3.200 \(\int \frac{\csc (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=38 \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]

[Out]

-(ArcTanh[Cos[c + d*x]]/(a*d)) + Cos[c + d*x]/(d*(a + a*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0537851, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2747, 3770, 2648} \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[Cos[c + d*x]]/(a*d)) + Cos[c + d*x]/(d*(a + a*Sin[c + d*x]))

Rule 2747

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[b/(
b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; Fre
eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\csc (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \csc (c+d x) \, dx}{a}-\int \frac{1}{a+a \sin (c+d x)} \, dx\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\cos (c+d x)}{d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0660189, size = 48, normalized size = 1.26 \[ -\frac{\sec (c+d x) \left (\sin (c+d x)+\sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\cos ^2(c+d x)}\right )-1\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sec[c + d*x]*(-1 + ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2] + Sin[c + d*x]))/(a*d))

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 40, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{da \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}+{\frac{1}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/a/d/(tan(1/2*d*x+1/2*c)+1)+1/a/d*ln(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [A]  time = 1.00464, size = 69, normalized size = 1.82 \begin{align*} \frac{\frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{2}{a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(sin(d*x + c)/(cos(d*x + c) + 1))/a + 2/(a + a*sin(d*x + c)/(cos(d*x + c) + 1)))/d

________________________________________________________________________________________

Fricas [B]  time = 1.60375, size = 293, normalized size = 7.71 \begin{align*} -\frac{{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) - 2}{2 \,{\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((cos(d*x + c) + sin(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) - (cos(d*x + c) + sin(d*x + c) + 1)*log(-1
/2*cos(d*x + c) + 1/2) - 2*cos(d*x + c) + 2*sin(d*x + c) - 2)/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(c + d*x)/(sin(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.18004, size = 51, normalized size = 1.34 \begin{align*} \frac{\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(abs(tan(1/2*d*x + 1/2*c)))/a + 2/(a*(tan(1/2*d*x + 1/2*c) + 1)))/d