)/(b*cos(b*x + a)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec ^{4}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**4,x)

[Out]

Integral(sec(a + b*x)**4, x)

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Giac [A]  time = 1.32413, size = 30, normalized size = 1.15 \begin{align*} \frac{\tan \left (b x + a\right )^{3} + 3 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^4,x, algorithm="giac")

[Out]

1/3*(tan(b*x + a)^3 + 3*tan(b*x + a))/b