\(\int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx\) [318]

Optimal result

Integrand size = 24, antiderivative size = 341 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {3 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b} \]

[Out]

-3*I*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b+2*d^2*x*arctanh(exp(I*(b*x+a)))/b^2-6*d*(d*x+c)*arctanh(exp(I*(b*x+a))
)/b^2-d^2*x*arctanh(cos(b*x+a))/b^2+d*(d*x+c)*arctanh(cos(b*x+a))/b^2+d^2*arctanh(sin(b*x+a))/b^3-3/2*(d*x+c)^
2*csc(b*x+a)/b+2*I*d^2*polylog(2,-exp(I*(b*x+a)))/b^3+3*I*d*(d*x+c)*polylog(2,-I*exp(I*(b*x+a)))/b^2-3*I*d*(d*
x+c)*polylog(2,I*exp(I*(b*x+a)))/b^2-2*I*d^2*polylog(2,exp(I*(b*x+a)))/b^3-3*d^2*polylog(3,-I*exp(I*(b*x+a)))/
b^3+3*d^2*polylog(3,I*exp(I*(b*x+a)))/b^3-d*(d*x+c)*sec(b*x+a)/b^2+1/2*(d*x+c)^2*csc(b*x+a)*sec(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {2701, 294, 327, 213, 4505, 6820, 12, 6874, 6408, 4266, 2611, 2320, 6724, 4268, 2317, 2438, 2702, 6406, 3855} \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {3 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \sec (a+b x)}{b^2}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b} \]

[In]

Int[(c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x]^3,x]

[Out]

((-3*I)*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b + (2*d^2*x*ArcTanh[E^(I*(a + b*x))])/b^2 - (6*d*(c + d*x)*ArcTa
nh[E^(I*(a + b*x))])/b^2 - (d^2*x*ArcTanh[Cos[a + b*x]])/b^2 + (d*(c + d*x)*ArcTanh[Cos[a + b*x]])/b^2 + (d^2*
ArcTanh[Sin[a + b*x]])/b^3 - (3*(c + d*x)^2*Csc[a + b*x])/(2*b) + ((2*I)*d^2*PolyLog[2, -E^(I*(a + b*x))])/b^3
+ ((3*I)*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - ((3*I)*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))]
)/b^2 - ((2*I)*d^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (3*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (3*d^2*Po
lyLog[3, I*E^(I*(a + b*x))])/b^3 - (d*(c + d*x)*Sec[a + b*x])/b^2 + ((c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^2)/
(2*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 2320

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 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 2611

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 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3855

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

Rule 4266

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

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 4505

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Modul
e[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u
, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 6406

Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 - u^2)), x], x] /; I
nverseFunctionFreeQ[u, x]

Rule 6408

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan
h[u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 - u^2)), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 6724

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]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-(2 d) \int (c+d x) \left (\frac {3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 \csc (a+b x)}{2 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{2 b}\right ) \, dx \\ & = \frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-(2 d) \int \frac {(c+d x) \left (3 \text {arctanh}(\sin (a+b x))+\csc (a+b x) \left (-3+\sec ^2(a+b x)\right )\right )}{2 b} \, dx \\ & = \frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {d \int (c+d x) \left (3 \text {arctanh}(\sin (a+b x))+\csc (a+b x) \left (-3+\sec ^2(a+b x)\right )\right ) \, dx}{b} \\ & = \frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {d \int \left (3 (c+d x) (\text {arctanh}(\sin (a+b x))-\csc (a+b x))+(c+d x) \csc (a+b x) \sec ^2(a+b x)\right ) \, dx}{b} \\ & = \frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {d \int (c+d x) \csc (a+b x) \sec ^2(a+b x) \, dx}{b}-\frac {(3 d) \int (c+d x) (\text {arctanh}(\sin (a+b x))-\csc (a+b x)) \, dx}{b} \\ & = \frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {(3 d) \int ((c+d x) \text {arctanh}(\sin (a+b x))-(c+d x) \csc (a+b x)) \, dx}{b}+\frac {d^2 \int \left (-\frac {\text {arctanh}(\cos (a+b x))}{b}+\frac {\sec (a+b x)}{b}\right ) \, dx}{b} \\ & = \frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {3 (c+d x)^2 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x) \text {arctanh}(\sin (a+b x)) \, dx}{b}+\frac {(3 d) \int (c+d x) \csc (a+b x) \, dx}{b}-\frac {d^2 \int \text {arctanh}(\cos (a+b x)) \, dx}{b^2}+\frac {d^2 \int \sec (a+b x) \, dx}{b^2} \\ & = -\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}+\frac {3 \int b (c+d x)^2 \sec (a+b x) \, dx}{2 b}-\frac {d^2 \int b x \csc (a+b x) \, dx}{b^2}-\frac {\left (3 d^2\right ) \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 d^2\right ) \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}+\frac {3}{2} \int (c+d x)^2 \sec (a+b x) \, dx+\frac {\left (3 i d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (3 i d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {d^2 \int x \csc (a+b x) \, dx}{b} \\ & = -\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {3 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x) \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x) \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}+\frac {d^2 \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {d^2 \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {3 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {\left (i d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\frac {\left (i d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (3 i d^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 i d^2\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3} \\ & = -\frac {3 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d^2 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {d^2 x \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d (c+d x) \text {arctanh}(\cos (a+b x))}{b^2}+\frac {d^2 \text {arctanh}(\sin (a+b x))}{b^3}-\frac {3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {3 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {d (c+d x) \sec (a+b x)}{b^2}+\frac {(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(889\) vs. \(2(341)=682\).

Time = 7.33 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.61 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {6 i b^2 c^2 \arctan \left (e^{i (a+b x)}\right )+4 i d^2 \arctan \left (e^{i (a+b x)}\right )-6 b^2 c d x \log \left (1-i e^{i (a+b x)}\right )-3 b^2 d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+6 b^2 c d x \log \left (1+i e^{i (a+b x)}\right )+3 b^2 d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-6 i b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+6 i b d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+6 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{2 b^3}-\frac {(c+d x) \csc (a) \sec (a) (b c \cos (a)+b d x \cos (a)+d \sin (a))}{b^2}+\frac {4 i c d \arctan \left (\frac {i \cos (a)-i \sin (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{b^2 \sqrt {\cos ^2(a)+\sin ^2(a)}}+\frac {\sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (-c^2 \sin \left (\frac {b x}{2}\right )-2 c d x \sin \left (\frac {b x}{2}\right )-d^2 x^2 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c^2 \sin \left (\frac {b x}{2}\right )+2 c d x \sin \left (\frac {b x}{2}\right )+d^2 x^2 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {c^2+2 c d x+d^2 x^2}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {-c d \sin \left (\frac {b x}{2}\right )-d^2 x \sin \left (\frac {b x}{2}\right )}{b^2 \left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}+\frac {-c^2-2 c d x-d^2 x^2}{4 b \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )^2}+\frac {c d \sin \left (\frac {b x}{2}\right )+d^2 x \sin \left (\frac {b x}{2}\right )}{b^2 \left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )+\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}+\frac {2 d^2 \left (-\frac {2 \arctan (\tan (a)) \text {arctanh}\left (\frac {-\cos (a)+\sin (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}+\frac {\left ((b x+\arctan (\tan (a))) \left (\log \left (1-e^{i (b x+\arctan (\tan (a)))}\right )-\log \left (1+e^{i (b x+\arctan (\tan (a)))}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (b x+\arctan (\tan (a)))}\right )-\operatorname {PolyLog}\left (2,e^{i (b x+\arctan (\tan (a)))}\right )\right )\right ) \sec (a)}{\sqrt {1+\tan ^2(a)}}\right )}{b^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 769 vs. \(2 (310 ) = 620\).

Time = 1.05 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.26

method	result	size
risch	\(\frac {3 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{2 b}-\frac {3 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{2 b}-\frac {2 i d^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 d c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{2}}+\frac {3 i d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {3 c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {3 d^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{2}}+\frac {6 i c d a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {2 i d^{2} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 a^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{3}}+\frac {3 a^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{3}}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}+\frac {2 d c \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}-\frac {3 i c^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {2 i d^{2} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{3}}-\frac {3 i d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {3 c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {3 i c d \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {3 d^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {i \left (3 x^{2} d^{2} b \,{\mathrm e}^{5 i \left (x b +a \right )}+6 c d x b \,{\mathrm e}^{5 i \left (x b +a \right )}+3 c^{2} b \,{\mathrm e}^{5 i \left (x b +a \right )}+2 x^{2} d^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}+4 c d x b \,{\mathrm e}^{3 i \left (x b +a \right )}-2 i d^{2} x \,{\mathrm e}^{5 i \left (x b +a \right )}+2 c^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}+3 x^{2} d^{2} b \,{\mathrm e}^{i \left (x b +a \right )}-2 i c d \,{\mathrm e}^{5 i \left (x b +a \right )}+6 c d x b \,{\mathrm e}^{i \left (x b +a \right )}+3 c^{2} b \,{\mathrm e}^{i \left (x b +a \right )}+2 i d^{2} x \,{\mathrm e}^{i \left (x b +a \right )}+2 i d c \,{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}-\frac {3 i d^{2} a^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 i c d \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {3 c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}+\frac {3 c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}\)	\(770\)

[In]

int((d*x+c)^2*csc(b*x+a)^2*sec(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-2*d^2/b^3*a*ln(exp(I*(b*x+a))-1)-3*I/b*c^2*arctan(exp(I*(b*x+a)))+3/2/b*d^2*ln(1-I*exp(I*(b*x+a)))*x^2-3/2/b*
d^2*ln(1+I*exp(I*(b*x+a)))*x^2-2*d/b^2*c*ln(exp(I*(b*x+a))+1)+2*d/b^2*c*ln(exp(I*(b*x+a))-1)-2*d^2/b^2*ln(exp(
I*(b*x+a))+1)*x-3*d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3+3*d^2*polylog(3,I*exp(I*(b*x+a)))/b^3+6*I/b^2*c*d*a*arc
tan(exp(I*(b*x+a)))-2*I/b^3*d^2*arctan(exp(I*(b*x+a)))+2*I/b^3*d^2*dilog(exp(I*(b*x+a)))+2*I/b^3*d^2*dilog(exp
(I*(b*x+a))+1)-3*I/b^3*d^2*a^2*arctan(exp(I*(b*x+a)))-3*I/b^2*d^2*polylog(2,I*exp(I*(b*x+a)))*x-3*I/b^2*c*d*po
lylog(2,I*exp(I*(b*x+a)))+3*I/b^2*c*d*polylog(2,-I*exp(I*(b*x+a)))+3*I/b^2*d^2*polylog(2,-I*exp(I*(b*x+a)))*x+
3/b^2*c*d*ln(1-I*exp(I*(b*x+a)))*a-3/b^2*c*d*ln(1+I*exp(I*(b*x+a)))*a-3/b*c*d*ln(1+I*exp(I*(b*x+a)))*x+3/b*c*d
*ln(1-I*exp(I*(b*x+a)))*x-I/b^2/(exp(2*I*(b*x+a))+1)^2/(exp(2*I*(b*x+a))-1)*(3*x^2*d^2*b*exp(5*I*(b*x+a))+6*c*
d*x*b*exp(5*I*(b*x+a))+3*c^2*b*exp(5*I*(b*x+a))+2*x^2*d^2*b*exp(3*I*(b*x+a))+4*c*d*x*b*exp(3*I*(b*x+a))-2*I*d^
2*x*exp(5*I*(b*x+a))+2*c^2*b*exp(3*I*(b*x+a))+3*x^2*d^2*b*exp(I*(b*x+a))-2*I*c*d*exp(5*I*(b*x+a))+6*c*d*x*b*ex
p(I*(b*x+a))+3*c^2*b*exp(I*(b*x+a))+2*I*d^2*x*exp(I*(b*x+a))+2*I*d*c*exp(I*(b*x+a)))-3/2/b^3*a^2*d^2*ln(1-I*ex
p(I*(b*x+a)))+3/2/b^3*a^2*d^2*ln(1+I*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. 1366 vs. \(2 (297) = 594\).

Time = 0.37 (sec) , antiderivative size = 1366, normalized size of antiderivative = 4.01 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\int \left (c + d x\right )^{2} \csc ^{2}{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral((c + d*x)**2*csc(a + b*x)**2*sec(a + b*x)**3, 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. 3828 vs. \(2 (297) = 594\).

Time = 1.12 (sec) , antiderivative size = 3828, normalized size of antiderivative = 11.23 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(c^2*(2*(3*sin(b*x + a)^2 - 2)/(sin(b*x + a)^3 - sin(b*x + a)) - 3*log(sin(b*x + a) + 1) + 3*log(sin(b*x
+ a) - 1)) - 2*a*c*d*(2*(3*sin(b*x + a)^2 - 2)/(sin(b*x + a)^3 - sin(b*x + a)) - 3*log(sin(b*x + a) + 1) + 3*l
og(sin(b*x + a) - 1))/b + a^2*d^2*(2*(3*sin(b*x + a)^2 - 2)/(sin(b*x + a)^3 - sin(b*x + a)) - 3*log(sin(b*x +
a) + 1) + 3*log(sin(b*x + a) - 1))/b^2 - 4*(2*(3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2 - (3*(b
*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(6*b*x + 6*a) - (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)
*(b*x + a) + 2*d^2)*cos(4*b*x + 4*a) + (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(2*b*x + 2
*a) - (3*I*(b*x + a)^2*d^2 + 6*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2)*sin(6*b*x + 6*a) - (3*I*(b*x + a)^2*d^
2 + 6*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2)*sin(4*b*x + 4*a) - (-3*I*(b*x + a)^2*d^2 + 6*(-I*b*c*d + I*a*d^
2)*(b*x + a) - 2*I*d^2)*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 1) + 2*(3*(b*x + a)^2*d^2 + 6*(
b*c*d - a*d^2)*(b*x + a) + 2*d^2 - (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(6*b*x + 6*a)
- (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*cos(4*b*x + 4*a) + (3*(b*x + a)^2*d^2 + 6*(b*c*d -
a*d^2)*(b*x + a) + 2*d^2)*cos(2*b*x + 2*a) - (3*I*(b*x + a)^2*d^2 + 6*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2
)*sin(6*b*x + 6*a) - (3*I*(b*x + a)^2*d^2 + 6*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2)*sin(4*b*x + 4*a) - (-3*
I*(b*x + a)^2*d^2 + 6*(-I*b*c*d + I*a*d^2)*(b*x + a) - 2*I*d^2)*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), -sin(b
*x + a) + 1) + 8*(b*c*d + (b*x + a)*d^2 - a*d^2 - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(6*b*x + 6*a) - (b*c*d +
(b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) - (I*b*c*d + I*(b*x
+ a)*d^2 - I*a*d^2)*sin(6*b*x + 6*a) - (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(4*b*x + 4*a) - (-I*b*c*d - I
*(b*x + a)*d^2 + I*a*d^2)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 8*(b*c*d - a*d^2 - (b*c*
d - a*d^2)*cos(6*b*x + 6*a) - (b*c*d - a*d^2)*cos(4*b*x + 4*a) + (b*c*d - a*d^2)*cos(2*b*x + 2*a) + (-I*b*c*d
+ I*a*d^2)*sin(6*b*x + 6*a) + (-I*b*c*d + I*a*d^2)*sin(4*b*x + 4*a) + (I*b*c*d - I*a*d^2)*sin(2*b*x + 2*a))*ar
ctan2(sin(b*x + a), cos(b*x + a) - 1) - 8*((b*x + a)*d^2*cos(6*b*x + 6*a) + (b*x + a)*d^2*cos(4*b*x + 4*a) - (
b*x + a)*d^2*cos(2*b*x + 2*a) + I*(b*x + a)*d^2*sin(6*b*x + 6*a) + I*(b*x + a)*d^2*sin(4*b*x + 4*a) - I*(b*x +
a)*d^2*sin(2*b*x + 2*a) - (b*x + a)*d^2)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 4*(3*(b*x + a)^2*d^2 - 2*
I*b*c*d + 2*I*a*d^2 + 2*(3*b*c*d - (3*a + I)*d^2)*(b*x + a))*cos(5*b*x + 5*a) - 8*((b*x + a)^2*d^2 + 2*(b*c*d
- a*d^2)*(b*x + a))*cos(3*b*x + 3*a) - 4*(3*(b*x + a)^2*d^2 + 2*I*b*c*d - 2*I*a*d^2 + 2*(3*b*c*d - (3*a - I)*d
^2)*(b*x + a))*cos(b*x + a) + 12*(b*c*d + (b*x + a)*d^2 - a*d^2 - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(6*b*x +
6*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) - (
I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(6*b*x + 6*a) - (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(4*b*x + 4*a)
- (-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 12*(b*c*d + (b*x + a)*d
^2 - a*d^2 - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(6*b*x + 6*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*
a) + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) + (-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(6*b*x + 6*a
) + (-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(4*b*x + 4*a) + (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(2*b*x
+ 2*a))*dilog(-I*e^(I*b*x + I*a)) + 8*(d^2*cos(6*b*x + 6*a) + d^2*cos(4*b*x + 4*a) - d^2*cos(2*b*x + 2*a) + I*
d^2*sin(6*b*x + 6*a) + I*d^2*sin(4*b*x + 4*a) - I*d^2*sin(2*b*x + 2*a) - d^2)*dilog(-e^(I*b*x + I*a)) - 8*(d^2
*cos(6*b*x + 6*a) + d^2*cos(4*b*x + 4*a) - d^2*cos(2*b*x + 2*a) + I*d^2*sin(6*b*x + 6*a) + I*d^2*sin(4*b*x + 4
*a) - I*d^2*sin(2*b*x + 2*a) - d^2)*dilog(e^(I*b*x + I*a)) - 4*(I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2 + (-I*b*c*
d - I*(b*x + a)*d^2 + I*a*d^2)*cos(6*b*x + 6*a) + (-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*cos(4*b*x + 4*a) + (I
*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*cos(2*b*x + 2*a) + (b*c*d + (b*x + a)*d^2 - a*d^2)*sin(6*b*x + 6*a) + (b*c
*d + (b*x + a)*d^2 - a*d^2)*sin(4*b*x + 4*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*sin(2*b*x + 2*a))*log(cos(b*x +
a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - 4*(-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2 + (I*b*c*d + I*(b*x + a
)*d^2 - I*a*d^2)*cos(6*b*x + 6*a) + (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*cos(4*b*x + 4*a) + (-I*b*c*d - I*(b*
x + a)*d^2 + I*a*d^2)*cos(2*b*x + 2*a) - (b*c*d + (b*x + a)*d^2 - a*d^2)*sin(6*b*x + 6*a) - (b*c*d + (b*x + a)
*d^2 - a*d^2)*sin(4*b*x + 4*a) + (b*c*d + (b*x + a)*d^2 - a*d^2)*sin(2*b*x + 2*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^2 - 6*(-I*b*c*d + I*a*d^2)*(b*x + a) + 2*I*d^2 + (-3*I*(b*
x + a)^2*d^2 - 6*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d^2)*cos(6*b*x + 6*a) + (-3*I*(b*x + a)^2*d^2 - 6*(I*b*c*
d - I*a*d^2)*(b*x + a) - 2*I*d^2)*cos(4*b*x + 4*a) + (3*I*(b*x + a)^2*d^2 - 6*(-I*b*c*d + I*a*d^2)*(b*x + a) +
2*I*d^2)*cos(2*b*x + 2*a) + (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*sin(6*b*x + 6*a) + (3*(
b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*sin(4*b*x + 4*a) - (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2
)*(b*x + a) + 2*d^2)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + (-3*I*(b*x
+ a)^2*d^2 - 6*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d^2 + (3*I*(b*x + a)^2*d^2 - 6*(-I*b*c*d + I*a*d^2)*(b*x +
a) + 2*I*d^2)*cos(6*b*x + 6*a) + (3*I*(b*x + a)^2*d^2 - 6*(-I*b*c*d + I*a*d^2)*(b*x + a) + 2*I*d^2)*cos(4*b*x
+ 4*a) + (-3*I*(b*x + a)^2*d^2 - 6*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d^2)*cos(2*b*x + 2*a) - (3*(b*x + a)^2*
d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*sin(6*b*x + 6*a) - (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a)
+ 2*d^2)*sin(4*b*x + 4*a) + (3*(b*x + a)^2*d^2 + 6*(b*c*d - a*d^2)*(b*x + a) + 2*d^2)*sin(2*b*x + 2*a))*log(c
os(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) - 12*(I*d^2*cos(6*b*x + 6*a) + I*d^2*cos(4*b*x + 4*a) - I
*d^2*cos(2*b*x + 2*a) - d^2*sin(6*b*x + 6*a) - d^2*sin(4*b*x + 4*a) + d^2*sin(2*b*x + 2*a) - I*d^2)*polylog(3,
I*e^(I*b*x + I*a)) - 12*(-I*d^2*cos(6*b*x + 6*a) - I*d^2*cos(4*b*x + 4*a) + I*d^2*cos(2*b*x + 2*a) + d^2*sin(
6*b*x + 6*a) + d^2*sin(4*b*x + 4*a) - d^2*sin(2*b*x + 2*a) + I*d^2)*polylog(3, -I*e^(I*b*x + I*a)) - 4*(3*I*(b
*x + a)^2*d^2 + 2*b*c*d - 2*a*d^2 + 2*(3*I*b*c*d + (-3*I*a + 1)*d^2)*(b*x + a))*sin(5*b*x + 5*a) - 8*(I*(b*x +
a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*sin(3*b*x + 3*a) - 4*(3*I*(b*x + a)^2*d^2 - 2*b*c*d + 2*a*d^2 + 2
*(3*I*b*c*d + (-3*I*a - 1)*d^2)*(b*x + a))*sin(b*x + a))/(-4*I*b^2*cos(6*b*x + 6*a) - 4*I*b^2*cos(4*b*x + 4*a)
+ 4*I*b^2*cos(2*b*x + 2*a) + 4*b^2*sin(6*b*x + 6*a) + 4*b^2*sin(4*b*x + 4*a) - 4*b^2*sin(2*b*x + 2*a) + 4*I*b
^2))/b

Giac [F]

\[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \csc \left (b x + a\right )^{2} \sec \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \]

[In]

int((c + d*x)^2/(cos(a + b*x)^3*sin(a + b*x)^2),x)

[Out]

\text{Hanged}