3.9 \(\int \csc (c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx\)

Optimal. Leaf size=38 \[ \frac{a A \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a A \tanh ^{-1}(\cos (c+d x))}{2 d} \]

[Out]

-(a*A*ArcTanh[Cos[c + d*x]])/(2*d) + (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)

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Rubi [A]  time = 0.0509353, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3958, 2611, 3770} \[ \frac{a A \cot (c+d x) \csc (c+d x)}{2 d}-\frac{a A \tanh ^{-1}(\cos (c+d x))}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]*(a - a*Csc[c + d*x])*(A + A*Csc[c + d*x]),x]

[Out]

-(a*A*ArcTanh[Cos[c + d*x]])/(2*d) + (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)

Rule 3958

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_.), x_Symbol] :> Dist[(-(a*c))^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n
- m), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m,
n] && GeQ[n - m, 0] && GtQ[m*n, 0]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

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

Rubi steps

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

Mathematica [B]  time = 0.03455, size = 79, normalized size = 2.08 \[ -a A \left (-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]*(a - a*Csc[c + d*x])*(A + A*Csc[c + d*x]),x]

[Out]

-(a*A*(-Csc[(c + d*x)/2]^2/(8*d) + Log[Cos[(c + d*x)/2]]/(2*d) - Log[Sin[(c + d*x)/2]]/(2*d) + Sec[(c + d*x)/2
]^2/(8*d)))

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Maple [A]  time = 0.023, size = 44, normalized size = 1.2 \begin{align*}{\frac{Aa\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{Aa\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x)

[Out]

1/2/d*A*a*ln(csc(d*x+c)-cot(d*x+c))+1/2*a*A*cot(d*x+c)*csc(d*x+c)/d

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Maxima [A]  time = 1.11618, size = 92, normalized size = 2.42 \begin{align*} -\frac{A a{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="maxima")

[Out]

-1/4*(A*a*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 4*A*a*log(co
t(d*x + c) + csc(d*x + c)))/d

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Fricas [B]  time = 0.495874, size = 223, normalized size = 5.87 \begin{align*} -\frac{2 \, A a \cos \left (d x + c\right ) +{\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{4 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - A a \left (\int - \csc{\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x)

[Out]

-A*a*(Integral(-csc(c + d*x), x) + Integral(csc(c + d*x)**3, x))

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Giac [B]  time = 1.33333, size = 136, normalized size = 3.58 \begin{align*} \frac{2 \, A a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac{A a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{{\left (A a + \frac{2 \, A a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a-a*csc(d*x+c))*(A+A*csc(d*x+c)),x, algorithm="giac")

[Out]

1/8*(2*A*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) + A*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - (A*
a + 2*A*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/(cos(d*x + c) - 1))/d