3.2.26 \(\int \csc (a+b x) \sec (a+b x) \, dx\) [126]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {2700, 29} \begin {gather*} \frac {\log (\tan (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Sec[a + b*x],x]

[Out]

Log[Tan[a + b*x]]/b

Rule 29

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

Rule 2700

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(31\) vs. \(2(11)=22\).
time = 0.02, size = 31, normalized size = 2.82 \begin {gather*} 2 \left (-\frac {\log (\cos (a+b x))}{2 b}+\frac {\log (\sin (a+b x))}{2 b}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-1/2*Log[Cos[a + b*x]]/b + Log[Sin[a + b*x]]/(2*b))

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Maple [A]
time = 0.05, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}\)	\(12\)
default	\(\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}\)	\(12\)
risch	\(\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}\)	\(35\)
norman	\(\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
time = 0.29, size = 28, normalized size = 2.55 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (11) = 22\).
time = 0.36, size = 30, normalized size = 2.73 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec {\left (a + b x \right )}}{\sin {\left (a + b x \right )}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (11) = 22\).
time = 4.11, size = 56, normalized size = 5.09 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.39, size = 11, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(a + b*x)*sin(a + b*x)),x)

[Out]

log(tan(a + b*x))/b

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