\(\int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx\) [220]

Optimal result

Integrand size = 22, antiderivative size = 16 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\text {arctanh}\left (\sin ^2(c+d x)\right )}{2 d} \]

[Out]

1/2*arctanh(sin(d*x+c)^2)/d

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {281, 212} \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\text {arctanh}\left (\sin ^2(c+d x)\right )}{2 d} \]

[In]

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

[Out]

ArcTanh[Sin[c + d*x]^2]/(2*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{1-x^4} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {arctanh}\left (\sin ^2(c+d x)\right )}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {-2 \log (\cos (c+d x))+\log \left (2-\cos ^2(c+d x)\right )}{4 d} \]

[In]

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

[Out]

(-2*Log[Cos[c + d*x]] + Log[2 - Cos[c + d*x]^2])/(4*d)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {\ln \left (2 \sec \left (d x +c \right )^{2}-1\right )}{4 d}\)	\(19\)
default	\(\frac {\ln \left (2 \sec \left (d x +c \right )^{2}-1\right )}{4 d}\)	\(19\)
risch	\(-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{4 d}\)	\(47\)
parallelrisch	\(\frac {-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\ln \left (-4+4 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}\right )}{4 d}\)	\(62\)
norman	\(-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )}{4 d}\)	\(68\)

[In]

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

[Out]

1/4/d*ln(2*sec(d*x+c)^2-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\log \left (-\cos \left (d x + c\right )^{2} + 2\right ) - 2 \, \log \left (-\cos \left (d x + c\right )\right )}{4 \, d} \]

[In]

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

[Out]

1/4*(log(-cos(d*x + c)^2 + 2) - 2*log(-cos(d*x + c)))/d

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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).

Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\log \left (\sin \left (d x + c\right )^{2} + 1\right ) - \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{4 \, d} \]

[In]

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

[Out]

1/4*(log(sin(d*x + c)^2 + 1) - log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (14) = 28\).

Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 4.94 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=-\frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \log \left ({\left | -\frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1 \right |}\right )}{4 \, d} \]

[In]

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

[Out]

-1/4*(2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - log(abs(-6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1
) + (cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1)))/d

Mupad [B] (verification not implemented)

Time = 23.74 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\mathrm {atanh}\left ({\sin \left (c+d\,x\right )}^2\right )}{2\,d} \]

[In]

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

[Out]

atanh(sin(c + d*x)^2)/(2*d)