3.228 \(\int (c+d x)^4 \csc (a+b x) \sec (a+b x) \, dx\)

Optimal. Leaf size=247 \[ -\frac{3 i d^3 (c+d x) \text{PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac{3 i d^3 (c+d x) \text{PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac{3 d^2 (c+d x)^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 d^2 (c+d x)^2 \text{PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac{2 i d (c+d x)^3 \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac{3 d^4 \text{PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac{3 d^4 \text{PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b} \]

[Out]

(-2*(c + d*x)^4*ArcTanh[E^((2*I)*(a + b*x))])/b + ((2*I)*d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 -
((2*I)*d*(c + d*x)^3*PolyLog[2, E^((2*I)*(a + b*x))])/b^2 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^((2*I)*(a + b*x)
)])/b^3 + (3*d^2*(c + d*x)^2*PolyLog[3, E^((2*I)*(a + b*x))])/b^3 - ((3*I)*d^3*(c + d*x)*PolyLog[4, -E^((2*I)*
(a + b*x))])/b^4 + ((3*I)*d^3*(c + d*x)*PolyLog[4, E^((2*I)*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^((2*I)*(a
+ b*x))])/(2*b^5) - (3*d^4*PolyLog[5, E^((2*I)*(a + b*x))])/(2*b^5)

________________________________________________________________________________________

Rubi [A]  time = 0.229067, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4419, 4183, 2531, 6609, 2282, 6589} \[ -\frac{3 i d^3 (c+d x) \text{PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac{3 i d^3 (c+d x) \text{PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac{3 d^2 (c+d x)^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 d^2 (c+d x)^2 \text{PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac{2 i d (c+d x)^3 \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac{3 d^4 \text{PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac{3 d^4 \text{PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x],x]

[Out]

(-2*(c + d*x)^4*ArcTanh[E^((2*I)*(a + b*x))])/b + ((2*I)*d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 -
((2*I)*d*(c + d*x)^3*PolyLog[2, E^((2*I)*(a + b*x))])/b^2 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^((2*I)*(a + b*x)
)])/b^3 + (3*d^2*(c + d*x)^2*PolyLog[3, E^((2*I)*(a + b*x))])/b^3 - ((3*I)*d^3*(c + d*x)*PolyLog[4, -E^((2*I)*
(a + b*x))])/b^4 + ((3*I)*d^3*(c + d*x)*PolyLog[4, E^((2*I)*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^((2*I)*(a
+ b*x))])/(2*b^5) - (3*d^4*PolyLog[5, E^((2*I)*(a + b*x))])/(2*b^5)

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 4183

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 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 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[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 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)^4 \csc (a+b x) \sec (a+b x) \, dx &=2 \int (c+d x)^4 \csc (2 a+2 b x) \, dx\\ &=-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}-\frac{(4 d) \int (c+d x)^3 \log \left (1-e^{i (2 a+2 b x)}\right ) \, dx}{b}+\frac{(4 d) \int (c+d x)^3 \log \left (1+e^{i (2 a+2 b x)}\right ) \, dx}{b}\\ &=-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac{2 i d (c+d x)^3 \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{\left (6 i d^2\right ) \int (c+d x)^2 \text{Li}_2\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^2}+\frac{\left (6 i d^2\right ) \int (c+d x)^2 \text{Li}_2\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^2}\\ &=-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac{2 i d (c+d x)^3 \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}+\frac{\left (6 d^3\right ) \int (c+d x) \text{Li}_3\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^3}-\frac{\left (6 d^3\right ) \int (c+d x) \text{Li}_3\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^3}\\ &=-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac{2 i d (c+d x)^3 \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac{3 i d^3 (c+d x) \text{Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac{3 i d^3 (c+d x) \text{Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}+\frac{\left (3 i d^4\right ) \int \text{Li}_4\left (-e^{i (2 a+2 b x)}\right ) \, dx}{b^4}-\frac{\left (3 i d^4\right ) \int \text{Li}_4\left (e^{i (2 a+2 b x)}\right ) \, dx}{b^4}\\ &=-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac{2 i d (c+d x)^3 \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac{3 i d^3 (c+d x) \text{Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac{3 i d^3 (c+d x) \text{Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}+\frac{\left (3 d^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4(-x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{2 b^5}-\frac{\left (3 d^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4(x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{2 b^5}\\ &=-\frac{2 (c+d x)^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )}{b}+\frac{2 i d (c+d x)^3 \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}-\frac{3 i d^3 (c+d x) \text{Li}_4\left (-e^{2 i (a+b x)}\right )}{b^4}+\frac{3 i d^3 (c+d x) \text{Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}+\frac{3 d^4 \text{Li}_5\left (-e^{2 i (a+b x)}\right )}{2 b^5}-\frac{3 d^4 \text{Li}_5\left (e^{2 i (a+b x)}\right )}{2 b^5}\\ \end{align*}

Mathematica [B]  time = 1.37853, size = 578, normalized size = 2.34 \[ \frac{-6 b^2 c^2 d^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )+6 b^2 c^2 d^2 \text{PolyLog}\left (3,e^{2 i (a+b x)}\right )-12 b^2 c d^3 x \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )+12 b^2 c d^3 x \text{PolyLog}\left (3,e^{2 i (a+b x)}\right )+4 i b^3 d (c+d x)^3 \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )-4 i b^3 d (c+d x)^3 \text{PolyLog}\left (2,e^{2 i (a+b x)}\right )-6 b^2 d^4 x^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )+6 b^2 d^4 x^2 \text{PolyLog}\left (3,e^{2 i (a+b x)}\right )-6 i b c d^3 \text{PolyLog}\left (4,-e^{2 i (a+b x)}\right )+6 i b c d^3 \text{PolyLog}\left (4,e^{2 i (a+b x)}\right )-6 i b d^4 x \text{PolyLog}\left (4,-e^{2 i (a+b x)}\right )+6 i b d^4 x \text{PolyLog}\left (4,e^{2 i (a+b x)}\right )+3 d^4 \text{PolyLog}\left (5,-e^{2 i (a+b x)}\right )-3 d^4 \text{PolyLog}\left (5,e^{2 i (a+b x)}\right )+12 b^4 c^2 d^2 x^2 \log \left (1-e^{2 i (a+b x)}\right )-12 b^4 c^2 d^2 x^2 \log \left (1+e^{2 i (a+b x)}\right )+8 b^4 c^3 d x \log \left (1-e^{2 i (a+b x)}\right )-8 b^4 c^3 d x \log \left (1+e^{2 i (a+b x)}\right )-4 b^4 c^4 \tanh ^{-1}\left (e^{2 i (a+b x)}\right )+8 b^4 c d^3 x^3 \log \left (1-e^{2 i (a+b x)}\right )-8 b^4 c d^3 x^3 \log \left (1+e^{2 i (a+b x)}\right )+2 b^4 d^4 x^4 \log \left (1-e^{2 i (a+b x)}\right )-2 b^4 d^4 x^4 \log \left (1+e^{2 i (a+b x)}\right )}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x],x]

[Out]

(-4*b^4*c^4*ArcTanh[E^((2*I)*(a + b*x))] + 8*b^4*c^3*d*x*Log[1 - E^((2*I)*(a + b*x))] + 12*b^4*c^2*d^2*x^2*Log
[1 - E^((2*I)*(a + b*x))] + 8*b^4*c*d^3*x^3*Log[1 - E^((2*I)*(a + b*x))] + 2*b^4*d^4*x^4*Log[1 - E^((2*I)*(a +
b*x))] - 8*b^4*c^3*d*x*Log[1 + E^((2*I)*(a + b*x))] - 12*b^4*c^2*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] - 8*b^4
*c*d^3*x^3*Log[1 + E^((2*I)*(a + b*x))] - 2*b^4*d^4*x^4*Log[1 + E^((2*I)*(a + b*x))] + (4*I)*b^3*d*(c + d*x)^3
*PolyLog[2, -E^((2*I)*(a + b*x))] - (4*I)*b^3*d*(c + d*x)^3*PolyLog[2, E^((2*I)*(a + b*x))] - 6*b^2*c^2*d^2*Po
lyLog[3, -E^((2*I)*(a + b*x))] - 12*b^2*c*d^3*x*PolyLog[3, -E^((2*I)*(a + b*x))] - 6*b^2*d^4*x^2*PolyLog[3, -E
^((2*I)*(a + b*x))] + 6*b^2*c^2*d^2*PolyLog[3, E^((2*I)*(a + b*x))] + 12*b^2*c*d^3*x*PolyLog[3, E^((2*I)*(a +
b*x))] + 6*b^2*d^4*x^2*PolyLog[3, E^((2*I)*(a + b*x))] - (6*I)*b*c*d^3*PolyLog[4, -E^((2*I)*(a + b*x))] - (6*I
)*b*d^4*x*PolyLog[4, -E^((2*I)*(a + b*x))] + (6*I)*b*c*d^3*PolyLog[4, E^((2*I)*(a + b*x))] + (6*I)*b*d^4*x*Pol
yLog[4, E^((2*I)*(a + b*x))] + 3*d^4*PolyLog[5, -E^((2*I)*(a + b*x))] - 3*d^4*PolyLog[5, E^((2*I)*(a + b*x))])
/(2*b^5)

________________________________________________________________________________________

Maple [B]  time = 0.329, size = 1242, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*csc(b*x+a)*sec(b*x+a),x)

[Out]

3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5+1/b*d^4*ln(1-exp(I*(b*x+a)))*x^4-1/b^5*d^4*ln(1-exp(I*(b*x+a)))*a^4-3
*I/b^4*c*d^3*polylog(4,-exp(2*I*(b*x+a)))-3/b^3*c^2*d^2*polylog(3,-exp(2*I*(b*x+a)))-3/b^3*d^4*polylog(3,-exp(
2*I*(b*x+a)))*x^2-4/b*c*d^3*ln(exp(2*I*(b*x+a))+1)*x^3+4/b*c*d^3*ln(1-exp(I*(b*x+a)))*x^3-1/b*c^4*ln(exp(2*I*(
b*x+a))+1)+4/b^4*c*d^3*ln(1-exp(I*(b*x+a)))*a^3-6/b*c^2*d^2*ln(exp(2*I*(b*x+a))+1)*x^2+4/b*c*d^3*ln(exp(I*(b*x
+a))+1)*x^3+6*I/b^2*c*d^3*polylog(2,-exp(2*I*(b*x+a)))*x^2+6*I/b^2*c^2*d^2*polylog(2,-exp(2*I*(b*x+a)))*x+12/b
^3*d^4*polylog(3,exp(I*(b*x+a)))*x^2+12/b^3*c^2*d^2*polylog(3,exp(I*(b*x+a)))+12/b^3*c^2*d^2*polylog(3,-exp(I*
(b*x+a)))+12/b^3*d^4*polylog(3,-exp(I*(b*x+a)))*x^2-6/b^3*c*d^3*polylog(3,-exp(2*I*(b*x+a)))*x-1/b*d^4*ln(exp(
2*I*(b*x+a))+1)*x^4+6/b*c^2*d^2*ln(1-exp(I*(b*x+a)))*x^2+1/b*d^4*ln(exp(I*(b*x+a))+1)*x^4-6/b^3*c^2*d^2*a^2*ln
(1-exp(I*(b*x+a)))+4/b*c^3*d*ln(1-exp(I*(b*x+a)))*x+4/b^2*c^3*d*ln(1-exp(I*(b*x+a)))*a+4/b*c^3*d*ln(exp(I*(b*x
+a))+1)*x+24/b^3*c*d^3*polylog(3,-exp(I*(b*x+a)))*x+6/b*c^2*d^2*ln(exp(I*(b*x+a))+1)*x^2+24/b^3*c*d^3*polylog(
3,exp(I*(b*x+a)))*x-4/b*c^3*d*ln(exp(2*I*(b*x+a))+1)*x+1/b^5*d^4*a^4*ln(exp(I*(b*x+a))-1)+6/b^3*c^2*d^2*a^2*ln
(exp(I*(b*x+a))-1)-4/b^4*c*d^3*a^3*ln(exp(I*(b*x+a))-1)-4/b^2*c^3*d*a*ln(exp(I*(b*x+a))-1)-4*I/b^2*c^3*d*polyl
og(2,exp(I*(b*x+a)))-4*I/b^2*c^3*d*polylog(2,-exp(I*(b*x+a)))+24*I/b^4*d^4*polylog(4,exp(I*(b*x+a)))*x-4*I/b^2
*d^4*polylog(2,exp(I*(b*x+a)))*x^3-4*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)))*x^3+24*I/b^4*d^4*polylog(4,-exp(I*(b
*x+a)))*x+24*I/b^4*c*d^3*polylog(4,exp(I*(b*x+a)))+24*I/b^4*c*d^3*polylog(4,-exp(I*(b*x+a)))-24*d^4*polylog(5,
-exp(I*(b*x+a)))/b^5-24*d^4*polylog(5,exp(I*(b*x+a)))/b^5-12*I/b^2*c*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-12*I/b
^2*c^2*d^2*polylog(2,exp(I*(b*x+a)))*x-12*I/b^2*c^2*d^2*polylog(2,-exp(I*(b*x+a)))*x+1/b*c^4*ln(exp(I*(b*x+a))
+1)+1/b*c^4*ln(exp(I*(b*x+a))-1)-12*I/b^2*c*d^3*polylog(2,exp(I*(b*x+a)))*x^2+2*I/b^2*c^3*d*polylog(2,-exp(2*I
*(b*x+a)))-3*I/b^4*d^4*polylog(4,-exp(2*I*(b*x+a)))*x+2*I/b^2*d^4*polylog(2,-exp(2*I*(b*x+a)))*x^3

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Maxima [B]  time = 2.50605, size = 2402, normalized size = 9.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a),x, algorithm="maxima")

[Out]

-1/6*(3*c^4*(log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2)) - 12*a*c^3*d*(log(sin(b*x + a)^2 - 1) - log(sin(b*
x + a)^2))/b + 18*a^2*c^2*d^2*(log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^2 - 12*a^3*c*d^3*(log(sin(b*x
+ a)^2 - 1) - log(sin(b*x + a)^2))/b^3 + 3*a^4*d^4*(log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^4 - (18*d
^4*polylog(5, -e^(2*I*b*x + 2*I*a)) - 144*d^4*polylog(5, -e^(I*b*x + I*a)) - 144*d^4*polylog(5, e^(I*b*x + I*a
)) - (12*I*(b*x + a)^4*d^4 + (32*I*b*c*d^3 - 32*I*a*d^4)*(b*x + a)^3 + (36*I*b^2*c^2*d^2 - 72*I*a*b*c*d^3 + 36
*I*a^2*d^4)*(b*x + a)^2 + (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 - 24*I*a^3*d^4)*(b*x + a))*a
rctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - (-6*I*(b*x + a)^4*d^4 + (-24*I*b*c*d^3 + 24*I*a*d^4)*(b*x + a
)^3 + (-36*I*b^2*c^2*d^2 + 72*I*a*b*c*d^3 - 36*I*a^2*d^4)*(b*x + a)^2 + (-24*I*b^3*c^3*d + 72*I*a*b^2*c^2*d^2
- 72*I*a^2*b*c*d^3 + 24*I*a^3*d^4)*(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - (6*I*(b*x + a)^4*d^4 +
(24*I*b*c*d^3 - 24*I*a*d^4)*(b*x + a)^3 + (36*I*b^2*c^2*d^2 - 72*I*a*b*c*d^3 + 36*I*a^2*d^4)*(b*x + a)^2 + (2
4*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 - 24*I*a^3*d^4)*(b*x + a))*arctan2(sin(b*x + a), -cos(b*
x + a) + 1) - (-12*I*b^3*c^3*d + 36*I*a*b^2*c^2*d^2 - 36*I*a^2*b*c*d^3 - 24*I*(b*x + a)^3*d^4 + 12*I*a^3*d^4 +
(-48*I*b*c*d^3 + 48*I*a*d^4)*(b*x + a)^2 + (-36*I*b^2*c^2*d^2 + 72*I*a*b*c*d^3 - 36*I*a^2*d^4)*(b*x + a))*dil
og(-e^(2*I*b*x + 2*I*a)) - (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 + 24*I*(b*x + a)^3*d^4 - 24
*I*a^3*d^4 + (72*I*b*c*d^3 - 72*I*a*d^4)*(b*x + a)^2 + (72*I*b^2*c^2*d^2 - 144*I*a*b*c*d^3 + 72*I*a^2*d^4)*(b*
x + a))*dilog(-e^(I*b*x + I*a)) - (24*I*b^3*c^3*d - 72*I*a*b^2*c^2*d^2 + 72*I*a^2*b*c*d^3 + 24*I*(b*x + a)^3*d
^4 - 24*I*a^3*d^4 + (72*I*b*c*d^3 - 72*I*a*d^4)*(b*x + a)^2 + (72*I*b^2*c^2*d^2 - 144*I*a*b*c*d^3 + 72*I*a^2*d
^4)*(b*x + a))*dilog(e^(I*b*x + I*a)) - 2*(3*(b*x + a)^4*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^
2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + 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*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^
4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*
b*c*d^3 - a^3*d^4)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + 3*((b*x + a)^4*d^4 +
4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^
2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - (2
4*I*b*c*d^3 + 36*I*(b*x + a)*d^4 - 24*I*a*d^4)*polylog(4, -e^(2*I*b*x + 2*I*a)) - (-144*I*b*c*d^3 - 144*I*(b*x
+ a)*d^4 + 144*I*a*d^4)*polylog(4, -e^(I*b*x + I*a)) - (-144*I*b*c*d^3 - 144*I*(b*x + a)*d^4 + 144*I*a*d^4)*p
olylog(4, e^(I*b*x + I*a)) - 6*(3*b^2*c^2*d^2 - 6*a*b*c*d^3 + 6*(b*x + a)^2*d^4 + 3*a^2*d^4 + 8*(b*c*d^3 - a*d
^4)*(b*x + a))*polylog(3, -e^(2*I*b*x + 2*I*a)) + 72*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 +
2*(b*c*d^3 - a*d^4)*(b*x + a))*polylog(3, -e^(I*b*x + I*a)) + 72*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4
+ a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*polylog(3, e^(I*b*x + I*a)))/b^4)/b

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Fricas [C]  time = 1.03994, size = 6280, normalized size = 25.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(24*d^4*polylog(5, cos(b*x + a) + I*sin(b*x + a)) + 24*d^4*polylog(5, cos(b*x + a) - I*sin(b*x + a)) - 24
*d^4*polylog(5, I*cos(b*x + a) + sin(b*x + a)) - 24*d^4*polylog(5, I*cos(b*x + a) - sin(b*x + a)) - 24*d^4*pol
ylog(5, -I*cos(b*x + a) + sin(b*x + a)) - 24*d^4*polylog(5, -I*cos(b*x + a) - sin(b*x + a)) + 24*d^4*polylog(5
, -cos(b*x + a) + I*sin(b*x + a)) + 24*d^4*polylog(5, -cos(b*x + a) - I*sin(b*x + a)) - (-4*I*b^3*d^4*x^3 - 12
*I*b^3*c*d^3*x^2 - 12*I*b^3*c^2*d^2*x - 4*I*b^3*c^3*d)*dilog(cos(b*x + a) + I*sin(b*x + a)) - (4*I*b^3*d^4*x^3
+ 12*I*b^3*c*d^3*x^2 + 12*I*b^3*c^2*d^2*x + 4*I*b^3*c^3*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) - (-4*I*b^3*d
^4*x^3 - 12*I*b^3*c*d^3*x^2 - 12*I*b^3*c^2*d^2*x - 4*I*b^3*c^3*d)*dilog(I*cos(b*x + a) + sin(b*x + a)) - (4*I*
b^3*d^4*x^3 + 12*I*b^3*c*d^3*x^2 + 12*I*b^3*c^2*d^2*x + 4*I*b^3*c^3*d)*dilog(I*cos(b*x + a) - sin(b*x + a)) -
(4*I*b^3*d^4*x^3 + 12*I*b^3*c*d^3*x^2 + 12*I*b^3*c^2*d^2*x + 4*I*b^3*c^3*d)*dilog(-I*cos(b*x + a) + sin(b*x +
a)) - (-4*I*b^3*d^4*x^3 - 12*I*b^3*c*d^3*x^2 - 12*I*b^3*c^2*d^2*x - 4*I*b^3*c^3*d)*dilog(-I*cos(b*x + a) - sin
(b*x + a)) - (4*I*b^3*d^4*x^3 + 12*I*b^3*c*d^3*x^2 + 12*I*b^3*c^2*d^2*x + 4*I*b^3*c^3*d)*dilog(-cos(b*x + a) +
I*sin(b*x + a)) - (-4*I*b^3*d^4*x^3 - 12*I*b^3*c*d^3*x^2 - 12*I*b^3*c^2*d^2*x - 4*I*b^3*c^3*d)*dilog(-cos(b*x
+ a) - I*sin(b*x + a)) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*log(co
s(b*x + a) + I*sin(b*x + a) + 1) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log
(cos(b*x + a) + I*sin(b*x + a) + I) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4
*c^4)*log(cos(b*x + a) - I*sin(b*x + a) + 1) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)*log(cos(b*x + a) - I*sin(b*x + a) + I) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c
^3*d*x + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(I*cos(b*x + a) + sin(b*x + a) + 1) +
(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^
3*b*c*d^3 - a^4*d^4)*log(I*cos(b*x + a) - sin(b*x + a) + 1) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x
^2 + 4*b^4*c^3*d*x + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(-I*cos(b*x + a) + sin(b*
x + a) + 1) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + 4*a*b^3*c^3*d - 6*a^2*b^2*c
^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b
^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - (b^4*c^4 - 4*a*b^3*c
^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - (b^4*d
^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d
^3 - a^4*d^4)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b
*c*d^3 + a^4*d^4)*log(-cos(b*x + a) + I*sin(b*x + a) + I) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2
+ 4*b^4*c^3*d*x + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(-cos(b*x + a) - I*sin(b*x
+ a) + 1) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(-cos(b*x + a) - I*sin(
b*x + a) + I) - (24*I*b*d^4*x + 24*I*b*c*d^3)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - (-24*I*b*d^4*x - 24*
I*b*c*d^3)*polylog(4, cos(b*x + a) - I*sin(b*x + a)) - (24*I*b*d^4*x + 24*I*b*c*d^3)*polylog(4, I*cos(b*x + a)
+ sin(b*x + a)) - (-24*I*b*d^4*x - 24*I*b*c*d^3)*polylog(4, I*cos(b*x + a) - sin(b*x + a)) - (-24*I*b*d^4*x -
24*I*b*c*d^3)*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) - (24*I*b*d^4*x + 24*I*b*c*d^3)*polylog(4, -I*cos(b*
x + a) - sin(b*x + a)) - (-24*I*b*d^4*x - 24*I*b*c*d^3)*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) - (24*I*b*d
^4*x + 24*I*b*c*d^3)*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^
2)*polylog(3, cos(b*x + a) + I*sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, cos(b
*x + a) - I*sin(b*x + a)) + 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, I*cos(b*x + a) + sin(b*x
+ a)) + 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, I*cos(b*x + a) - sin(b*x + a)) + 12*(b^2*d^
4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) + 12*(b^2*d^4*x^2 + 2*b^2*c*d^
3*x + b^2*c^2*d^2)*polylog(3, -I*cos(b*x + a) - sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)
*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, -cos(b
*x + a) - I*sin(b*x + a)))/b^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4*csc(b*x+a)*sec(b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{4} \csc \left (b x + a\right ) \sec \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^4*csc(b*x + a)*sec(b*x + a), x)