\(\int x^5 (a+b \csc (c+d x^2))^2 \, dx\) [8]

Optimal result

Integrand size = 18, antiderivative size = 228 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=-\frac {i b^2 x^4}{2 d}+\frac {a^2 x^6}{6}-\frac {2 a b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b^2 x^4 \cot \left (c+d x^2\right )}{2 d}+\frac {b^2 x^2 \log \left (1-e^{2 i \left (c+d x^2\right )}\right )}{d^2}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b^2 \operatorname {PolyLog}\left (2,e^{2 i \left (c+d x^2\right )}\right )}{2 d^3}-\frac {2 a b \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,e^{i \left (c+d x^2\right )}\right )}{d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {4290, 4275, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438} \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^6}{6}-\frac {2 a b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {2 a b \operatorname {PolyLog}\left (3,-e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {2 a b \operatorname {PolyLog}\left (3,e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {2 i a b x^2 \operatorname {PolyLog}\left (2,e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {i b^2 \operatorname {PolyLog}\left (2,e^{2 i \left (d x^2+c\right )}\right )}{2 d^3}+\frac {b^2 x^2 \log \left (1-e^{2 i \left (c+d x^2\right )}\right )}{d^2}-\frac {b^2 x^4 \cot \left (c+d x^2\right )}{2 d}-\frac {i b^2 x^4}{2 d} \]

[In]

Int[x^5*(a + b*Csc[c + d*x^2])^2,x]

[Out]

((-1/2*I)*b^2*x^4)/d + (a^2*x^6)/6 - (2*a*b*x^4*ArcTanh[E^(I*(c + d*x^2))])/d - (b^2*x^4*Cot[c + d*x^2])/(2*d)
+ (b^2*x^2*Log[1 - E^((2*I)*(c + d*x^2))])/d^2 + ((2*I)*a*b*x^2*PolyLog[2, -E^(I*(c + d*x^2))])/d^2 - ((2*I)*
a*b*x^2*PolyLog[2, E^(I*(c + d*x^2))])/d^2 - ((I/2)*b^2*PolyLog[2, E^((2*I)*(c + d*x^2))])/d^3 - (2*a*b*PolyLo
g[3, -E^(I*(c + d*x^2))])/d^3 + (2*a*b*PolyLog[3, E^(I*(c + d*x^2))])/d^3

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 3798

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 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 4269

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 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4290

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

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]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(639\) vs. \(2(228)=456\).

Time = 3.30 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.80 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {-12 i b^2 d^2 x^4-2 a^2 d^3 x^6+2 a^2 d^3 e^{2 i c} x^6-12 b^2 d x^2 \log \left (1-e^{-i \left (c+d x^2\right )}\right )+12 b^2 d e^{2 i c} x^2 \log \left (1-e^{-i \left (c+d x^2\right )}\right )-12 a b d^2 x^4 \log \left (1-e^{-i \left (c+d x^2\right )}\right )+12 a b d^2 e^{2 i c} x^4 \log \left (1-e^{-i \left (c+d x^2\right )}\right )-12 b^2 d x^2 \log \left (1+e^{-i \left (c+d x^2\right )}\right )+12 b^2 d e^{2 i c} x^2 \log \left (1+e^{-i \left (c+d x^2\right )}\right )+12 a b d^2 x^4 \log \left (1+e^{-i \left (c+d x^2\right )}\right )-12 a b d^2 e^{2 i c} x^4 \log \left (1+e^{-i \left (c+d x^2\right )}\right )+12 i b \left (-1+e^{2 i c}\right ) \left (b-2 a d x^2\right ) \operatorname {PolyLog}\left (2,-e^{-i \left (c+d x^2\right )}\right )+12 i b \left (-1+e^{2 i c}\right ) \left (b+2 a d x^2\right ) \operatorname {PolyLog}\left (2,e^{-i \left (c+d x^2\right )}\right )+24 a b \operatorname {PolyLog}\left (3,-e^{-i \left (c+d x^2\right )}\right )-24 a b e^{2 i c} \operatorname {PolyLog}\left (3,-e^{-i \left (c+d x^2\right )}\right )-24 a b \operatorname {PolyLog}\left (3,e^{-i \left (c+d x^2\right )}\right )+24 a b e^{2 i c} \operatorname {PolyLog}\left (3,e^{-i \left (c+d x^2\right )}\right )-3 b^2 d^2 x^4 \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} \left (c+d x^2\right )\right ) \sin \left (\frac {d x^2}{2}\right )+3 b^2 d^2 e^{2 i c} x^4 \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} \left (c+d x^2\right )\right ) \sin \left (\frac {d x^2}{2}\right )-3 b^2 d^2 x^4 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} \left (c+d x^2\right )\right ) \sin \left (\frac {d x^2}{2}\right )+3 b^2 d^2 e^{2 i c} x^4 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} \left (c+d x^2\right )\right ) \sin \left (\frac {d x^2}{2}\right )}{12 d^3 \left (-1+e^{2 i c}\right )} \]

[In]

Integrate[x^5*(a + b*Csc[c + d*x^2])^2,x]

[Out]

((-12*I)*b^2*d^2*x^4 - 2*a^2*d^3*x^6 + 2*a^2*d^3*E^((2*I)*c)*x^6 - 12*b^2*d*x^2*Log[1 - E^((-I)*(c + d*x^2))]
+ 12*b^2*d*E^((2*I)*c)*x^2*Log[1 - E^((-I)*(c + d*x^2))] - 12*a*b*d^2*x^4*Log[1 - E^((-I)*(c + d*x^2))] + 12*a
*b*d^2*E^((2*I)*c)*x^4*Log[1 - E^((-I)*(c + d*x^2))] - 12*b^2*d*x^2*Log[1 + E^((-I)*(c + d*x^2))] + 12*b^2*d*E
^((2*I)*c)*x^2*Log[1 + E^((-I)*(c + d*x^2))] + 12*a*b*d^2*x^4*Log[1 + E^((-I)*(c + d*x^2))] - 12*a*b*d^2*E^((2
*I)*c)*x^4*Log[1 + E^((-I)*(c + d*x^2))] + (12*I)*b*(-1 + E^((2*I)*c))*(b - 2*a*d*x^2)*PolyLog[2, -E^((-I)*(c
+ d*x^2))] + (12*I)*b*(-1 + E^((2*I)*c))*(b + 2*a*d*x^2)*PolyLog[2, E^((-I)*(c + d*x^2))] + 24*a*b*PolyLog[3,
-E^((-I)*(c + d*x^2))] - 24*a*b*E^((2*I)*c)*PolyLog[3, -E^((-I)*(c + d*x^2))] - 24*a*b*PolyLog[3, E^((-I)*(c +
d*x^2))] + 24*a*b*E^((2*I)*c)*PolyLog[3, E^((-I)*(c + d*x^2))] - 3*b^2*d^2*x^4*Csc[c/2]*Csc[(c + d*x^2)/2]*Si
n[(d*x^2)/2] + 3*b^2*d^2*E^((2*I)*c)*x^4*Csc[c/2]*Csc[(c + d*x^2)/2]*Sin[(d*x^2)/2] - 3*b^2*d^2*x^4*Sec[c/2]*S
ec[(c + d*x^2)/2]*Sin[(d*x^2)/2] + 3*b^2*d^2*E^((2*I)*c)*x^4*Sec[c/2]*Sec[(c + d*x^2)/2]*Sin[(d*x^2)/2])/(12*d
^3*(-1 + E^((2*I)*c)))

Maple [F]

\[\int x^{5} {\left (a +b \csc \left (d \,x^{2}+c \right )\right )}^{2}d x\]

[In]

int(x^5*(a+b*csc(d*x^2+c))^2,x)

[Out]

int(x^5*(a+b*csc(d*x^2+c))^2,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. 687 vs. \(2 (195) = 390\).

Time = 0.28 (sec) , antiderivative size = 687, normalized size of antiderivative = 3.01 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {a^{2} d^{3} x^{6} \sin \left (d x^{2} + c\right ) - 3 \, b^{2} d^{2} x^{4} \cos \left (d x^{2} + c\right ) + 6 \, a b {\rm polylog}\left (3, \cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) + 6 \, a b {\rm polylog}\left (3, \cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 6 \, a b {\rm polylog}\left (3, -\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 6 \, a b {\rm polylog}\left (3, -\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 3 \, {\left (2 i \, a b d x^{2} + i \, b^{2}\right )} {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 3 \, {\left (-2 i \, a b d x^{2} - i \, b^{2}\right )} {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 3 \, {\left (2 i \, a b d x^{2} - i \, b^{2}\right )} {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 3 \, {\left (-2 i \, a b d x^{2} + i \, b^{2}\right )} {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 3 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right ) - 3 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right ) + 3 \, {\left (a b c^{2} - b^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) + 3 \, {\left (a b c^{2} - b^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) + 3 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right ) + 3 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - a b c^{2} + b^{2} c\right )} \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right )}{6 \, d^{3} \sin \left (d x^{2} + c\right )} \]

[In]

integrate(x^5*(a+b*csc(d*x^2+c))^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^{5} \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}\, dx \]

[In]

integrate(x**5*(a+b*csc(d*x**2+c))**2,x)

[Out]

Integral(x**5*(a + b*csc(c + d*x**2))**2, 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. 800 vs. \(2 (195) = 390\).

Time = 0.32 (sec) , antiderivative size = 800, normalized size of antiderivative = 3.51 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {1}{6} \, a^{2} x^{6} - \frac {2 \, b^{2} d^{2} x^{4} \cos \left (2 \, d x^{2} + 2 \, c\right ) + 2 i \, b^{2} d^{2} x^{4} \sin \left (2 \, d x^{2} + 2 \, c\right ) - 2 \, {\left (a b d^{2} x^{4} - b^{2} d x^{2} - {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (-i \, a b d^{2} x^{4} + i \, b^{2} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \arctan \left (\sin \left (d x^{2} + c\right ), \cos \left (d x^{2} + c\right ) + 1\right ) - 2 \, {\left (a b d^{2} x^{4} + b^{2} d x^{2} - {\left (a b d^{2} x^{4} + b^{2} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (-i \, a b d^{2} x^{4} - i \, b^{2} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \arctan \left (\sin \left (d x^{2} + c\right ), -\cos \left (d x^{2} + c\right ) + 1\right ) + 2 \, {\left (2 \, a b d x^{2} - b^{2} - {\left (2 \, a b d x^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (2 i \, a b d x^{2} - i \, b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} {\rm Li}_2\left (-e^{\left (i \, d x^{2} + i \, c\right )}\right ) - 2 \, {\left (2 \, a b d x^{2} + b^{2} - {\left (2 \, a b d x^{2} + b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (-2 i \, a b d x^{2} - i \, b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} {\rm Li}_2\left (e^{\left (i \, d x^{2} + i \, c\right )}\right ) + {\left (i \, a b d^{2} x^{4} - i \, b^{2} d x^{2} + {\left (-i \, a b d^{2} x^{4} + i \, b^{2} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (a b d^{2} x^{4} - b^{2} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left (\cos \left (d x^{2} + c\right )^{2} + \sin \left (d x^{2} + c\right )^{2} + 2 \, \cos \left (d x^{2} + c\right ) + 1\right ) + {\left (-i \, a b d^{2} x^{4} - i \, b^{2} d x^{2} + {\left (i \, a b d^{2} x^{4} + i \, b^{2} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) - {\left (a b d^{2} x^{4} + b^{2} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left (\cos \left (d x^{2} + c\right )^{2} + \sin \left (d x^{2} + c\right )^{2} - 2 \, \cos \left (d x^{2} + c\right ) + 1\right ) - 4 \, {\left (i \, a b \cos \left (2 \, d x^{2} + 2 \, c\right ) - a b \sin \left (2 \, d x^{2} + 2 \, c\right ) - i \, a b\right )} {\rm Li}_{3}(-e^{\left (i \, d x^{2} + i \, c\right )}) - 4 \, {\left (-i \, a b \cos \left (2 \, d x^{2} + 2 \, c\right ) + a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + i \, a b\right )} {\rm Li}_{3}(e^{\left (i \, d x^{2} + i \, c\right )})}{-2 i \, d^{3} \cos \left (2 \, d x^{2} + 2 \, c\right ) + 2 \, d^{3} \sin \left (2 \, d x^{2} + 2 \, c\right ) + 2 i \, d^{3}} \]

[In]

integrate(x^5*(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/6*a^2*x^6 - (2*b^2*d^2*x^4*cos(2*d*x^2 + 2*c) + 2*I*b^2*d^2*x^4*sin(2*d*x^2 + 2*c) - 2*(a*b*d^2*x^4 - b^2*d*
x^2 - (a*b*d^2*x^4 - b^2*d*x^2)*cos(2*d*x^2 + 2*c) + (-I*a*b*d^2*x^4 + I*b^2*d*x^2)*sin(2*d*x^2 + 2*c))*arctan
2(sin(d*x^2 + c), cos(d*x^2 + c) + 1) - 2*(a*b*d^2*x^4 + b^2*d*x^2 - (a*b*d^2*x^4 + b^2*d*x^2)*cos(2*d*x^2 + 2
*c) + (-I*a*b*d^2*x^4 - I*b^2*d*x^2)*sin(2*d*x^2 + 2*c))*arctan2(sin(d*x^2 + c), -cos(d*x^2 + c) + 1) + 2*(2*a
*b*d*x^2 - b^2 - (2*a*b*d*x^2 - b^2)*cos(2*d*x^2 + 2*c) - (2*I*a*b*d*x^2 - I*b^2)*sin(2*d*x^2 + 2*c))*dilog(-e
^(I*d*x^2 + I*c)) - 2*(2*a*b*d*x^2 + b^2 - (2*a*b*d*x^2 + b^2)*cos(2*d*x^2 + 2*c) + (-2*I*a*b*d*x^2 - I*b^2)*s
in(2*d*x^2 + 2*c))*dilog(e^(I*d*x^2 + I*c)) + (I*a*b*d^2*x^4 - I*b^2*d*x^2 + (-I*a*b*d^2*x^4 + I*b^2*d*x^2)*co
s(2*d*x^2 + 2*c) + (a*b*d^2*x^4 - b^2*d*x^2)*sin(2*d*x^2 + 2*c))*log(cos(d*x^2 + c)^2 + sin(d*x^2 + c)^2 + 2*c
os(d*x^2 + c) + 1) + (-I*a*b*d^2*x^4 - I*b^2*d*x^2 + (I*a*b*d^2*x^4 + I*b^2*d*x^2)*cos(2*d*x^2 + 2*c) - (a*b*d
^2*x^4 + b^2*d*x^2)*sin(2*d*x^2 + 2*c))*log(cos(d*x^2 + c)^2 + sin(d*x^2 + c)^2 - 2*cos(d*x^2 + c) + 1) - 4*(I
*a*b*cos(2*d*x^2 + 2*c) - a*b*sin(2*d*x^2 + 2*c) - I*a*b)*polylog(3, -e^(I*d*x^2 + I*c)) - 4*(-I*a*b*cos(2*d*x
^2 + 2*c) + a*b*sin(2*d*x^2 + 2*c) + I*a*b)*polylog(3, e^(I*d*x^2 + I*c)))/(-2*I*d^3*cos(2*d*x^2 + 2*c) + 2*d^
3*sin(2*d*x^2 + 2*c) + 2*I*d^3)

Giac [F]

\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{5} \,d x } \]

[In]

integrate(x^5*(a+b*csc(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate((b*csc(d*x^2 + c) + a)^2*x^5, x)

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^5\,{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2 \,d x \]

[In]

int(x^5*(a + b/sin(c + d*x^2))^2,x)

[Out]

int(x^5*(a + b/sin(c + d*x^2))^2, x)