\(\int x^2 (a+b \csc (c+d \sqrt {x})) \, dx\) [32]

Optimal result

Integrand size = 18, antiderivative size = 316 \[ \int x^2 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {120 i b x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {120 i b x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {240 i b \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {240 i b \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6} \]

[Out]

1/3*a*x^3-4*b*x^(5/2)*arctanh(exp(I*(c+d*x^(1/2))))/d+10*I*b*x^2*polylog(2,-exp(I*(c+d*x^(1/2))))/d^2-10*I*b*x
^2*polylog(2,exp(I*(c+d*x^(1/2))))/d^2-40*b*x^(3/2)*polylog(3,-exp(I*(c+d*x^(1/2))))/d^3+40*b*x^(3/2)*polylog(
3,exp(I*(c+d*x^(1/2))))/d^3-120*I*b*x*polylog(4,-exp(I*(c+d*x^(1/2))))/d^4+120*I*b*x*polylog(4,exp(I*(c+d*x^(1
/2))))/d^4+240*I*b*polylog(6,-exp(I*(c+d*x^(1/2))))/d^6-240*I*b*polylog(6,exp(I*(c+d*x^(1/2))))/d^6+240*b*poly
log(5,-exp(I*(c+d*x^(1/2))))*x^(1/2)/d^5-240*b*polylog(5,exp(I*(c+d*x^(1/2))))*x^(1/2)/d^5

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {14, 4290, 4268, 2611, 6744, 2320, 6724} \[ \int x^2 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {240 i b \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {240 i b \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {120 i b x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {120 i b x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2} \]

[In]

Int[x^2*(a + b*Csc[c + d*Sqrt[x]]),x]

[Out]

(a*x^3)/3 - (4*b*x^(5/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((10*I)*b*x^2*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])
/d^2 - ((10*I)*b*x^2*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (40*b*x^(3/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))]
)/d^3 + (40*b*x^(3/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((120*I)*b*x*PolyLog[4, -E^(I*(c + d*Sqrt[x]))]
)/d^4 + ((120*I)*b*x*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (240*b*Sqrt[x]*PolyLog[5, -E^(I*(c + d*Sqrt[x]))
])/d^5 - (240*b*Sqrt[x]*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 + ((240*I)*b*PolyLog[6, -E^(I*(c + d*Sqrt[x]))]
)/d^6 - ((240*I)*b*PolyLog[6, E^(I*(c + d*Sqrt[x]))])/d^6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 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 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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^2+b x^2 \csc \left (c+d \sqrt {x}\right )\right ) \, dx \\ & = \frac {a x^3}{3}+b \int x^2 \csc \left (c+d \sqrt {x}\right ) \, dx \\ & = \frac {a x^3}{3}+(2 b) \text {Subst}\left (\int x^5 \csc (c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {(10 b) \text {Subst}\left (\int x^4 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(10 b) \text {Subst}\left (\int x^4 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(40 i b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(40 i b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(120 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(120 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {120 i b x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {120 i b x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(240 i b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(240 i b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {120 i b x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {120 i b x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {(240 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(240 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (5,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5} \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {120 i b x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {120 i b x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {(240 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(5,-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {(240 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(5,x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6} \\ & = \frac {a x^3}{3}-\frac {4 b x^{5/2} \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {10 i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {10 i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {40 b x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {120 i b x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {120 i b x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {240 b \sqrt {x} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {240 i b \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {240 i b \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.05 \[ \int x^2 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^3}{3}+\frac {2 b \left (d^5 x^{5/2} \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )-d^5 x^{5/2} \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )+5 i d^4 x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )-5 i d^4 x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )-20 d^3 x^{3/2} \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )+20 d^3 x^{3/2} \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-60 i d^2 x \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )+60 i d^2 x \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )+120 d \sqrt {x} \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )-120 d \sqrt {x} \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )+120 i \operatorname {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )-120 i \operatorname {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )\right )}{d^6} \]

[In]

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

[Out]

(a*x^3)/3 + (2*b*(d^5*x^(5/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] - d^5*x^(5/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (5
*I)*d^4*x^2*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (5*I)*d^4*x^2*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 20*d^3*x^(3
/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 20*d^3*x^(3/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (60*I)*d^2*x*PolyL
og[4, -E^(I*(c + d*Sqrt[x]))] + (60*I)*d^2*x*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + 120*d*Sqrt[x]*PolyLog[5, -E^(
I*(c + d*Sqrt[x]))] - 120*d*Sqrt[x]*PolyLog[5, E^(I*(c + d*Sqrt[x]))] + (120*I)*PolyLog[6, -E^(I*(c + d*Sqrt[x
]))] - (120*I)*PolyLog[6, E^(I*(c + d*Sqrt[x]))]))/d^6

Maple [F]

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

[In]

int(x^2*(a+b*csc(c+d*x^(1/2))),x)

[Out]

int(x^2*(a+b*csc(c+d*x^(1/2))),x)

Fricas [F]

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

[In]

integrate(x^2*(a+b*csc(c+d*x^(1/2))),x, algorithm="fricas")

[Out]

integral(b*x^2*csc(d*sqrt(x) + c) + a*x^2, x)

Sympy [F]

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

[In]

integrate(x**2*(a+b*csc(c+d*x**(1/2))),x)

[Out]

Integral(x**2*(a + b*csc(c + d*sqrt(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. 956 vs. \(2 (246) = 492\).

Time = 0.29 (sec) , antiderivative size = 956, normalized size of antiderivative = 3.03 \[ \int x^2 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x^2*(a+b*csc(c+d*x^(1/2))),x, algorithm="maxima")

[Out]

1/3*((d*sqrt(x) + c)^6*a - 6*(d*sqrt(x) + c)^5*a*c + 15*(d*sqrt(x) + c)^4*a*c^2 - 20*(d*sqrt(x) + c)^3*a*c^3 +
15*(d*sqrt(x) + c)^2*a*c^4 - 6*(d*sqrt(x) + c)*a*c^5 + 6*b*c^5*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c)) +
6*(-I*(d*sqrt(x) + c)^5*b + 5*I*(d*sqrt(x) + c)^4*b*c - 10*I*(d*sqrt(x) + c)^3*b*c^2 + 10*I*(d*sqrt(x) + c)^2
*b*c^3 - 5*I*(d*sqrt(x) + c)*b*c^4)*arctan2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) + 1) + 6*(-I*(d*sqrt(x) + c
)^5*b + 5*I*(d*sqrt(x) + c)^4*b*c - 10*I*(d*sqrt(x) + c)^3*b*c^2 + 10*I*(d*sqrt(x) + c)^2*b*c^3 - 5*I*(d*sqrt(
x) + c)*b*c^4)*arctan2(sin(d*sqrt(x) + c), -cos(d*sqrt(x) + c) + 1) + 30*(I*(d*sqrt(x) + c)^4*b - 4*I*(d*sqrt(
x) + c)^3*b*c + 6*I*(d*sqrt(x) + c)^2*b*c^2 - 4*I*(d*sqrt(x) + c)*b*c^3 + I*b*c^4)*dilog(-e^(I*d*sqrt(x) + I*c
)) + 30*(-I*(d*sqrt(x) + c)^4*b + 4*I*(d*sqrt(x) + c)^3*b*c - 6*I*(d*sqrt(x) + c)^2*b*c^2 + 4*I*(d*sqrt(x) + c
)*b*c^3 - I*b*c^4)*dilog(e^(I*d*sqrt(x) + I*c)) - 3*((d*sqrt(x) + c)^5*b - 5*(d*sqrt(x) + c)^4*b*c + 10*(d*sqr
t(x) + c)^3*b*c^2 - 10*(d*sqrt(x) + c)^2*b*c^3 + 5*(d*sqrt(x) + c)*b*c^4)*log(cos(d*sqrt(x) + c)^2 + sin(d*sqr
t(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1) + 3*((d*sqrt(x) + c)^5*b - 5*(d*sqrt(x) + c)^4*b*c + 10*(d*sqrt(x) + c
)^3*b*c^2 - 10*(d*sqrt(x) + c)^2*b*c^3 + 5*(d*sqrt(x) + c)*b*c^4)*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c
)^2 - 2*cos(d*sqrt(x) + c) + 1) + 720*I*b*polylog(6, -e^(I*d*sqrt(x) + I*c)) - 720*I*b*polylog(6, e^(I*d*sqrt(
x) + I*c)) + 720*((d*sqrt(x) + c)*b - b*c)*polylog(5, -e^(I*d*sqrt(x) + I*c)) - 720*((d*sqrt(x) + c)*b - b*c)*
polylog(5, e^(I*d*sqrt(x) + I*c)) + 360*(-I*(d*sqrt(x) + c)^2*b + 2*I*(d*sqrt(x) + c)*b*c - I*b*c^2)*polylog(4
, -e^(I*d*sqrt(x) + I*c)) + 360*(I*(d*sqrt(x) + c)^2*b - 2*I*(d*sqrt(x) + c)*b*c + I*b*c^2)*polylog(4, e^(I*d*
sqrt(x) + I*c)) - 120*((d*sqrt(x) + c)^3*b - 3*(d*sqrt(x) + c)^2*b*c + 3*(d*sqrt(x) + c)*b*c^2 - b*c^3)*polylo
g(3, -e^(I*d*sqrt(x) + I*c)) + 120*((d*sqrt(x) + c)^3*b - 3*(d*sqrt(x) + c)^2*b*c + 3*(d*sqrt(x) + c)*b*c^2 -
b*c^3)*polylog(3, e^(I*d*sqrt(x) + I*c)))/d^6

Giac [F]

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

[In]

integrate(x^2*(a+b*csc(c+d*x^(1/2))),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^2*(a + b/sin(c + d*x^(1/2))),x)

[Out]

int(x^2*(a + b/sin(c + d*x^(1/2))), x)