3.177 \(\int \csc (c+d x) (a+b \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=74 \[ -\frac{(a-b)^2 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^2 \log (1-\cos (c+d x))}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.180278, antiderivative size = 74, 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, 1802} \[ -\frac{(a-b)^2 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^2 \log (1-\cos (c+d x))}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)^2*Log[1 - Cos[c + d*x]])/(2*d) - (2*a*b*Log[Cos[c + d*x]])/d - ((a - b)^2*Log[1 + Cos[c + d*x]])/(2*d
) + (b^2*Sec[c + d*x])/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 1802

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

Rubi steps

\begin{align*} \int \csc (c+d x) (a+b \sec (c+d x))^2 \, dx &=\int (-b-a \cos (c+d x))^2 \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{a^2 (-b+x)^2}{x^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{(-b+x)^2}{x^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{(a-b)^2}{2 a^3 (a-x)}+\frac{b^2}{a^2 x^2}-\frac{2 b}{a^2 x}+\frac{(a+b)^2}{2 a^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{(a+b)^2 \log (1-\cos (c+d x))}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}-\frac{(a-b)^2 \log (1+\cos (c+d x))}{2 d}+\frac{b^2 \sec (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.151346, size = 91, normalized size = 1.23 \[ \frac{a^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 a b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 a b \log (\cos (c+d x))-(a-b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+b^2 \sec (c+d x)+b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 77, normalized size = 1. \begin{align*}{\frac{{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{ab\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}}{d\cos \left ( dx+c \right ) }}+{\frac{{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \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*a^2*ln(csc(d*x+c)-cot(d*x+c))+2/d*a*b*ln(tan(d*x+c))+1/d*b^2/cos(d*x+c)+1/d*b^2*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 0.944199, size = 99, normalized size = 1.34 \begin{align*} -\frac{4 \, a b \log \left (\cos \left (d x + c\right )\right ) +{\left (a^{2} - 2 \, a b + b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) -{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \, b^{2}}{\cos \left (d x + c\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="maxima")

[Out]

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

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Fricas [A]  time = 1.79424, size = 267, normalized size = 3.61 \begin{align*} -\frac{4 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) +{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right ) \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \, b^{2}}{2 \, d \cos \left (d x + c\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*(4*a*b*cos(d*x + c)*log(-cos(d*x + c)) + (a^2 - 2*a*b + b^2)*cos(d*x + c)*log(1/2*cos(d*x + c) + 1/2) - (
a^2 + 2*a*b + b^2)*cos(d*x + c)*log(-1/2*cos(d*x + c) + 1/2) - 2*b^2)/(d*cos(d*x + c))

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

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Giac [A]  time = 1.36733, size = 167, normalized size = 2.26 \begin{align*} -\frac{4 \, a b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) -{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - \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 )}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{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(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - (a^2 + 2*a*b + b^2)*log(abs(-cos(d*x + c) +
1)/abs(cos(d*x + c) + 1)) - 4*(a*b + b^2 + a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c) - 1)/(co
s(d*x + c) + 1) + 1))/d