\(\int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx\) [227]

Optimal result

Integrand size = 24, antiderivative size = 15 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sec ^2(c+d x)}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {267} \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sec ^2(c+d x)}{2 d} \]

[In]

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

[Out]

Sec[c + d*x]^2/(2*d)

Rule 267

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*} \text {integral}& = \frac {\text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\sec ^2(c+d x)}{2 d} \]

[In]

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

[Out]

Sec[c + d*x]^2/(2*d)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\sec \left (d x +c \right )^{2}}{2 d}\)	\(14\)
default	\(\frac {\sec \left (d x +c \right )^{2}}{2 d}\)	\(14\)
risch	\(\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}\)	\(28\)
norman	\(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\)	\(32\)
parallelrisch	\(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}\)	\(43\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {1}{2 \, d \cos \left (d x + c\right )^{2}} \]

[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)

Sympy [F]

\[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {1}{2 \, {\left (\sin \left (d x + c\right )^{2} - 1\right )} d} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (13) = 26\).

Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 3.07 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=-\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2} {\left (\cos \left (d x + c\right ) + 1\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {1}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]

[In]

int(-1/(cos(c + d*x)*(sin(c + d*x) - 1/sin(c + d*x))),x)

[Out]

1/(2*d*cos(c + d*x)^2)