3.106 \(\int (c+d x)^3 \cot ^2(a+b x) \, dx\)

Optimal. Leaf size=127 \[ \frac {3 d^3 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d} \]

[Out]

-I*(d*x+c)^3/b-1/4*(d*x+c)^4/d-(d*x+c)^3*cot(b*x+a)/b+3*d*(d*x+c)^2*ln(1-exp(2*I*(b*x+a)))/b^2-3*I*d^2*(d*x+c)
*polylog(2,exp(2*I*(b*x+a)))/b^3+3/2*d^3*polylog(3,exp(2*I*(b*x+a)))/b^4

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Rubi [A]  time = 0.20, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3720, 3717, 2190, 2531, 2282, 6589, 32} \[ -\frac {3 i d^2 (c+d x) \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cot[a + b*x]^2,x]

[Out]

((-I)*(c + d*x)^3)/b - (c + d*x)^4/(4*d) - ((c + d*x)^3*Cot[a + b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^((2*I)*(a
+ b*x))])/b^2 - ((3*I)*d^2*(c + d*x)*PolyLog[2, E^((2*I)*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, E^((2*I)*(a + b
*x))])/(2*b^4)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int (c+d x)^3 \cot ^2(a+b x) \, dx &=-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {(3 d) \int (c+d x)^2 \cot (a+b x) \, dx}{b}-\int (c+d x)^3 \, dx\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(6 i d) \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1-e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {\left (3 i d^3\right ) \int \text {Li}_2\left (e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {\left (3 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac {i (c+d x)^3}{b}-\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \cot (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^3 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{2 b^4}\\ \end {align*}

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Mathematica [B]  time = 6.15, size = 374, normalized size = 2.94 \[ -\frac {3 c^2 d (b x \cot (a)-\log (\sin (a+b x)))}{b^2}+\frac {3 c d^2 \left (-b^2 x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt {\sec ^2(a)}-i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+i b x \left (\pi -2 \tan ^{-1}(\tan (a))\right )+2 \left (\tan ^{-1}(\tan (a))+b x\right ) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (a))+b x\right )}\right )-2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (\tan ^{-1}(\tan (a))+b x\right )\right )+\pi \log \left (1+e^{-2 i b x}\right )-\pi \log (\cos (b x))\right )}{b^3}+\frac {e^{-i a} d^3 \sin (a) (\cot (a)+i) \left (-b^3 x^3 \cot (a)+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \text {Li}_2\left (-e^{-i (a+b x)}\right )+6 i b x \text {Li}_2\left (e^{-i (a+b x)}\right )+6 \text {Li}_3\left (-e^{-i (a+b x)}\right )+6 \text {Li}_3\left (e^{-i (a+b x)}\right )+i b^3 x^3\right )}{b^4}+\frac {\csc (a) \sin (b x) (c+d x)^3 \csc (a+b x)}{b}-\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)^3*Cot[a + b*x]^2,x]

[Out]

-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)) - (3*c^2*d*(b*x*Cot[a] - Log[Sin[a + b*x]]))/b^2 + (3*c*d
^2*(I*b*x*(Pi - 2*ArcTan[Tan[a]]) + Pi*Log[1 + E^((-2*I)*b*x)] + 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*
x + ArcTan[Tan[a]]))] - Pi*Log[Cos[b*x]] - 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] - I*PolyLog[2, E^((
2*I)*(b*x + ArcTan[Tan[a]]))] - b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2]))/b^3 + (d^3*(I + Cot[a])*(
I*b^3*x^3 - b^3*x^3*Cot[a] + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^((-I)*(a + b*x))] + (
6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a
+ b*x))] + 6*PolyLog[3, E^((-I)*(a + b*x))])*Sin[a])/(b^4*E^(I*a)) + ((c + d*x)^3*Csc[a]*Csc[a + b*x]*Sin[b*x
])/b

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fricas [C]  time = 0.78, size = 599, normalized size = 4.72 \[ -\frac {4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x + 4 \, b^{3} c^{3} - 3 \, d^{3} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, d^{3} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - {\left (-6 i \, b d^{3} x - 6 i \, b c d^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - {\left (6 i \, b d^{3} x + 6 i \, b c d^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b^{4} d^{3} x^{4} + 4 \, b^{4} c d^{2} x^{3} + 6 \, b^{4} c^{2} d x^{2} + 4 \, b^{4} c^{3} x\right )} \sin \left (2 \, b x + 2 \, a\right )}{4 \, b^{4} \sin \left (2 \, b x + 2 \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cot(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*d^3*x^3 + 12*b^3*c*d^2*x^2 + 12*b^3*c^2*d*x + 4*b^3*c^3 - 3*d^3*polylog(3, cos(2*b*x + 2*a) + I*si
n(2*b*x + 2*a))*sin(2*b*x + 2*a) - 3*d^3*polylog(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) -
(-6*I*b*d^3*x - 6*I*b*c*d^2)*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) - (6*I*b*d^3*x + 6*
I*b*c*d^2)*dilog(cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) - 6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^
3)*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2)*sin(2*b*x + 2*a) - 6*(b^2*c^2*d - 2*a*b*c*d^2 + a
^2*d^3)*log(-1/2*cos(2*b*x + 2*a) - 1/2*I*sin(2*b*x + 2*a) + 1/2)*sin(2*b*x + 2*a) - 6*(b^2*d^3*x^2 + 2*b^2*c*
d^2*x + 2*a*b*c*d^2 - a^2*d^3)*log(-cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a) + 1)*sin(2*b*x + 2*a) - 6*(b^2*d^3*x
^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*log(-cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1)*sin(2*b*x + 2*a) +
4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(2*b*x + 2*a) + (b^4*d^3*x^4 + 4*b^4*c*d^2*x^3
+ 6*b^4*c^2*d*x^2 + 4*b^4*c^3*x)*sin(2*b*x + 2*a))/(b^4*sin(2*b*x + 2*a))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \cot \left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cot(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*cot(b*x + a)^2, x)

________________________________________________________________________________________

maple [B]  time = 0.11, size = 573, normalized size = 4.51 \[ \frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {2 i d^{3} x^{3}}{b}+\frac {4 i d^{3} a^{3}}{b^{4}}-\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {12 i d^{2} c a x}{b^{2}}-\frac {3 c^{2} d \,x^{2}}{2}-c \,d^{2} x^{3}-\frac {d^{3} x^{4}}{4}-c^{3} x +\frac {3 d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{4}}+\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {6 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 i d^{2} c \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 i d^{2} c \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}+\frac {12 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {6 i d^{3} a^{2} x}{b^{3}}-\frac {6 i d^{2} c \,x^{2}}{b}-\frac {6 i d^{2} c \,a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*cot(b*x+a)^2,x)

[Out]

3/b^4*d^3*a^2*ln(exp(I*(b*x+a))-1)-6/b^4*d^3*a^2*ln(exp(I*(b*x+a)))+3/b^2*d*c^2*ln(exp(I*(b*x+a))-1)+3/b^2*d*c
^2*ln(exp(I*(b*x+a))+1)-6/b^2*d*c^2*ln(exp(I*(b*x+a)))-2*I/b*d^3*x^3+4*I/b^4*d^3*a^3-12*I/b^2*d^2*c*a*x+6*d^3*
polylog(3,-exp(I*(b*x+a)))/b^4+6*d^3*polylog(3,exp(I*(b*x+a)))/b^4-3/2*c^2*d*x^2-c*d^2*x^3-1/4*d^3*x^4-c^3*x+3
/b^2*d^3*ln(exp(I*(b*x+a))+1)*x^2+3/b^2*d^3*ln(1-exp(I*(b*x+a)))*x^2-3/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^2-2*I*(d
^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/b/(exp(2*I*(b*x+a))-1)-6*I/b^3*d^3*polylog(2,exp(I*(b*x+a)))*x+6/b^2*d^2*c*l
n(exp(I*(b*x+a))+1)*x-6*I/b^3*d^2*c*polylog(2,exp(I*(b*x+a)))+12/b^3*d^2*c*a*ln(exp(I*(b*x+a)))-6/b^3*d^2*c*a*
ln(exp(I*(b*x+a))-1)-6*I/b^3*d^2*c*polylog(2,-exp(I*(b*x+a)))+6*I/b^3*d^3*a^2*x-6*I/b*d^2*c*x^2-6*I/b^3*d^2*c*
a^2-6*I/b^3*d^3*polylog(2,-exp(I*(b*x+a)))*x+6/b^2*d^2*c*ln(1-exp(I*(b*x+a)))*x+6/b^3*d^2*c*ln(1-exp(I*(b*x+a)
))*a

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maxima [B]  time = 0.74, size = 1945, normalized size = 15.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cot(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/2*(2*(b*x + a + 1/tan(b*x + a))*c^3 - 6*(b*x + a + 1/tan(b*x + a))*a*c^2*d/b + 6*(b*x + a + 1/tan(b*x + a))
*a^2*c*d^2/b^2 - 2*(b*x + a + 1/tan(b*x + a))*a^3*d^3/b^3 + 3*((b*x + a)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*si
n(2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2
*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2
*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a
)*sin(2*b*x + 2*a))*c^2*d/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b) - 6*((b*x + a
)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin(2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)^2 - (cos(
2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x
+ a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x +
a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(2*b*x + 2*a))*a*c*d^2/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 -
2*cos(2*b*x + 2*a) + 1)*b^2) + 3*((b*x + a)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin(2*b*x + 2*a)^2 - 2*(b*x +
a)^2*cos(2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(c
os(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x
+ 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(2*b*x + 2*a))*a^2*d^3/
((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b^3) - 2*(-I*(b*x + a)^4*d^3 + (-4*I*b*c*d
^2 + 4*I*a*d^3)*(b*x + a)^3 - (12*(b*x + a)^2*d^3 + 24*(b*c*d^2 - a*d^3)*(b*x + a) - 12*((b*x + a)^2*d^3 + 2*(
b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (12*I*(b*x + a)^2*d^3 + (24*I*b*c*d^2 - 24*I*a*d^3)*(b*x + a))*
sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + (12*(b*x + a)^2*d^3 + 24*(b*c*d^2 - a*d^3)*(b*x +
a) - 12*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (-12*I*(b*x + a)^2*d^3 + (-24*I*b
*c*d^2 + 24*I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + (I*(b*x + a)^4*d^
3 + (4*I*b*c*d^2 - 4*(I*a + 2)*d^3)*(b*x + a)^3 - 24*(b*c*d^2 - a*d^3)*(b*x + a)^2)*cos(2*b*x + 2*a) + (24*b*c
*d^2 + 24*(b*x + a)*d^3 - 24*a*d^3 - 24*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(2*b*x + 2*a) + (-24*I*b*c*d^2 -
24*I*(b*x + a)*d^3 + 24*I*a*d^3)*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)) + (24*b*c*d^2 + 24*(b*x + a)*d^3 -
24*a*d^3 - 24*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(2*b*x + 2*a) + (-24*I*b*c*d^2 - 24*I*(b*x + a)*d^3 + 24*I*
a*d^3)*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) + (6*I*(b*x + a)^2*d^3 + (12*I*b*c*d^2 - 12*I*a*d^3)*(b*x + a)
+ (-6*I*(b*x + a)^2*d^3 + (-12*I*b*c*d^2 + 12*I*a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + 6*((b*x + a)^2*d^3 + 2*(
b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (6*I
*(b*x + a)^2*d^3 + (12*I*b*c*d^2 - 12*I*a*d^3)*(b*x + a) + (-6*I*(b*x + a)^2*d^3 + (-12*I*b*c*d^2 + 12*I*a*d^3
)*(b*x + a))*cos(2*b*x + 2*a) + 6*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(
b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + (-24*I*d^3*cos(2*b*x + 2*a) + 24*d^3*sin(2*b*x + 2*a) + 24
*I*d^3)*polylog(3, -e^(I*b*x + I*a)) + (-24*I*d^3*cos(2*b*x + 2*a) + 24*d^3*sin(2*b*x + 2*a) + 24*I*d^3)*polyl
og(3, e^(I*b*x + I*a)) - ((b*x + a)^4*d^3 + (4*b*c*d^2 - (4*a - 8*I)*d^3)*(b*x + a)^3 - (-24*I*b*c*d^2 + 24*I*
a*d^3)*(b*x + a)^2)*sin(2*b*x + 2*a))/(-4*I*b^3*cos(2*b*x + 2*a) + 4*b^3*sin(2*b*x + 2*a) + 4*I*b^3))/b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + b*x)^2*(c + d*x)^3,x)

[Out]

int(cot(a + b*x)^2*(c + d*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{3} \cot ^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*cot(b*x+a)**2,x)

[Out]

Integral((c + d*x)**3*cot(a + b*x)**2, x)

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