3.126 \(\int \csc (a+b x) \sec (a+b x) \, dx\)

Optimal. Leaf size=11 \[ \frac{\log (\tan (a+b x))}{b} \]

[Out]

Log[Tan[a + b*x]]/b

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Rubi [A]  time = 0.0099491, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2620, 29} \[ \frac{\log (\tan (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Sec[a + b*x],x]

[Out]

Log[Tan[a + b*x]]/b

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \csc (a+b x) \sec (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\log (\tan (a+b x))}{b}\\ \end{align*}

Mathematica [B]  time = 0.0214335, size = 31, normalized size = 2.82 \[ 2 \left (\frac{\log (\sin (a+b x))}{2 b}-\frac{\log (\cos (a+b x))}{2 b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]*Sec[a + b*x],x]

[Out]

2*(-Log[Cos[a + b*x]]/(2*b) + Log[Sin[a + b*x]]/(2*b))

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Maple [A]  time = 0.017, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)/sin(b*x+a),x)

[Out]

ln(tan(b*x+a))/b

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Maxima [B]  time = 0.977127, size = 38, normalized size = 3.45 \begin{align*} -\frac{\log \left (\sin \left (b x + a\right )^{2} - 1\right ) - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b

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Fricas [B]  time = 1.89355, size = 85, normalized size = 7.73 \begin{align*} -\frac{\log \left (\cos \left (b x + a\right )^{2}\right ) - \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(cos(b*x + a)^2) - log(-1/4*cos(b*x + a)^2 + 1/4))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1987, size = 76, normalized size = 6.91 \begin{align*} \frac{\log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) - 2*log(abs(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)
))/b