\(\int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx\) [236]

Optimal result

Integrand size = 22, antiderivative size = 226 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3} \]

[Out]

-2*I*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b-4*d*(d*x+c)*arctanh(exp(I*(b*x+a)))/b^2-(d*x+c)^2*csc(b*x+a)/b+2*I*d^2
*polylog(2,-exp(I*(b*x+a)))/b^3+2*I*d*(d*x+c)*polylog(2,-I*exp(I*(b*x+a)))/b^2-2*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^3-2*d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3+2*d^2*polylog(3,I*
exp(I*(b*x+a)))/b^3

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {2701, 327, 213, 4505, 6873, 12, 6874, 6408, 4266, 2611, 2320, 6724, 4268, 2317, 2438} \[ \int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b} \]

[In]

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

[Out]

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

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 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 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 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 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2 \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b}-(2 d) \int (c+d x) \left (\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\csc (a+b x)}{b}\right ) \, dx \\ & = \frac {(c+d x)^2 \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b}-(2 d) \int \frac {(c+d x) (\text {arctanh}(\sin (a+b x))-\csc (a+b x))}{b} \, dx \\ & = \frac {(c+d x)^2 \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b}-\frac {(2 d) \int (c+d x) (\text {arctanh}(\sin (a+b x))-\csc (a+b x)) \, dx}{b} \\ & = \frac {(c+d x)^2 \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b}-\frac {(2 d) \int ((c+d x) \text {arctanh}(\sin (a+b x))-(c+d x) \csc (a+b x)) \, dx}{b} \\ & = \frac {(c+d x)^2 \text {arctanh}(\sin (a+b x))}{b}-\frac {(c+d x)^2 \csc (a+b x)}{b}-\frac {(2 d) \int (c+d x) \text {arctanh}(\sin (a+b x)) \, dx}{b}+\frac {(2 d) \int (c+d x) \csc (a+b x) \, dx}{b} \\ & = -\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {\int b (c+d x)^2 \sec (a+b x) \, dx}{b}-\frac {\left (2 d^2\right ) \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (2 d^2\right ) \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {\left (2 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 (2 i d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\int (c+d x)^2 \sec (a+b x) \, dx \\ & = -\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\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 {(2 d) \int (c+d x) \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac {(2 d) \int (c+d x) \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b} \\ & = -\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {2 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 {\left (2 i d^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (2 i d^2\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {2 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 {\left (2 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 (2 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 {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {4 d (c+d x) \text {arctanh}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \csc (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {2 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 {2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(593\) vs. \(2(226)=452\).

Time = 6.28 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.62 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx=-\frac {(c+d x)^2 \csc (a)}{b}+\frac {-2 i b^2 c^2 \arctan \left (e^{i (a+b x)}\right )+2 b^2 c d x \log \left (1-i e^{i (a+b x)}\right )+b^2 d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )-2 b^2 c d x \log \left (1+i e^{i (a+b x)}\right )-b^2 d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )+2 i b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-2 i b d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {4 i c d \arctan \left (\frac {i \cos (a)-i \sin (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{b^2 \sqrt {\cos ^2(a)+\sin ^2(a)}}+\frac {\sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (-c^2 \sin \left (\frac {b x}{2}\right )-2 c d x \sin \left (\frac {b x}{2}\right )-d^2 x^2 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c^2 \sin \left (\frac {b x}{2}\right )+2 c d x \sin \left (\frac {b x}{2}\right )+d^2 x^2 \sin \left (\frac {b x}{2}\right )\right )}{2 b}+\frac {2 d^2 \left (-\frac {2 \arctan (\tan (a)) \text {arctanh}\left (\frac {-\cos (a)+\sin (a) \tan \left (\frac {b x}{2}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}\right )}{\sqrt {\cos ^2(a)+\sin ^2(a)}}+\frac {\left ((b x+\arctan (\tan (a))) \left (\log \left (1-e^{i (b x+\arctan (\tan (a)))}\right )-\log \left (1+e^{i (b x+\arctan (\tan (a)))}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (b x+\arctan (\tan (a)))}\right )-\operatorname {PolyLog}\left (2,e^{i (b x+\arctan (\tan (a)))}\right )\right )\right ) \sec (a)}{\sqrt {1+\tan ^2(a)}}\right )}{b^3} \]

[In]

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

[Out]

-(((c + d*x)^2*Csc[a])/b) + ((-2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] +
b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*
(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a
+ b*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + ((4*I)*c*d*ArcT
an[(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) + (2*d^2*((-2*ArcTan[Ta
n[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 + ArcTan[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. 555 vs. \(2 (201 ) = 402\).

Time = 1.61 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.46

method	result	size
risch	\(\frac {2 i d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {2 c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{3}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b^{2}}+\frac {2 c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {2 i c d \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}-\frac {2 i c^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b}-\frac {2 d^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b}-\frac {2 i c d \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {a^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {a^{2} d^{2} \ln \left (1-i {\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 {2 d^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {2 i d^{2} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{3}}-\frac {2 i d^{2} a^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {4 i c d a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 i \left (x^{2} d^{2}+2 c d x +c^{2}\right ) {\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}+\frac {2 i d^{2} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 i d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}\)	\(556\)

[In]

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

[Out]

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

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. 1067 vs. \(2 (188) = 376\).

Time = 0.32 (sec) , antiderivative size = 1067, normalized size of antiderivative = 4.72 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1638 vs. \(2 (188) = 376\).

Time = 0.50 (sec) , antiderivative size = 1638, normalized size of antiderivative = 7.25 \[ \int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(c^2*(2/sin(b*x + a) - log(sin(b*x + a) + 1) + log(sin(b*x + a) - 1)) - 2*a*c*d*(2/sin(b*x + a) - log(sin
(b*x + a) + 1) + log(sin(b*x + a) - 1))/b + a^2*d^2*(2/sin(b*x + a) - log(sin(b*x + a) + 1) + log(sin(b*x + a)
- 1))/b^2 - 2*(2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x +
a))*cos(2*b*x + 2*a) - (I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*
x + a), sin(b*x + a) + 1) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - ((b*x + a)^2*d^2 + 2*(b*c*d - a
*d^2)*(b*x + a))*cos(2*b*x + 2*a) - (I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*sin(2*b*x + 2*a))*ar
ctan2(cos(b*x + a), -sin(b*x + a) + 1) + 4*(b*c*d + (b*x + a)*d^2 - a*d^2 - (b*c*d + (b*x + a)*d^2 - a*d^2)*co
s(2*b*x + 2*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) - 4*(b*c*d - a*d^2 - (b*c*d - a*d^2)*cos(2*b*x + 2*a) + (-I*b*c*d + I*a*d^2)*sin(2*b*x + 2*a))*arctan2(sin(
b*x + a), cos(b*x + a) - 1) - 4*((b*x + a)*d^2*cos(2*b*x + 2*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*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(b*x + a
) + 4*(b*c*d + (b*x + a)*d^2 - a*d^2 - (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(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 4*(b*c*d + (b*x + a)*d^2 - a*d^2 - (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(2*b*x + 2*a))*dilog(-I*e^(I*
b*x + I*a)) + 4*(d^2*cos(2*b*x + 2*a) + I*d^2*sin(2*b*x + 2*a) - d^2)*dilog(-e^(I*b*x + I*a)) - 4*(d^2*cos(2*b
*x + 2*a) + I*d^2*sin(2*b*x + 2*a) - d^2)*dilog(e^(I*b*x + I*a)) - 2*(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(2*b*x + 2*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) - 2*(-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(2*b*x + 2*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) + (I*(b*x + a)^2*d^2 - 2*(-I*b*c*d + I*a*d^2)*(b*x + a) + (-I*(b*x
+ a)^2*d^2 - 2*(I*b*c*d - I*a*d^2)*(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a
))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + (-I*(b*x + a)^2*d^2 - 2*(I*b*
c*d - I*a*d^2)*(b*x + a) + (I*(b*x + a)^2*d^2 - 2*(-I*b*c*d + I*a*d^2)*(b*x + a))*cos(2*b*x + 2*a) - ((b*x + a
)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a)
+ 1) - 4*(I*d^2*cos(2*b*x + 2*a) - d^2*sin(2*b*x + 2*a) - I*d^2)*polylog(3, I*e^(I*b*x + I*a)) - 4*(-I*d^2*cos
(2*b*x + 2*a) + d^2*sin(2*b*x + 2*a) + I*d^2)*polylog(3, -I*e^(I*b*x + I*a)) - 4*(I*(b*x + a)^2*d^2 + 2*(I*b*c
*d - I*a*d^2)*(b*x + a))*sin(b*x + a))/(-2*I*b^2*cos(2*b*x + 2*a) + 2*b^2*sin(2*b*x + 2*a) + 2*I*b^2))/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}