3.273 \(\int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx\)
Optimal. Leaf size=118 \[ \frac {3 d^3 \text {Li}_3\left (e^{4 i (a+b x)}\right )}{8 b^4}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{4 i (a+b x)}\right )}{2 b^3}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}-\frac {2 i (c+d x)^3}{b} \]
[Out]
-2*I*(d*x+c)^3/b-2*(d*x+c)^3*cot(2*b*x+2*a)/b+3*d*(d*x+c)^2*ln(1-exp(4*I*(b*x+a)))/b^2-3/2*I*d^2*(d*x+c)*polyl
og(2,exp(4*I*(b*x+a)))/b^3+3/8*d^3*polylog(3,exp(4*I*(b*x+a)))/b^4
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Rubi [A] time = 0.28, antiderivative size = 118, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used =
{4419, 4184, 3717, 2190, 2531, 2282, 6589} \[ -\frac {3 i d^2 (c+d x) \text {PolyLog}\left (2,e^{4 i (a+b x)}\right )}{2 b^3}+\frac {3 d^3 \text {PolyLog}\left (3,e^{4 i (a+b x)}\right )}{8 b^4}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}-\frac {2 i (c+d x)^3}{b} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^2,x]
[Out]
((-2*I)*(c + d*x)^3)/b - (2*(c + d*x)^3*Cot[2*a + 2*b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^((4*I)*(a + b*x))])/b
^2 - (((3*I)/2)*d^2*(c + d*x)*PolyLog[2, E^((4*I)*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, E^((4*I)*(a + b*x))])/(
8*b^4)
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 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 4419
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
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 \csc ^2(a+b x) \sec ^2(a+b x) \, dx &=4 \int (c+d x)^3 \csc ^2(2 a+2 b x) \, dx\\ &=-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}+\frac {(6 d) \int (c+d x)^2 \cot (2 a+2 b x) \, dx}{b}\\ &=-\frac {2 i (c+d x)^3}{b}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}-\frac {(12 i d) \int \frac {e^{2 i (2 a+2 b x)} (c+d x)^2}{1-e^{2 i (2 a+2 b x)}} \, dx}{b}\\ &=-\frac {2 i (c+d x)^3}{b}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1-e^{2 i (2 a+2 b x)}\right ) \, dx}{b^2}\\ &=-\frac {2 i (c+d x)^3}{b}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{4 i (a+b x)}\right )}{2 b^3}+\frac {\left (3 i d^3\right ) \int \text {Li}_2\left (e^{2 i (2 a+2 b x)}\right ) \, dx}{2 b^3}\\ &=-\frac {2 i (c+d x)^3}{b}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{4 i (a+b x)}\right )}{2 b^3}+\frac {\left (3 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (2 a+2 b x)}\right )}{8 b^4}\\ &=-\frac {2 i (c+d x)^3}{b}-\frac {2 (c+d x)^3 \cot (2 a+2 b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{4 i (a+b x)}\right )}{b^2}-\frac {3 i d^2 (c+d x) \text {Li}_2\left (e^{4 i (a+b x)}\right )}{2 b^3}+\frac {3 d^3 \text {Li}_3\left (e^{4 i (a+b x)}\right )}{8 b^4}\\ \end {align*}
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Mathematica [B] time = 2.13, size = 285, normalized size = 2.42 \[ \frac {-\frac {8 i b^3 (c+d x)^3}{-1+e^{4 i a}}+4 b^3 \csc (2 a) \sin (2 b x) (c+d x)^3 \csc (2 (a+b x))+6 b^2 d (c+d x)^2 \log \left (1-e^{-i (a+b x)}\right )+6 b^2 d (c+d x)^2 \log \left (1+e^{-i (a+b x)}\right )+6 b^2 d (c+d x)^2 \log \left (1+e^{-2 i (a+b x)}\right )+12 i b d^2 (c+d x) \text {Li}_2\left (-e^{-i (a+b x)}\right )+12 i b d^2 (c+d x) \text {Li}_2\left (e^{-i (a+b x)}\right )+6 i b d^2 (c+d x) \text {Li}_2\left (-e^{-2 i (a+b x)}\right )+12 d^3 \text {Li}_3\left (-e^{-i (a+b x)}\right )+12 d^3 \text {Li}_3\left (e^{-i (a+b x)}\right )+3 d^3 \text {Li}_3\left (-e^{-2 i (a+b x)}\right )}{2 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^2,x]
[Out]
(((-8*I)*b^3*(c + d*x)^3)/(-1 + E^((4*I)*a)) + 6*b^2*d*(c + d*x)^2*Log[1 - E^((-I)*(a + b*x))] + 6*b^2*d*(c +
d*x)^2*Log[1 + E^((-I)*(a + b*x))] + 6*b^2*d*(c + d*x)^2*Log[1 + E^((-2*I)*(a + b*x))] + (12*I)*b*d^2*(c + d*x
)*PolyLog[2, -E^((-I)*(a + b*x))] + (12*I)*b*d^2*(c + d*x)*PolyLog[2, E^((-I)*(a + b*x))] + (6*I)*b*d^2*(c + d
*x)*PolyLog[2, -E^((-2*I)*(a + b*x))] + 12*d^3*PolyLog[3, -E^((-I)*(a + b*x))] + 12*d^3*PolyLog[3, E^((-I)*(a
+ b*x))] + 3*d^3*PolyLog[3, -E^((-2*I)*(a + b*x))] + 4*b^3*(c + d*x)^3*Csc[2*a]*Csc[2*(a + b*x)]*Sin[2*b*x])/(
2*b^4)
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fricas [C] time = 0.62, size = 1627, normalized size = 13.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="fricas")
[Out]
1/2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 + 6*d^3*cos(b*x + a)*polylog(3, cos(b*x + a)
+ I*sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylog(3, cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 6
*d^3*cos(b*x + a)*polylog(3, I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylog(3, I*cos
(b*x + a) - sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylog(3, -I*cos(b*x + a) + sin(b*x + a))*sin(b*x
+ a) + 6*d^3*cos(b*x + a)*polylog(3, -I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylo
g(3, -cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 6*d^3*cos(b*x + a)*polylog(3, -cos(b*x + a) - I*sin(b*x +
a))*sin(b*x + a) + (-6*I*b*d^3*x - 6*I*b*c*d^2)*cos(b*x + a)*dilog(cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a)
+ (6*I*b*d^3*x + 6*I*b*c*d^2)*cos(b*x + a)*dilog(cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + (6*I*b*d^3*x +
6*I*b*c*d^2)*cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) + (-6*I*b*d^3*x - 6*I*b*c*d^2)*co
s(b*x + a)*dilog(I*cos(b*x + a) - sin(b*x + a))*sin(b*x + a) + (-6*I*b*d^3*x - 6*I*b*c*d^2)*cos(b*x + a)*dilog
(-I*cos(b*x + a) + sin(b*x + a))*sin(b*x + a) + (6*I*b*d^3*x + 6*I*b*c*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a)
- sin(b*x + a))*sin(b*x + a) + (6*I*b*d^3*x + 6*I*b*c*d^2)*cos(b*x + a)*dilog(-cos(b*x + a) + I*sin(b*x + a))
*sin(b*x + a) + (-6*I*b*d^3*x - 6*I*b*c*d^2)*cos(b*x + a)*dilog(-cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) +
3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos(b*x + a)*log(cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a)
+ 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log(cos(b*x + a) + I*sin(b*x + a) + I)*sin(b*x + a) + 3*(
b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos(b*x + a)*log(cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) + 3*
(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log(cos(b*x + a) - I*sin(b*x + a) + I)*sin(b*x + a) + 3*(b^2*
d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)*log(I*cos(b*x + a) + sin(b*x + a) + 1)*sin(b*x +
a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)*log(I*cos(b*x + a) - sin(b*x + a) +
1)*sin(b*x + a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)*log(-I*cos(b*x + a) +
sin(b*x + a) + 1)*sin(b*x + a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)*log(-I*c
os(b*x + a) - sin(b*x + a) + 1)*sin(b*x + a) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log(-1/2*cos
(b*x + a) + 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log(-1
/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*
d^3)*cos(b*x + a)*log(-cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)
*cos(b*x + a)*log(-cos(b*x + a) + I*sin(b*x + a) + I)*sin(b*x + a) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*
d^2 - a^2*d^3)*cos(b*x + a)*log(-cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) + 3*(b^2*c^2*d - 2*a*b*c*d^2
+ a^2*d^3)*cos(b*x + a)*log(-cos(b*x + a) - I*sin(b*x + a) + I)*sin(b*x + 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(b*x + a)^2)/(b^4*cos(b*x + a)*sin(b*x + a))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="giac")
[Out]
integrate((d*x + c)^3*csc(b*x + a)^2*sec(b*x + a)^2, x)
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maple [B] time = 0.13, size = 687, normalized size = 5.82 \[ -\frac {24 i d^{2} c a x}{b^{2}}+\frac {3 d \,c^{2} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{2}}-\frac {12 d \,c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {12 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {3 d^{3} \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x^{2}}{b^{2}}+\frac {8 i d^{3} a^{3}}{b^{4}}-\frac {4 i d^{3} x^{3}}{b}+\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 {3 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}+\frac {3 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}-\frac {6 c \,d^{2} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\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 {3 d^{3} \polylog \left (3, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{4}}-\frac {4 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\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 {24 d^{2} c a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {3 i d^{2} c \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b^{3}}-\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {6 d^{2} c \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {12 i d^{2} c \,a^{2}}{b^{3}}-\frac {12 i d^{2} c \,x^{2}}{b}+\frac {12 i d^{3} a^{2} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x)
[Out]
3/2*d^3*polylog(3,-exp(2*I*(b*x+a)))/b^4+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*d/b^2*c^2*ln(1+exp(2*I*(b*x+a)))-12*d/b^2*c^2*ln(exp(I*(b*x+a)))-12*d^3/b^4*a^2*ln(exp(I*(b*x+a)))+3*d^3
/b^2*ln(1+exp(2*I*(b*x+a)))*x^2+3/b^2*c^2*d*ln(exp(I*(b*x+a))-1)+3/b^2*c^2*d*ln(exp(I*(b*x+a))+1)+3/b^4*d^3*a^
2*ln(exp(I*(b*x+a))-1)+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-ex
p(I*(b*x+a)))*a^2-6/b^3*c*d^2*a*ln(exp(I*(b*x+a))-1)+8*I*d^3/b^4*a^3-4*I*d^3/b*x^3-6*I*d^2/b^3*c*polylog(2,-ex
p(I*(b*x+a)))+12*I*d^3/b^3*a^2*x-12*I*d^2/b*c*x^2-12*I*d^2/b^3*c*a^2-6*I*d^3/b^3*polylog(2,-exp(I*(b*x+a)))*x-
6*I/b^3*d^3*polylog(2,exp(I*(b*x+a)))*x+6/b^2*d^2*c*ln(exp(I*(b*x+a))+1)*x-6*I/b^3*d^2*c*polylog(2,exp(I*(b*x+
a)))-24*I*d^2/b^2*c*a*x-4*I*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/b/(1+exp(2*I*(b*x+a)))/(exp(2*I*(b*x+a))-1)+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+24*d^2/b^3*c*a*ln(exp(I*(b*x+a)))+6*d^2/b^
2*c*ln(1+exp(2*I*(b*x+a)))*x-3*I*d^3/b^3*polylog(2,-exp(2*I*(b*x+a)))*x-3*I*d^2/b^3*c*polylog(2,-exp(2*I*(b*x+
a)))
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maxima [B] time = 0.74, size = 2355, normalized size = 19.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="maxima")
[Out]
-1/2*(2*c^3*(1/tan(b*x + a) - tan(b*x + a)) - 6*a*c^2*d*(1/tan(b*x + a) - tan(b*x + a))/b + 6*a^2*c*d^2*(1/tan
(b*x + a) - tan(b*x + a))/b^2 - 2*a^3*d^3*(1/tan(b*x + a) - tan(b*x + a))/b^3 - 3*((cos(4*b*x + 4*a)^2 + sin(4
*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)
+ (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*c
os(b*x + a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin
(b*x + a)^2 - 2*cos(b*x + a) + 1) - 8*(b*x + a)*sin(4*b*x + 4*a))*c^2*d/((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a
)^2 - 2*cos(4*b*x + 4*a) + 1)*b) + 6*((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(c
os(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 -
2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (cos(4*b*x + 4*a)^2 + sin(
4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 8*(b*x +
a)*sin(4*b*x + 4*a))*a*c*d^2/((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*b^2) - 3*((co
s(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 +
2*cos(2*b*x + 2*a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^
2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*
log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 8*(b*x + a)*sin(4*b*x + 4*a))*a^2*d^3/((cos(4*b*x
+ 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*b^3) + 2*((6*(b*x + a)^2*d^3 + 12*(b*c*d^2 - a*d^3)*(b
*x + a) - 6*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - (6*I*(b*x + a)^2*d^3 + (12*I*
b*c*d^2 - 12*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a))*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) + (6*(b*x +
a)^2*d^3 + 12*(b*c*d^2 - a*d^3)*(b*x + a) - 6*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(4*b*x + 4
*a) - (6*I*(b*x + a)^2*d^3 + (12*I*b*c*d^2 - 12*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a))*arctan2(sin(b*x + a), co
s(b*x + a) + 1) - (6*(b*x + a)^2*d^3 + 12*(b*c*d^2 - a*d^3)*(b*x + a) - 6*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^
3)*(b*x + a))*cos(4*b*x + 4*a) + (-6*I*(b*x + a)^2*d^3 + (-12*I*b*c*d^2 + 12*I*a*d^3)*(b*x + a))*sin(4*b*x + 4
*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + 8*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2)*cos(4*b*
x + 4*a) - (6*b*c*d^2 + 6*(b*x + a)*d^3 - 6*a*d^3 - 6*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(4*b*x + 4*a) + (-6
*I*b*c*d^2 - 6*I*(b*x + a)*d^3 + 6*I*a*d^3)*sin(4*b*x + 4*a))*dilog(-e^(2*I*b*x + 2*I*a)) - (12*b*c*d^2 + 12*(
b*x + a)*d^3 - 12*a*d^3 - 12*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(4*b*x + 4*a) + (-12*I*b*c*d^2 - 12*I*(b*x +
a)*d^3 + 12*I*a*d^3)*sin(4*b*x + 4*a))*dilog(-e^(I*b*x + I*a)) - (12*b*c*d^2 + 12*(b*x + a)*d^3 - 12*a*d^3 -
12*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(4*b*x + 4*a) + (-12*I*b*c*d^2 - 12*I*(b*x + a)*d^3 + 12*I*a*d^3)*sin(
4*b*x + 4*a))*dilog(e^(I*b*x + I*a)) - (3*I*(b*x + a)^2*d^3 + (6*I*b*c*d^2 - 6*I*a*d^3)*(b*x + a) + (-3*I*(b*x
+ a)^2*d^3 + (-6*I*b*c*d^2 + 6*I*a*d^3)*(b*x + a))*cos(4*b*x + 4*a) + 3*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3
)*(b*x + a))*sin(4*b*x + 4*a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) - (3*I*(b
*x + a)^2*d^3 + (6*I*b*c*d^2 - 6*I*a*d^3)*(b*x + a) + (-3*I*(b*x + a)^2*d^3 + (-6*I*b*c*d^2 + 6*I*a*d^3)*(b*x
+ a))*cos(4*b*x + 4*a) + 3*((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(4*b*x + 4*a))*log(cos(b*x + a
)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (3*I*(b*x + a)^2*d^3 + (6*I*b*c*d^2 - 6*I*a*d^3)*(b*x + a) + (-3*
I*(b*x + a)^2*d^3 + (-6*I*b*c*d^2 + 6*I*a*d^3)*(b*x + a))*cos(4*b*x + 4*a) + 3*((b*x + a)^2*d^3 + 2*(b*c*d^2 -
a*d^3)*(b*x + a))*sin(4*b*x + 4*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - (-3*I*d^3*cos
(4*b*x + 4*a) + 3*d^3*sin(4*b*x + 4*a) + 3*I*d^3)*polylog(3, -e^(2*I*b*x + 2*I*a)) - (-12*I*d^3*cos(4*b*x + 4*
a) + 12*d^3*sin(4*b*x + 4*a) + 12*I*d^3)*polylog(3, -e^(I*b*x + I*a)) - (-12*I*d^3*cos(4*b*x + 4*a) + 12*d^3*s
in(4*b*x + 4*a) + 12*I*d^3)*polylog(3, e^(I*b*x + I*a)) - (-8*I*(b*x + a)^3*d^3 + (-24*I*b*c*d^2 + 24*I*a*d^3)
*(b*x + a)^2)*sin(4*b*x + 4*a))/(-2*I*b^3*cos(4*b*x + 4*a) + 2*b^3*sin(4*b*x + 4*a) + 2*I*b^3))/b
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^3}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^3/(cos(a + b*x)^2*sin(a + b*x)^2),x)
[Out]
int((c + d*x)^3/(cos(a + b*x)^2*sin(a + b*x)^2), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**3*csc(b*x+a)**2*sec(b*x+a)**2,x)
[Out]
Timed out
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