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

Optimal. Leaf size=258 \[ \frac{8 i b x^{3/2} \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 b x \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 b x \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{48 i b \sqrt{x} \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 i b \sqrt{x} \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 b \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{48 b \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d} \]

[Out]

(2*a*x^(5/2))/5 - (4*b*x^2*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((8*I)*b*x^(3/2)*PolyLog[2, -E^(I*(c + d*Sqrt[x
]))])/d^2 - ((8*I)*b*x^(3/2)*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (24*b*x*PolyLog[3, -E^(I*(c + d*Sqrt[x])
)])/d^3 + (24*b*x*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((48*I)*b*Sqrt[x]*PolyLog[4, -E^(I*(c + d*Sqrt[x]))
])/d^4 + ((48*I)*b*Sqrt[x]*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (48*b*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/
d^5 - (48*b*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5

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Rubi [A]  time = 0.238078, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {14, 4205, 4183, 2531, 6609, 2282, 6589} \[ \frac{8 i b x^{3/2} \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 b x \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 b x \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{48 i b \sqrt{x} \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 i b \sqrt{x} \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 b \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{48 b \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x^(5/2))/5 - (4*b*x^2*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((8*I)*b*x^(3/2)*PolyLog[2, -E^(I*(c + d*Sqrt[x
]))])/d^2 - ((8*I)*b*x^(3/2)*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (24*b*x*PolyLog[3, -E^(I*(c + d*Sqrt[x])
)])/d^3 + (24*b*x*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((48*I)*b*Sqrt[x]*PolyLog[4, -E^(I*(c + d*Sqrt[x]))
])/d^4 + ((48*I)*b*Sqrt[x]*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (48*b*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/
d^5 - (48*b*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5

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 4205

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 4183

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 2531

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 6609

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]

Rule 2282

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 6589

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*} \int x^{3/2} \left (a+b \csc \left (c+d \sqrt{x}\right )\right ) \, dx &=\int \left (a x^{3/2}+b x^{3/2} \csc \left (c+d \sqrt{x}\right )\right ) \, dx\\ &=\frac{2}{5} a x^{5/2}+b \int x^{3/2} \csc \left (c+d \sqrt{x}\right ) \, dx\\ &=\frac{2}{5} a x^{5/2}+(2 b) \operatorname{Subst}\left (\int x^4 \csc (c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}-\frac{(8 b) \operatorname{Subst}\left (\int x^3 \log \left (1-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(8 b) \operatorname{Subst}\left (\int x^3 \log \left (1+e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d}\\ &=\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{8 i b x^{3/2} \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{(24 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(24 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{8 i b x^{3/2} \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 b x \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 b x \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{(48 b) \operatorname{Subst}\left (\int x \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{(48 b) \operatorname{Subst}\left (\int x \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{8 i b x^{3/2} \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 b x \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 b x \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{48 i b \sqrt{x} \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 i b \sqrt{x} \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(48 i b) \operatorname{Subst}\left (\int \text{Li}_4\left (-e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{(48 i b) \operatorname{Subst}\left (\int \text{Li}_4\left (e^{i (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}\\ &=\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{8 i b x^{3/2} \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 b x \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 b x \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{48 i b \sqrt{x} \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 i b \sqrt{x} \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{(48 b) \operatorname{Subst}\left (\int \frac{\text{Li}_4(-x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{(48 b) \operatorname{Subst}\left (\int \frac{\text{Li}_4(x)}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}\\ &=\frac{2}{5} a x^{5/2}-\frac{4 b x^2 \tanh ^{-1}\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d}+\frac{8 i b x^{3/2} \text{Li}_2\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{24 b x \text{Li}_3\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{24 b x \text{Li}_3\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{48 i b \sqrt{x} \text{Li}_4\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 i b \sqrt{x} \text{Li}_4\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{48 b \text{Li}_5\left (-e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{48 b \text{Li}_5\left (e^{i \left (c+d \sqrt{x}\right )}\right )}{d^5}\\ \end{align*}

Mathematica [A]  time = 0.480295, size = 286, normalized size = 1.11 \[ \frac{2 \left (20 i b d^3 x^{3/2} \text{PolyLog}\left (2,-e^{i \left (c+d \sqrt{x}\right )}\right )-20 i b d^3 x^{3/2} \text{PolyLog}\left (2,e^{i \left (c+d \sqrt{x}\right )}\right )-60 b d^2 x \text{PolyLog}\left (3,-e^{i \left (c+d \sqrt{x}\right )}\right )+60 b d^2 x \text{PolyLog}\left (3,e^{i \left (c+d \sqrt{x}\right )}\right )-120 i b d \sqrt{x} \text{PolyLog}\left (4,-e^{i \left (c+d \sqrt{x}\right )}\right )+120 i b d \sqrt{x} \text{PolyLog}\left (4,e^{i \left (c+d \sqrt{x}\right )}\right )+120 b \text{PolyLog}\left (5,-e^{i \left (c+d \sqrt{x}\right )}\right )-120 b \text{PolyLog}\left (5,e^{i \left (c+d \sqrt{x}\right )}\right )+a d^5 x^{5/2}+5 b d^4 x^2 \log \left (1-e^{i \left (c+d \sqrt{x}\right )}\right )-5 b d^4 x^2 \log \left (1+e^{i \left (c+d \sqrt{x}\right )}\right )\right )}{5 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \begin{align*} \int{x}^{{\frac{3}{2}}} \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.50595, size = 986, normalized size = 3.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(2*(d*sqrt(x) + c)^5*a - 10*(d*sqrt(x) + c)^4*a*c + 20*(d*sqrt(x) + c)^3*a*c^2 - 20*(d*sqrt(x) + c)^2*a*c^
3 + 10*(d*sqrt(x) + c)*a*c^4 - 10*b*c^4*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c)) - 5*(2*I*(d*sqrt(x) + c)^
4*b - 8*I*(d*sqrt(x) + c)^3*b*c + 12*I*(d*sqrt(x) + c)^2*b*c^2 - 8*I*(d*sqrt(x) + c)*b*c^3)*arctan2(sin(d*sqrt
(x) + c), cos(d*sqrt(x) + c) + 1) - 5*(2*I*(d*sqrt(x) + c)^4*b - 8*I*(d*sqrt(x) + c)^3*b*c + 12*I*(d*sqrt(x) +
c)^2*b*c^2 - 8*I*(d*sqrt(x) + c)*b*c^3)*arctan2(sin(d*sqrt(x) + c), -cos(d*sqrt(x) + c) + 1) - 5*(-8*I*(d*sqr
t(x) + c)^3*b + 24*I*(d*sqrt(x) + c)^2*b*c - 24*I*(d*sqrt(x) + c)*b*c^2 + 8*I*b*c^3)*dilog(-e^(I*d*sqrt(x) + I
*c)) - 5*(8*I*(d*sqrt(x) + c)^3*b - 24*I*(d*sqrt(x) + c)^2*b*c + 24*I*(d*sqrt(x) + c)*b*c^2 - 8*I*b*c^3)*dilog
(e^(I*d*sqrt(x) + I*c)) - 5*((d*sqrt(x) + c)^4*b - 4*(d*sqrt(x) + c)^3*b*c + 6*(d*sqrt(x) + c)^2*b*c^2 - 4*(d*
sqrt(x) + c)*b*c^3)*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1) + 5*((d*sqrt(x
) + c)^4*b - 4*(d*sqrt(x) + c)^3*b*c + 6*(d*sqrt(x) + c)^2*b*c^2 - 4*(d*sqrt(x) + c)*b*c^3)*log(cos(d*sqrt(x)
+ c)^2 + sin(d*sqrt(x) + c)^2 - 2*cos(d*sqrt(x) + c) + 1) + 240*b*polylog(5, -e^(I*d*sqrt(x) + I*c)) - 240*b*p
olylog(5, e^(I*d*sqrt(x) + I*c)) - 5*(48*I*(d*sqrt(x) + c)*b - 48*I*b*c)*polylog(4, -e^(I*d*sqrt(x) + I*c)) -
5*(-48*I*(d*sqrt(x) + c)*b + 48*I*b*c)*polylog(4, e^(I*d*sqrt(x) + I*c)) - 120*((d*sqrt(x) + c)^2*b - 2*(d*sqr
t(x) + c)*b*c + b*c^2)*polylog(3, -e^(I*d*sqrt(x) + I*c)) + 120*((d*sqrt(x) + c)^2*b - 2*(d*sqrt(x) + c)*b*c +
b*c^2)*polylog(3, e^(I*d*sqrt(x) + I*c)))/d^5

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x^{\frac{3}{2}} \csc \left (d \sqrt{x} + c\right ) + a x^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \left (a + b \csc{\left (c + d \sqrt{x} \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d \sqrt{x} + c\right ) + a\right )} x^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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