3.47 \(\int \sec ^2(a+b x) \tan (a+b x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\sec ^2(a+b x)}{2 b} \]

[Out]

Sec[a + b*x]^2/(2*b)

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Rubi [A]  time = 0.0194794, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2606, 30} \[ \frac{\sec ^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^2*Tan[a + b*x],x]

[Out]

Sec[a + b*x]^2/(2*b)

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.010365, size = 15, normalized size = 1. \[ \frac{\sec ^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^2*Tan[a + b*x],x]

[Out]

Sec[a + b*x]^2/(2*b)

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Maple [A]  time = 0.008, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sec \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^3*sin(b*x+a),x)

[Out]

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

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Maxima [A]  time = 0.989576, size = 23, normalized size = 1.53 \begin{align*} -\frac{1}{2 \,{\left (\sin \left (b x + a\right )^{2} - 1\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^3*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/2/((sin(b*x + a)^2 - 1)*b)

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Fricas [A]  time = 1.67507, size = 32, normalized size = 2.13 \begin{align*} \frac{1}{2 \, b \cos \left (b x + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^3*sin(b*x+a),x, algorithm="fricas")

[Out]

1/2/(b*cos(b*x + a)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**3*sin(b*x+a),x)

[Out]

Integral(sin(a + b*x)*sec(a + b*x)**3, x)

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Giac [A]  time = 1.15784, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2 \, b \cos \left (b x + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^3*sin(b*x+a),x, algorithm="giac")

[Out]

1/2/(b*cos(b*x + a)^2)