3.1.31 \(\int x^3 (a+b \csc (c+d \sqrt {x})) \, dx\) [31]

Optimal. Leaf size=432 \[ \frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \text {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \text {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \text {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \text {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \text {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {10080 i b \text {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {10080 i b \text {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8} \]

[Out]

1/4*a*x^4-4*b*x^(7/2)*arctanh(exp(I*(c+d*x^(1/2))))/d+5040*I*b*x*polylog(6,-exp(I*(c+d*x^(1/2))))/d^6-420*I*b*
x^2*polylog(4,-exp(I*(c+d*x^(1/2))))/d^4-84*b*x^(5/2)*polylog(3,-exp(I*(c+d*x^(1/2))))/d^3+84*b*x^(5/2)*polylo
g(3,exp(I*(c+d*x^(1/2))))/d^3-5040*I*b*x*polylog(6,exp(I*(c+d*x^(1/2))))/d^6+10080*I*b*polylog(8,exp(I*(c+d*x^
(1/2))))/d^8+1680*b*x^(3/2)*polylog(5,-exp(I*(c+d*x^(1/2))))/d^5-1680*b*x^(3/2)*polylog(5,exp(I*(c+d*x^(1/2)))
)/d^5-10080*I*b*polylog(8,-exp(I*(c+d*x^(1/2))))/d^8+14*I*b*x^3*polylog(2,-exp(I*(c+d*x^(1/2))))/d^2+420*I*b*x
^2*polylog(4,exp(I*(c+d*x^(1/2))))/d^4-14*I*b*x^3*polylog(2,exp(I*(c+d*x^(1/2))))/d^2-10080*b*polylog(7,-exp(I
*(c+d*x^(1/2))))*x^(1/2)/d^7+10080*b*polylog(7,exp(I*(c+d*x^(1/2))))*x^(1/2)/d^7

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {14, 4290, 4268, 2611, 6744, 2320, 6724} \begin {gather*} \frac {a x^4}{4}-\frac {10080 i b \text {Li}_8\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {10080 i b \text {Li}_8\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}-\frac {10080 b \sqrt {x} \text {Li}_7\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \text {Li}_7\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {5040 i b x \text {Li}_6\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \text {Li}_6\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {1680 b x^{3/2} \text {Li}_5\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {Li}_5\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Csc[c + d*Sqrt[x]]),x]

[Out]

(a*x^4)/4 - (4*b*x^(7/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((14*I)*b*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])
/d^2 - ((14*I)*b*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (84*b*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))]
)/d^3 + (84*b*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((420*I)*b*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x])
)])/d^4 + ((420*I)*b*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (1680*b*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt
[x]))])/d^5 - (1680*b*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 + ((5040*I)*b*x*PolyLog[6, -E^(I*(c + d*S
qrt[x]))])/d^6 - ((5040*I)*b*x*PolyLog[6, E^(I*(c + d*Sqrt[x]))])/d^6 - (10080*b*Sqrt[x]*PolyLog[7, -E^(I*(c +
d*Sqrt[x]))])/d^7 + (10080*b*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))])/d^7 - ((10080*I)*b*PolyLog[8, -E^(I*(
c + d*Sqrt[x]))])/d^8 + ((10080*I)*b*PolyLog[8, E^(I*(c + d*Sqrt[x]))])/d^8

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*} \int x^3 \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx &=\int \left (a x^3+b x^3 \csc \left (c+d \sqrt {x}\right )\right ) \, dx\\ &=\frac {a x^4}{4}+b \int x^3 \csc \left (c+d \sqrt {x}\right ) \, dx\\ &=\frac {a x^4}{4}+(2 b) \text {Subst}\left (\int x^7 \csc (c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {(14 b) \text {Subst}\left (\int x^6 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(14 b) \text {Subst}\left (\int x^6 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(84 i b) \text {Subst}\left (\int x^5 \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(84 i b) \text {Subst}\left (\int x^5 \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(420 b) \text {Subst}\left (\int x^4 \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(420 b) \text {Subst}\left (\int x^4 \text {Li}_3\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(1680 i b) \text {Subst}\left (\int x^3 \text {Li}_4\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(1680 i b) \text {Subst}\left (\int x^3 \text {Li}_4\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \text {Li}_5\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {Li}_5\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {(5040 b) \text {Subst}\left (\int x^2 \text {Li}_5\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(5040 b) \text {Subst}\left (\int x^2 \text {Li}_5\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \text {Li}_5\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {Li}_5\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \text {Li}_6\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \text {Li}_6\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {(10080 i b) \text {Subst}\left (\int x \text {Li}_6\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}+\frac {(10080 i b) \text {Subst}\left (\int x \text {Li}_6\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \text {Li}_5\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {Li}_5\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \text {Li}_6\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \text {Li}_6\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \text {Li}_7\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \text {Li}_7\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {(10080 b) \text {Subst}\left (\int \text {Li}_7\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}-\frac {(10080 b) \text {Subst}\left (\int \text {Li}_7\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \text {Li}_5\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {Li}_5\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \text {Li}_6\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \text {Li}_6\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \text {Li}_7\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \text {Li}_7\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {(10080 i b) \text {Subst}\left (\int \frac {\text {Li}_7(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {(10080 i b) \text {Subst}\left (\int \frac {\text {Li}_7(x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}\\ &=\frac {a x^4}{4}-\frac {4 b x^{7/2} \tanh ^{-1}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {14 i b x^3 \text {Li}_2\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {14 i b x^3 \text {Li}_2\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {84 b x^{5/2} \text {Li}_3\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {84 b x^{5/2} \text {Li}_3\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {420 i b x^2 \text {Li}_4\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {420 i b x^2 \text {Li}_4\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {1680 b x^{3/2} \text {Li}_5\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {1680 b x^{3/2} \text {Li}_5\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {5040 i b x \text {Li}_6\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {5040 i b x \text {Li}_6\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 b \sqrt {x} \text {Li}_7\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {10080 b \sqrt {x} \text {Li}_7\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {10080 i b \text {Li}_8\left (-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}+\frac {10080 i b \text {Li}_8\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d^8}\\ \end {align*}

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Mathematica [A]
time = 0.60, size = 445, normalized size = 1.03 \begin {gather*} \frac {a x^4}{4}+\frac {2 b \left (d^7 x^{7/2} \log \left (1-e^{i \left (c+d \sqrt {x}\right )}\right )-d^7 x^{7/2} \log \left (1+e^{i \left (c+d \sqrt {x}\right )}\right )+7 i d^6 x^3 \text {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )-7 i d^6 x^3 \text {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )-42 d^5 x^{5/2} \text {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )+42 d^5 x^{5/2} \text {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )-210 i d^4 x^2 \text {PolyLog}\left (4,-e^{i \left (c+d \sqrt {x}\right )}\right )+210 i d^4 x^2 \text {PolyLog}\left (4,e^{i \left (c+d \sqrt {x}\right )}\right )+840 d^3 x^{3/2} \text {PolyLog}\left (5,-e^{i \left (c+d \sqrt {x}\right )}\right )-840 d^3 x^{3/2} \text {PolyLog}\left (5,e^{i \left (c+d \sqrt {x}\right )}\right )+2520 i d^2 x \text {PolyLog}\left (6,-e^{i \left (c+d \sqrt {x}\right )}\right )-2520 i d^2 x \text {PolyLog}\left (6,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 d \sqrt {x} \text {PolyLog}\left (7,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 d \sqrt {x} \text {PolyLog}\left (7,e^{i \left (c+d \sqrt {x}\right )}\right )-5040 i \text {PolyLog}\left (8,-e^{i \left (c+d \sqrt {x}\right )}\right )+5040 i \text {PolyLog}\left (8,e^{i \left (c+d \sqrt {x}\right )}\right )\right )}{d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^4)/4 + (2*b*(d^7*x^(7/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] - d^7*x^(7/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (7
*I)*d^6*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (7*I)*d^6*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 42*d^5*x^(5
/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 42*d^5*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (210*I)*d^4*x^2*Po
lyLog[4, -E^(I*(c + d*Sqrt[x]))] + (210*I)*d^4*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + 840*d^3*x^(3/2)*PolyLog
[5, -E^(I*(c + d*Sqrt[x]))] - 840*d^3*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))] + (2520*I)*d^2*x*PolyLog[6, -E
^(I*(c + d*Sqrt[x]))] - (2520*I)*d^2*x*PolyLog[6, E^(I*(c + d*Sqrt[x]))] - 5040*d*Sqrt[x]*PolyLog[7, -E^(I*(c
+ d*Sqrt[x]))] + 5040*d*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))] - (5040*I)*PolyLog[8, -E^(I*(c + d*Sqrt[x]))
] + (5040*I)*PolyLog[8, E^(I*(c + d*Sqrt[x]))]))/d^8

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1498 vs. \(2 (338) = 676\).
time = 0.39, size = 1498, normalized size = 3.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((d*sqrt(x) + c)^8*a - 8*(d*sqrt(x) + c)^7*a*c + 28*(d*sqrt(x) + c)^6*a*c^2 - 56*(d*sqrt(x) + c)^5*a*c^3 +
70*(d*sqrt(x) + c)^4*a*c^4 - 56*(d*sqrt(x) + c)^3*a*c^5 + 28*(d*sqrt(x) + c)^2*a*c^6 - 8*(d*sqrt(x) + c)*a*c^
7 + 8*b*c^7*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c)) + 8*(-I*(d*sqrt(x) + c)^7*b + 7*I*(d*sqrt(x) + c)^6*b
*c - 21*I*(d*sqrt(x) + c)^5*b*c^2 + 35*I*(d*sqrt(x) + c)^4*b*c^3 - 35*I*(d*sqrt(x) + c)^3*b*c^4 + 21*I*(d*sqrt
(x) + c)^2*b*c^5 - 7*I*(d*sqrt(x) + c)*b*c^6)*arctan2(sin(d*sqrt(x) + c), cos(d*sqrt(x) + c) + 1) + 8*(-I*(d*s
qrt(x) + c)^7*b + 7*I*(d*sqrt(x) + c)^6*b*c - 21*I*(d*sqrt(x) + c)^5*b*c^2 + 35*I*(d*sqrt(x) + c)^4*b*c^3 - 35
*I*(d*sqrt(x) + c)^3*b*c^4 + 21*I*(d*sqrt(x) + c)^2*b*c^5 - 7*I*(d*sqrt(x) + c)*b*c^6)*arctan2(sin(d*sqrt(x) +
c), -cos(d*sqrt(x) + c) + 1) + 56*(I*(d*sqrt(x) + c)^6*b - 6*I*(d*sqrt(x) + c)^5*b*c + 15*I*(d*sqrt(x) + c)^4
*b*c^2 - 20*I*(d*sqrt(x) + c)^3*b*c^3 + 15*I*(d*sqrt(x) + c)^2*b*c^4 - 6*I*(d*sqrt(x) + c)*b*c^5 + I*b*c^6)*di
log(-e^(I*d*sqrt(x) + I*c)) + 56*(-I*(d*sqrt(x) + c)^6*b + 6*I*(d*sqrt(x) + c)^5*b*c - 15*I*(d*sqrt(x) + c)^4*
b*c^2 + 20*I*(d*sqrt(x) + c)^3*b*c^3 - 15*I*(d*sqrt(x) + c)^2*b*c^4 + 6*I*(d*sqrt(x) + c)*b*c^5 - I*b*c^6)*dil
og(e^(I*d*sqrt(x) + I*c)) - 4*((d*sqrt(x) + c)^7*b - 7*(d*sqrt(x) + c)^6*b*c + 21*(d*sqrt(x) + c)^5*b*c^2 - 35
*(d*sqrt(x) + c)^4*b*c^3 + 35*(d*sqrt(x) + c)^3*b*c^4 - 21*(d*sqrt(x) + c)^2*b*c^5 + 7*(d*sqrt(x) + c)*b*c^6)*
log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1) + 4*((d*sqrt(x) + c)^7*b - 7*(d*sq
rt(x) + c)^6*b*c + 21*(d*sqrt(x) + c)^5*b*c^2 - 35*(d*sqrt(x) + c)^4*b*c^3 + 35*(d*sqrt(x) + c)^3*b*c^4 - 21*(
d*sqrt(x) + c)^2*b*c^5 + 7*(d*sqrt(x) + c)*b*c^6)*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 - 2*cos(d*sq
rt(x) + c) + 1) - 40320*I*b*polylog(8, -e^(I*d*sqrt(x) + I*c)) + 40320*I*b*polylog(8, e^(I*d*sqrt(x) + I*c)) -
40320*((d*sqrt(x) + c)*b - b*c)*polylog(7, -e^(I*d*sqrt(x) + I*c)) + 40320*((d*sqrt(x) + c)*b - b*c)*polylog(
7, e^(I*d*sqrt(x) + I*c)) + 20160*(I*(d*sqrt(x) + c)^2*b - 2*I*(d*sqrt(x) + c)*b*c + I*b*c^2)*polylog(6, -e^(I
*d*sqrt(x) + I*c)) + 20160*(-I*(d*sqrt(x) + c)^2*b + 2*I*(d*sqrt(x) + c)*b*c - I*b*c^2)*polylog(6, e^(I*d*sqrt
(x) + I*c)) + 6720*((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(5
, -e^(I*d*sqrt(x) + I*c)) - 6720*((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(5, e^(I*d*sqrt(x) + I*c)) + 1680*(-I*(d*sqrt(x) + c)^4*b + 4*I*(d*sqrt(x) + c)^3*b*c - 6*I*(d*sqr
t(x) + c)^2*b*c^2 + 4*I*(d*sqrt(x) + c)*b*c^3 - I*b*c^4)*polylog(4, -e^(I*d*sqrt(x) + I*c)) + 1680*(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)*p
olylog(4, e^(I*d*sqrt(x) + I*c)) - 336*((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 - b*c^5)*polylog(3, -e^(I*d*sqrt(x) + I*c)) + 336*
((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 - b*c^5)*polylog(3, e^(I*d*sqrt(x) + I*c)))/d^8

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(a + b*csc(c + d*sqrt(x))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b/sin(c + d*x^(1/2))),x)

[Out]

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

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