\(\int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx\) [115]

Optimal result

Integrand size = 20, antiderivative size = 108 \[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\frac {(c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}-\frac {i d \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}+\frac {i d \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2} \]

[Out]

(d*x+c)*arctanh(exp(I*(b*x+a)))/b-1/2*d*csc(b*x+a)/b^2-1/2*(d*x+c)*cot(b*x+a)*csc(b*x+a)/b-1/2*I*d*polylog(2,-
exp(I*(b*x+a)))/b^2+1/2*I*d*polylog(2,exp(I*(b*x+a)))/b^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4500, 4268, 2317, 2438, 4270} \[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\frac {(c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}+\frac {i d \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b} \]

[In]

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

[Out]

((c + d*x)*ArcTanh[E^(I*(a + b*x))])/b - (d*Csc[a + b*x])/(2*b^2) - ((c + d*x)*Cot[a + b*x]*Csc[a + b*x])/(2*b
) - ((I/2)*d*PolyLog[2, -E^(I*(a + b*x))])/b^2 + ((I/2)*d*PolyLog[2, E^(I*(a + b*x))])/b^2

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4270

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]
*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)*(b*Csc[e + f*x])^(n -
2), x], x] - Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 4500

Int[Cot[(a_.) + (b_.)*(x_)]^(p_)*Csc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*
x)^m*Csc[a + b*x]*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^3*Cot[a + b*x]^(p - 2), x] /; FreeQ[
{a, b, c, d, m}, x] && IGtQ[p/2, 0]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x) \csc (a+b x) \, dx+\int (c+d x) \csc ^3(a+b x) \, dx \\ & = \frac {2 (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int (c+d x) \csc (a+b x) \, dx+\frac {d \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}-\frac {d \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b} \\ & = \frac {(c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}-\frac {(i d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}+\frac {(i d) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}-\frac {d \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{2 b}+\frac {d \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{2 b} \\ & = \frac {(c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}-\frac {i d \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {(i d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{2 b^2}-\frac {(i d) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{2 b^2} \\ & = \frac {(c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d \csc (a+b x)}{2 b^2}-\frac {(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}-\frac {i d \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}+\frac {i d \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(260\) vs. \(2(108)=216\).

Time = 1.94 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.41 \[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=-\frac {d \cot \left (\frac {1}{2} (a+b x)\right )}{4 b^2}-\frac {c \csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {d x \csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {c \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {c \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {a d \log \left (\tan \left (\frac {1}{2} (a+b x)\right )\right )}{2 b^2}-\frac {d \left ((a+b x) \left (\log \left (1-e^{i (a+b x)}\right )-\log \left (1+e^{i (a+b x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )\right )\right )}{2 b^2}+\frac {c \sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {d x \sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {d \tan \left (\frac {1}{2} (a+b x)\right )}{4 b^2} \]

[In]

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

[Out]

-1/4*(d*Cot[(a + b*x)/2])/b^2 - (c*Csc[(a + b*x)/2]^2)/(8*b) - (d*x*Csc[(a + b*x)/2]^2)/(8*b) + (c*Log[Cos[(a
+ b*x)/2]])/(2*b) - (c*Log[Sin[(a + b*x)/2]])/(2*b) + (a*d*Log[Tan[(a + b*x)/2]])/(2*b^2) - (d*((a + b*x)*(Log
[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x
))])))/(2*b^2) + (c*Sec[(a + b*x)/2]^2)/(8*b) + (d*x*Sec[(a + b*x)/2]^2)/(8*b) - (d*Tan[(a + b*x)/2])/(4*b^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (92 ) = 184\).

Time = 0.88 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.28

method	result	size
risch	\(\frac {d x b \,{\mathrm e}^{3 i \left (x b +a \right )}+c b \,{\mathrm e}^{3 i \left (x b +a \right )}+d x b \,{\mathrm e}^{i \left (x b +a \right )}+c b \,{\mathrm e}^{i \left (x b +a \right )}-i d \,{\mathrm e}^{3 i \left (x b +a \right )}+i d \,{\mathrm e}^{i \left (x b +a \right )}}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {c \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}-\frac {d \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{2 b}-\frac {d \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{2 b^{2}}+\frac {i d \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{2}}+\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{2 b}+\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a}{2 b^{2}}-\frac {i d \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{2}}-\frac {d a \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}\)	\(246\)

[In]

int((d*x+c)*cot(b*x+a)^2*csc(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b^2/(exp(2*I*(b*x+a))-1)^2*(d*x*b*exp(3*I*(b*x+a))+c*b*exp(3*I*(b*x+a))+d*x*b*exp(I*(b*x+a))+c*b*exp(I*(b*x+
a))-I*d*exp(3*I*(b*x+a))+I*d*exp(I*(b*x+a)))+1/b*c*arctanh(exp(I*(b*x+a)))-1/2/b*d*ln(1-exp(I*(b*x+a)))*x-1/2/
b^2*d*ln(1-exp(I*(b*x+a)))*a+1/2*I*d*polylog(2,exp(I*(b*x+a)))/b^2+1/2/b*d*ln(exp(I*(b*x+a))+1)*x+1/2/b^2*d*ln
(exp(I*(b*x+a))+1)*a-1/2*I*d*polylog(2,-exp(I*(b*x+a)))/b^2-1/b^2*d*a*arctanh(exp(I*(b*x+a)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (88) = 176\).

Time = 0.27 (sec) , antiderivative size = 454, normalized size of antiderivative = 4.20 \[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\frac {2 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) + {\left (i \, d \cos \left (b x + a\right )^{2} - i \, d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (-i \, d \cos \left (b x + a\right )^{2} + i \, d\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (i \, d \cos \left (b x + a\right )^{2} - i \, d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + {\left (-i \, d \cos \left (b x + a\right )^{2} + i \, d\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - {\left (b d x - {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + b c\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - {\left (b d x - {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + b c\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - {\left ({\left (b c - a d\right )} \cos \left (b x + a\right )^{2} - b c + a d\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) - {\left ({\left (b c - a d\right )} \cos \left (b x + a\right )^{2} - b c + a d\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (b d x - {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} + a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b d x - {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} + a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + 2 \, d \sin \left (b x + a\right )}{4 \, {\left (b^{2} \cos \left (b x + a\right )^{2} - b^{2}\right )}} \]

[In]

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

[Out]

1/4*(2*(b*d*x + b*c)*cos(b*x + a) + (I*d*cos(b*x + a)^2 - I*d)*dilog(cos(b*x + a) + I*sin(b*x + a)) + (-I*d*co
s(b*x + a)^2 + I*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) + (I*d*cos(b*x + a)^2 - I*d)*dilog(-cos(b*x + a) + I*
sin(b*x + a)) + (-I*d*cos(b*x + a)^2 + I*d)*dilog(-cos(b*x + a) - I*sin(b*x + a)) - (b*d*x - (b*d*x + b*c)*cos
(b*x + a)^2 + b*c)*log(cos(b*x + a) + I*sin(b*x + a) + 1) - (b*d*x - (b*d*x + b*c)*cos(b*x + a)^2 + b*c)*log(c
os(b*x + a) - I*sin(b*x + a) + 1) - ((b*c - a*d)*cos(b*x + a)^2 - b*c + a*d)*log(-1/2*cos(b*x + a) + 1/2*I*sin
(b*x + a) + 1/2) - ((b*c - a*d)*cos(b*x + a)^2 - b*c + a*d)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2)
+ (b*d*x - (b*d*x + a*d)*cos(b*x + a)^2 + a*d)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) + (b*d*x - (b*d*x + a*d
)*cos(b*x + a)^2 + a*d)*log(-cos(b*x + a) - I*sin(b*x + a) + 1) + 2*d*sin(b*x + a))/(b^2*cos(b*x + a)^2 - b^2)

Sympy [F]

\[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\int \left (c + d x\right ) \cot ^{2}{\left (a + b x \right )} \csc {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (88) = 176\).

Time = 0.35 (sec) , antiderivative size = 762, normalized size of antiderivative = 7.06 \[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\frac {2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (-i \, b d x - i \, b c\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (i \, b d x + i \, b c\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 2 \, {\left (b c \cos \left (4 \, b x + 4 \, a\right ) - 2 \, b c \cos \left (2 \, b x + 2 \, a\right ) + i \, b c \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, b c \sin \left (2 \, b x + 2 \, a\right ) + b c\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) - 1\right ) + 2 \, {\left (b d x \cos \left (4 \, b x + 4 \, a\right ) - 2 \, b d x \cos \left (2 \, b x + 2 \, a\right ) + i \, b d x \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, b d x \sin \left (2 \, b x + 2 \, a\right ) + b d x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 4 \, {\left (i \, b d x + i \, b c + d\right )} \cos \left (3 \, b x + 3 \, a\right ) - 4 \, {\left (i \, b d x + i \, b c - d\right )} \cos \left (b x + a\right ) - 2 \, {\left (d \cos \left (4 \, b x + 4 \, a\right ) - 2 \, d \cos \left (2 \, b x + 2 \, a\right ) + i \, d \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, d \sin \left (2 \, b x + 2 \, a\right ) + d\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 2 \, {\left (d \cos \left (4 \, b x + 4 \, a\right ) - 2 \, d \cos \left (2 \, b x + 2 \, a\right ) + i \, d \sin \left (4 \, b x + 4 \, a\right ) - 2 i \, d \sin \left (2 \, b x + 2 \, a\right ) + d\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + {\left (-i \, b d x - i \, b c + {\left (-i \, b d x - i \, b c\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (-i \, b d x - i \, b c\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b d x + b c\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b d x + b c\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + {\left (i \, b d x + i \, b c + {\left (i \, b d x + i \, b c\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (i \, b d x + i \, b c\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (b d x + b c\right )} \sin \left (4 \, b x + 4 \, a\right ) + 2 \, {\left (b d x + b c\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) + 4 \, {\left (b d x + b c - i \, d\right )} \sin \left (3 \, b x + 3 \, a\right ) + 4 \, {\left (b d x + b c + i \, d\right )} \sin \left (b x + a\right )}{-4 i \, b^{2} \cos \left (4 \, b x + 4 \, a\right ) + 8 i \, b^{2} \cos \left (2 \, b x + 2 \, a\right ) + 4 \, b^{2} \sin \left (4 \, b x + 4 \, a\right ) - 8 \, b^{2} \sin \left (2 \, b x + 2 \, a\right ) - 4 i \, b^{2}} \]

[In]

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

[Out]

(2*(b*d*x + b*c + (b*d*x + b*c)*cos(4*b*x + 4*a) - 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - (-I*b*d*x - I*b*c)*sin(4
*b*x + 4*a) - 2*(I*b*d*x + I*b*c)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 2*(b*c*cos(4*b*x
+ 4*a) - 2*b*c*cos(2*b*x + 2*a) + I*b*c*sin(4*b*x + 4*a) - 2*I*b*c*sin(2*b*x + 2*a) + b*c)*arctan2(sin(b*x +
a), cos(b*x + a) - 1) + 2*(b*d*x*cos(4*b*x + 4*a) - 2*b*d*x*cos(2*b*x + 2*a) + I*b*d*x*sin(4*b*x + 4*a) - 2*I*
b*d*x*sin(2*b*x + 2*a) + b*d*x)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 4*(I*b*d*x + I*b*c + d)*cos(3*b*x +
3*a) - 4*(I*b*d*x + I*b*c - d)*cos(b*x + a) - 2*(d*cos(4*b*x + 4*a) - 2*d*cos(2*b*x + 2*a) + I*d*sin(4*b*x +
4*a) - 2*I*d*sin(2*b*x + 2*a) + d)*dilog(-e^(I*b*x + I*a)) + 2*(d*cos(4*b*x + 4*a) - 2*d*cos(2*b*x + 2*a) + I*
d*sin(4*b*x + 4*a) - 2*I*d*sin(2*b*x + 2*a) + d)*dilog(e^(I*b*x + I*a)) + (-I*b*d*x - I*b*c + (-I*b*d*x - I*b*
c)*cos(4*b*x + 4*a) - 2*(-I*b*d*x - I*b*c)*cos(2*b*x + 2*a) + (b*d*x + b*c)*sin(4*b*x + 4*a) - 2*(b*d*x + b*c)
*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (I*b*d*x + I*b*c + (I*b*d*x + I
*b*c)*cos(4*b*x + 4*a) - 2*(I*b*d*x + I*b*c)*cos(2*b*x + 2*a) - (b*d*x + b*c)*sin(4*b*x + 4*a) + 2*(b*d*x + b*
c)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*d*x + b*c - I*d)*sin(3*b
*x + 3*a) + 4*(b*d*x + b*c + I*d)*sin(b*x + a))/(-4*I*b^2*cos(4*b*x + 4*a) + 8*I*b^2*cos(2*b*x + 2*a) + 4*b^2*
sin(4*b*x + 4*a) - 8*b^2*sin(2*b*x + 2*a) - 4*I*b^2)

Giac [F]

\[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

integrate((d*x + c)*cot(b*x + a)^2*csc(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx=\text {Hanged} \]

[In]

int((cot(a + b*x)^2*(c + d*x))/sin(a + b*x),x)

[Out]

\text{Hanged}