3.227 \(\int \frac{\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\)

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

[Out]

Sec[c + d*x]^2/(2*d)

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Rubi [A]  time = 0.0317387, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {261} \[ \frac{\sec ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]

[Out]

Sec[c + d*x]^2/(2*d)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\sec ^2(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0117939, size = 15, normalized size = 1. \[ \frac{\sec ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]

[Out]

Sec[c + d*x]^2/(2*d)

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Maple [A]  time = 0.046, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x)

[Out]

1/2*sec(d*x+c)^2/d

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Maxima [A]  time = 1.09814, size = 23, normalized size = 1.53 \begin{align*} -\frac{1}{2 \,{\left (\sin \left (d x + c\right )^{2} - 1\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2/((sin(d*x + c)^2 - 1)*d)

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Fricas [A]  time = 0.480102, size = 32, normalized size = 2.13 \begin{align*} \frac{1}{2 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2/(d*cos(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x)

[Out]

Integral(sec(c + d*x)/(-sin(c + d*x) + csc(c + d*x)), x)

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Giac [B]  time = 1.14365, size = 62, normalized size = 4.13 \begin{align*} -\frac{2 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(csc(d*x+c)-sin(d*x+c)),x, algorithm="giac")

[Out]

-2*(cos(d*x + c) - 1)/(d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2*(cos(d*x + c) + 1))