\(\int x^{3/2} (a+b \csc (c+d \sqrt {x}))^2 \, dx\) [56]

Optimal result

Integrand size = 22, antiderivative size = 421 \[ \int x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i b^2 x \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {6 i b^2 \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {96 a b \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {96 a b \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5} \]

[Out]

6*I*b^2*polylog(4,exp(2*I*(c+d*x^(1/2))))/d^5+2/5*a^2*x^(5/2)-8*a*b*x^2*arctanh(exp(I*(c+d*x^(1/2))))/d-2*b^2*
x^2*cot(c+d*x^(1/2))/d+8*b^2*x^(3/2)*ln(1-exp(2*I*(c+d*x^(1/2))))/d^2-16*I*a*b*x^(3/2)*polylog(2,exp(I*(c+d*x^
(1/2))))/d^2-12*I*b^2*x*polylog(2,exp(2*I*(c+d*x^(1/2))))/d^3+96*I*a*b*polylog(4,exp(I*(c+d*x^(1/2))))*x^(1/2)
/d^4-48*a*b*x*polylog(3,-exp(I*(c+d*x^(1/2))))/d^3+48*a*b*x*polylog(3,exp(I*(c+d*x^(1/2))))/d^3+16*I*a*b*x^(3/
2)*polylog(2,-exp(I*(c+d*x^(1/2))))/d^2+96*a*b*polylog(5,-exp(I*(c+d*x^(1/2))))/d^5-96*a*b*polylog(5,exp(I*(c+
d*x^(1/2))))/d^5+12*b^2*polylog(3,exp(2*I*(c+d*x^(1/2))))*x^(1/2)/d^4-96*I*a*b*polylog(4,-exp(I*(c+d*x^(1/2)))
)*x^(1/2)/d^4-2*I*b^2*x^2/d

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {4290, 4275, 4268, 2611, 6744, 2320, 6724, 4269, 3798, 2221} \[ \int x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {96 a b \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {96 a b \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {6 i b^2 \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {12 i b^2 x \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^2}{d} \]

[In]

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

[Out]

((-2*I)*b^2*x^2)/d + (2*a^2*x^(5/2))/5 - (8*a*b*x^2*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x^2*Cot[c + d*S
qrt[x]])/d + (8*b^2*x^(3/2)*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((16*I)*a*b*x^(3/2)*PolyLog[2, -E^(I*(c
+ d*Sqrt[x]))])/d^2 - ((16*I)*a*b*x^(3/2)*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((12*I)*b^2*x*PolyLog[2, E^
((2*I)*(c + d*Sqrt[x]))])/d^3 - (48*a*b*x*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (48*a*b*x*PolyLog[3, E^(I*
(c + d*Sqrt[x]))])/d^3 + (12*b^2*Sqrt[x]*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((96*I)*a*b*Sqrt[x]*Poly
Log[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((96*I)*a*b*Sqrt[x]*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + ((6*I)*b^2*
PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (96*a*b*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (96*a*b*PolyLog
[5, E^(I*(c + d*Sqrt[x]))])/d^5

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 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 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]

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}& = 2 \text {Subst}\left (\int x^4 (a+b \csc (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^4+2 a b x^4 \csc (c+d x)+b^2 x^4 \csc ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{5} a^2 x^{5/2}+(4 a b) \text {Subst}\left (\int x^4 \csc (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^4 \csc ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}-\frac {(16 a b) \text {Subst}\left (\int x^3 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(16 a b) \text {Subst}\left (\int x^3 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int x^3 \cot (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(48 i a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(48 i a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (16 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^3}{1-e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(96 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(96 a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (24 b^2\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i b^2 x \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(96 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(96 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (24 i b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i b^2 x \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(96 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {(96 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = -\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i b^2 x \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 a b \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {96 a b \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {\left (6 i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5} \\ & = -\frac {2 i b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {2 b^2 x^2 \cot \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i b^2 x \operatorname {PolyLog}\left (2,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {48 a b x \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {6 i b^2 \operatorname {PolyLog}\left (4,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {96 a b \operatorname {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {96 a b \operatorname {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.64 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.78 \[ \int x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2 a^2 x^{5/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sin ^2\left (c+d \sqrt {x}\right )}{5 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {4 b \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \left (-\frac {i b d^4 x^2}{-1+e^{2 i c}}+2 b d^3 x^{3/2} \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )+a d^4 x^2 \log \left (1-e^{-i \left (c+d \sqrt {x}\right )}\right )+2 b d^3 x^{3/2} \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )-a d^4 x^2 \log \left (1+e^{-i \left (c+d \sqrt {x}\right )}\right )-2 i d^2 \left (-3 b+2 a d \sqrt {x}\right ) x \operatorname {PolyLog}\left (2,-e^{-i \left (c+d \sqrt {x}\right )}\right )+2 i d^2 \left (3 b+2 a d \sqrt {x}\right ) x \operatorname {PolyLog}\left (2,e^{-i \left (c+d \sqrt {x}\right )}\right )+12 b d \sqrt {x} \operatorname {PolyLog}\left (3,-e^{-i \left (c+d \sqrt {x}\right )}\right )-12 a d^2 x \operatorname {PolyLog}\left (3,-e^{-i \left (c+d \sqrt {x}\right )}\right )+12 b d \sqrt {x} \operatorname {PolyLog}\left (3,e^{-i \left (c+d \sqrt {x}\right )}\right )+12 a d^2 x \operatorname {PolyLog}\left (3,e^{-i \left (c+d \sqrt {x}\right )}\right )-12 i b \operatorname {PolyLog}\left (4,-e^{-i \left (c+d \sqrt {x}\right )}\right )+24 i a d \sqrt {x} \operatorname {PolyLog}\left (4,-e^{-i \left (c+d \sqrt {x}\right )}\right )-12 i b \operatorname {PolyLog}\left (4,e^{-i \left (c+d \sqrt {x}\right )}\right )-24 i a d \sqrt {x} \operatorname {PolyLog}\left (4,e^{-i \left (c+d \sqrt {x}\right )}\right )+24 a \operatorname {PolyLog}\left (5,-e^{-i \left (c+d \sqrt {x}\right )}\right )-24 a \operatorname {PolyLog}\left (5,e^{-i \left (c+d \sqrt {x}\right )}\right )\right ) \sin ^2\left (c+d \sqrt {x}\right )}{d^5 \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^2 \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sin ^2\left (c+d \sqrt {x}\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^2 \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \sin ^2\left (c+d \sqrt {x}\right ) \sin \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )^2} \]

[In]

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

[Out]

(2*a^2*x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2)/(5*(b + a*Sin[c + d*Sqrt[x]])^2) + (4*b*(a +
b*Csc[c + d*Sqrt[x]])^2*(((-I)*b*d^4*x^2)/(-1 + E^((2*I)*c)) + 2*b*d^3*x^(3/2)*Log[1 - E^((-I)*(c + d*Sqrt[x]
))] + a*d^4*x^2*Log[1 - E^((-I)*(c + d*Sqrt[x]))] + 2*b*d^3*x^(3/2)*Log[1 + E^((-I)*(c + d*Sqrt[x]))] - a*d^4*
x^2*Log[1 + E^((-I)*(c + d*Sqrt[x]))] - (2*I)*d^2*(-3*b + 2*a*d*Sqrt[x])*x*PolyLog[2, -E^((-I)*(c + d*Sqrt[x])
)] + (2*I)*d^2*(3*b + 2*a*d*Sqrt[x])*x*PolyLog[2, E^((-I)*(c + d*Sqrt[x]))] + 12*b*d*Sqrt[x]*PolyLog[3, -E^((-
I)*(c + d*Sqrt[x]))] - 12*a*d^2*x*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] + 12*b*d*Sqrt[x]*PolyLog[3, E^((-I)*(c
+ d*Sqrt[x]))] + 12*a*d^2*x*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] - (12*I)*b*PolyLog[4, -E^((-I)*(c + d*Sqrt[x
]))] + (24*I)*a*d*Sqrt[x]*PolyLog[4, -E^((-I)*(c + d*Sqrt[x]))] - (12*I)*b*PolyLog[4, E^((-I)*(c + d*Sqrt[x]))
] - (24*I)*a*d*Sqrt[x]*PolyLog[4, E^((-I)*(c + d*Sqrt[x]))] + 24*a*PolyLog[5, -E^((-I)*(c + d*Sqrt[x]))] - 24*
a*PolyLog[5, E^((-I)*(c + d*Sqrt[x]))])*Sin[c + d*Sqrt[x]]^2)/(d^5*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^2*Cs
c[c/2]*Csc[c/2 + (d*Sqrt[x])/2]*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b +
a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^2*(a + b*Csc[c + d*Sqrt[x]])^2*Sec[c/2]*Sec[c/2 + (d*Sqrt[x])/2]*Sin[c + d*S
qrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2)

Maple [F]

\[\int x^{\frac {3}{2}} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}d x\]

[In]

int(x^(3/2)*(a+b*csc(c+d*x^(1/2)))^2,x)

[Out]

int(x^(3/2)*(a+b*csc(c+d*x^(1/2)))^2,x)

Fricas [F]

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

[In]

integrate(x^(3/2)*(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^(3/2)*csc(d*sqrt(x) + c)^2 + 2*a*b*x^(3/2)*csc(d*sqrt(x) + c) + a^2*x^(3/2), x)

Sympy [F]

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

[In]

integrate(x**(3/2)*(a+b*csc(c+d*x**(1/2)))**2,x)

[Out]

Integral(x**(3/2)*(a + b*csc(c + d*sqrt(x)))**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. 2836 vs. \(2 (334) = 668\).

Time = 0.37 (sec) , antiderivative size = 2836, normalized size of antiderivative = 6.74 \[ \int x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate(x^(3/2)*(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="maxima")

[Out]

2/5*((d*sqrt(x) + c)^5*a^2 - 5*(d*sqrt(x) + c)^4*a^2*c + 10*(d*sqrt(x) + c)^3*a^2*c^2 - 10*(d*sqrt(x) + c)^2*a
^2*c^3 + 5*(d*sqrt(x) + c)*a^2*c^4 - 10*a*b*c^4*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c)) - 5*(2*b^2*c^4 -
2*((d*sqrt(x) + c)^4*a*b + 2*b^2*c^3 - 2*(2*a*b*c + b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 + b^2*c)*(d*sqrt(x) +
c)^2 - 2*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c) - ((d*sqrt(x) + c)^4*a*b + 2*b^2*c^3 - 2*(2*a*b*c + b^2)*(d*s
qrt(x) + c)^3 + 6*(a*b*c^2 + b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c))*cos(2*d*sqr
t(x) + 2*c) + (-I*(d*sqrt(x) + c)^4*a*b - 2*I*b^2*c^3 + 2*(2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)^3 + 6*(-I*a*b*c^
2 - I*b^2*c)*(d*sqrt(x) + c)^2 + 2*(2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*arctan
2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) + 1) + 4*(b^2*c^3*cos(2*d*sqrt(x) + 2*c) + I*b^2*c^3*sin(2*d*sqrt(x)
+ 2*c) - b^2*c^3)*arctan2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) - 1) - 2*((d*sqrt(x) + c)^4*a*b - 2*(2*a*b*c
- b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c) -
((d*sqrt(x) + c)^4*a*b - 2*(2*a*b*c - b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*a
*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-I*(d*sqrt(x) + c)^4*a*b + 2*(2*I*a*b*c - I*b^2
)*(d*sqrt(x) + c)^3 + 6*(-I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c)^2 + 2*(2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) +
c))*sin(2*d*sqrt(x) + 2*c))*arctan2(sin(d*sqrt(x) + c), -cos(d*sqrt(x) + c) + 1) + 2*((d*sqrt(x) + c)^4*b^2 -
4*(d*sqrt(x) + c)^3*b^2*c + 6*(d*sqrt(x) + c)^2*b^2*c^2 - 4*(d*sqrt(x) + c)*b^2*c^3)*cos(2*d*sqrt(x) + 2*c) +
4*(2*(d*sqrt(x) + c)^3*a*b - 2*a*b*c^3 - 3*b^2*c^2 - 3*(2*a*b*c + b^2)*(d*sqrt(x) + c)^2 + 6*(a*b*c^2 + b^2*c)
*(d*sqrt(x) + c) - (2*(d*sqrt(x) + c)^3*a*b - 2*a*b*c^3 - 3*b^2*c^2 - 3*(2*a*b*c + b^2)*(d*sqrt(x) + c)^2 + 6*
(a*b*c^2 + b^2*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - (2*I*(d*sqrt(x) + c)^3*a*b - 2*I*a*b*c^3 - 3*I*b^2
*c^2 + 3*(-2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^2 + 6*(I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2
*c))*dilog(-e^(I*d*sqrt(x) + I*c)) - 4*(2*(d*sqrt(x) + c)^3*a*b - 2*a*b*c^3 + 3*b^2*c^2 - 3*(2*a*b*c - b^2)*(d
*sqrt(x) + c)^2 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c) - (2*(d*sqrt(x) + c)^3*a*b - 2*a*b*c^3 + 3*b^2*c^2 - 3*(
2*a*b*c - b^2)*(d*sqrt(x) + c)^2 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) + (-2*I*(d*sqrt
(x) + c)^3*a*b + 2*I*a*b*c^3 - 3*I*b^2*c^2 + 3*(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^2 + 6*(-I*a*b*c^2 + I*b^2*c
)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*dilog(e^(I*d*sqrt(x) + I*c)) + (I*(d*sqrt(x) + c)^4*a*b + 2*I*b^2*c
^3 - 2*(2*I*a*b*c + I*b^2)*(d*sqrt(x) + c)^3 - 6*(-I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*I*a*b*c^3 + 3
*I*b^2*c^2)*(d*sqrt(x) + c) + (-I*(d*sqrt(x) + c)^4*a*b - 2*I*b^2*c^3 - 2*(-2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)
^3 - 6*(I*a*b*c^2 + I*b^2*c)*(d*sqrt(x) + c)^2 - 2*(-2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(
x) + 2*c) + ((d*sqrt(x) + c)^4*a*b + 2*b^2*c^3 - 2*(2*a*b*c + b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 + b^2*c)*(d*
sqrt(x) + c)^2 - 2*(2*a*b*c^3 + 3*b^2*c^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 +
sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1) + (-I*(d*sqrt(x) + c)^4*a*b + 2*I*b^2*c^3 - 2*(-2*I*a*b*c +
I*b^2)*(d*sqrt(x) + c)^3 - 6*(I*a*b*c^2 - I*b^2*c)*(d*sqrt(x) + c)^2 - 2*(-2*I*a*b*c^3 + 3*I*b^2*c^2)*(d*sqrt(
x) + c) + (I*(d*sqrt(x) + c)^4*a*b - 2*I*b^2*c^3 - 2*(2*I*a*b*c - I*b^2)*(d*sqrt(x) + c)^3 - 6*(-I*a*b*c^2 + I
*b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*I*a*b*c^3 - 3*I*b^2*c^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) - ((d*sqrt(x
) + c)^4*a*b - 2*b^2*c^3 - 2*(2*a*b*c - b^2)*(d*sqrt(x) + c)^3 + 6*(a*b*c^2 - b^2*c)*(d*sqrt(x) + c)^2 - 2*(2*
a*b*c^3 - 3*b^2*c^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2
- 2*cos(d*sqrt(x) + c) + 1) - 48*(-I*a*b*cos(2*d*sqrt(x) + 2*c) + a*b*sin(2*d*sqrt(x) + 2*c) + I*a*b)*polylog(
5, -e^(I*d*sqrt(x) + I*c)) - 48*(I*a*b*cos(2*d*sqrt(x) + 2*c) - a*b*sin(2*d*sqrt(x) + 2*c) - I*a*b)*polylog(5,
e^(I*d*sqrt(x) + I*c)) - 24*(2*(d*sqrt(x) + c)*a*b - 2*a*b*c - b^2 - (2*(d*sqrt(x) + c)*a*b - 2*a*b*c - b^2)*
cos(2*d*sqrt(x) + 2*c) + (-2*I*(d*sqrt(x) + c)*a*b + 2*I*a*b*c + I*b^2)*sin(2*d*sqrt(x) + 2*c))*polylog(4, -e^
(I*d*sqrt(x) + I*c)) + 24*(2*(d*sqrt(x) + c)*a*b - 2*a*b*c + b^2 - (2*(d*sqrt(x) + c)*a*b - 2*a*b*c + b^2)*cos
(2*d*sqrt(x) + 2*c) - (2*I*(d*sqrt(x) + c)*a*b - 2*I*a*b*c + I*b^2)*sin(2*d*sqrt(x) + 2*c))*polylog(4, e^(I*d*
sqrt(x) + I*c)) - 24*(-I*(d*sqrt(x) + c)^2*a*b - I*a*b*c^2 - I*b^2*c + (2*I*a*b*c + I*b^2)*(d*sqrt(x) + c) + (
I*(d*sqrt(x) + c)^2*a*b + I*a*b*c^2 + I*b^2*c + (-2*I*a*b*c - I*b^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x) + 2*c) -
((d*sqrt(x) + c)^2*a*b + a*b*c^2 + b^2*c - (2*a*b*c + b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))*polylog(3
, -e^(I*d*sqrt(x) + I*c)) - 24*(I*(d*sqrt(x) + c)^2*a*b + I*a*b*c^2 - I*b^2*c + (-2*I*a*b*c + I*b^2)*(d*sqrt(x
) + c) + (-I*(d*sqrt(x) + c)^2*a*b - I*a*b*c^2 + I*b^2*c + (2*I*a*b*c - I*b^2)*(d*sqrt(x) + c))*cos(2*d*sqrt(x
) + 2*c) + ((d*sqrt(x) + c)^2*a*b + a*b*c^2 - b^2*c - (2*a*b*c - b^2)*(d*sqrt(x) + c))*sin(2*d*sqrt(x) + 2*c))
*polylog(3, e^(I*d*sqrt(x) + I*c)) - 2*(-I*(d*sqrt(x) + c)^4*b^2 + 4*I*(d*sqrt(x) + c)^3*b^2*c - 6*I*(d*sqrt(x
) + c)^2*b^2*c^2 + 4*I*(d*sqrt(x) + c)*b^2*c^3)*sin(2*d*sqrt(x) + 2*c))/(-I*cos(2*d*sqrt(x) + 2*c) + sin(2*d*s
qrt(x) + 2*c) + I))/d^5

Giac [F]

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

[In]

integrate(x^(3/2)*(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3/2}\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]

[In]

int(x^(3/2)*(a + b/sin(c + d*x^(1/2)))^2,x)

[Out]

int(x^(3/2)*(a + b/sin(c + d*x^(1/2)))^2, x)