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]

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

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\sec ^2(c+d x)}{2 d} \]

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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fricas [A]  time = 0.39, size = 13, normalized size = 0.87 \[ \frac {1}{2 \, d \cos \left (d x + c\right )^{2}} \]

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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giac [B]  time = 0.23, size = 46, normalized size = 3.07 \[ -\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2} {\left (\cos \left (d x + c\right ) + 1\right )}} \]

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))

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maple [A]  time = 0.08, size = 14, normalized size = 0.93 \[ \frac {\sec ^{2}\left (d x +c \right )}{2 d} \]

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 = 0.35, size = 17, normalized size = 1.13 \[ -\frac {1}{2 \, {\left (\sin \left (d x + c\right )^{2} - 1\right )} d} \]

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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mupad [B]  time = 0.04, size = 13, normalized size = 0.87 \[ \frac {1}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]

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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