3.213 \(\int \frac{\csc (c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=109 \[ \frac{b^2}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2} \]

[Out]

b^2/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(2*(a + b)^2*d) - Log[1 + Cos[c + d*x]]/(2*
(a - b)^2*d) + (2*a*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^2*d)

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Rubi [A]  time = 0.226432, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2837, 12, 1629} \[ \frac{b^2}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]/(a + b*Sec[c + d*x])^2,x]

[Out]

b^2/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(2*(a + b)^2*d) - Log[1 + Cos[c + d*x]]/(2*
(a - b)^2*d) + (2*a*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^2*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2837

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.281333, size = 165, normalized size = 1.51 \[ \frac{b \left (2 a^2 b \log (a \cos (c+d x)+b)+(a-b) \left (a (a-b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+b (a+b)\right )-a (a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-a^2 \cos (c+d x) \left ((a-b)^2 \left (-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )+(a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2 a b \log (a \cos (c+d x)+b)\right )}{a d (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]/(a + b*Sec[c + d*x])^2,x]

[Out]

(-(a^2*Cos[c + d*x]*((a + b)^2*Log[Cos[(c + d*x)/2]] - 2*a*b*Log[b + a*Cos[c + d*x]] - (a - b)^2*Log[Sin[(c +
d*x)/2]])) + b*(-(a*(a + b)^2*Log[Cos[(c + d*x)/2]]) + 2*a^2*b*Log[b + a*Cos[c + d*x]] + (a - b)*(b*(a + b) +
a*(a - b)*Log[Sin[(c + d*x)/2]])))/(a*(a - b)^2*(a + b)^2*d*(b + a*Cos[c + d*x]))

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Maple [A]  time = 0.067, size = 106, normalized size = 1. \begin{align*}{\frac{{b}^{2}}{d \left ( a+b \right ) \left ( a-b \right ) a \left ( b+a\cos \left ( dx+c \right ) \right ) }}+2\,{\frac{ab\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{2\, \left ( a-b \right ) ^{2}d}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)/(a+b*sec(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.0816, size = 166, normalized size = 1.52 \begin{align*} \frac{\frac{4 \, a b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \, b^{2}}{a^{3} b - a b^{3} +{\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )} - \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

1/2*(4*a*b*log(a*cos(d*x + c) + b)/(a^4 - 2*a^2*b^2 + b^4) + 2*b^2/(a^3*b - a*b^3 + (a^4 - a^2*b^2)*cos(d*x +
c)) - log(cos(d*x + c) + 1)/(a^2 - 2*a*b + b^2) + log(cos(d*x + c) - 1)/(a^2 + 2*a*b + b^2))/d

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Fricas [A]  time = 2.19723, size = 486, normalized size = 4.46 \begin{align*} \frac{2 \, a^{2} b^{2} - 2 \, b^{4} + 4 \,{\left (a^{3} b \cos \left (d x + c\right ) + a^{2} b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) -{\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3} +{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3} +{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d \cos \left (d x + c\right ) +{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/2*(2*a^2*b^2 - 2*b^4 + 4*(a^3*b*cos(d*x + c) + a^2*b^2)*log(a*cos(d*x + c) + b) - (a^3*b + 2*a^2*b^2 + a*b^3
+ (a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^3*b - 2*a^2*b^2 + a*b^3 + (a^4 - 2
*a^3*b + a^2*b^2)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^6 - 2*a^4*b^2 + a^2*b^4)*d*cos(d*x + c) + (a
^5*b - 2*a^3*b^3 + a*b^5)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sec(d*x+c))**2,x)

[Out]

Integral(csc(c + d*x)/(a + b*sec(c + d*x))**2, x)

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Giac [B]  time = 1.32882, size = 288, normalized size = 2.64 \begin{align*} \frac{\frac{4 \, a b \log \left ({\left | -a - b - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{4 \,{\left (a b + b^{2} + \frac{a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )}{\left (a + b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(4*a*b*log(abs(-a - b - a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))
)/(a^4 - 2*a^2*b^2 + b^4) + log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^2 + 2*a*b + b^2) - 4*(a*b + b
^2 + a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a^3 + a^2*b - a*b^2 - b^3)*(a + b + a*(cos(d*x + c) - 1)/(co
s(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))))/d