3.1.16 \(\int \frac {x^5}{a+b \csc (c+d x^2)} \, dx\) [16]

Optimal. Leaf size=396 \[ \frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {i b \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {i b \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3} \]

[Out]

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

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Rubi [A]
time = 0.70, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4290, 4276, 3404, 2296, 2221, 2611, 2320, 6724} \begin {gather*} \frac {i b \text {Li}_3\left (\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {i b \text {Li}_3\left (\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}+\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a d \sqrt {b^2-a^2}}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a d \sqrt {b^2-a^2}}+\frac {x^6}{6 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Csc[c + d*x^2]),x]

[Out]

x^6/(6*a) + ((I/2)*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((I
/2)*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*x^2*PolyLog[2,
(I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (b*x^2*PolyLog[2, (I*a*E^(I*(c + d
*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + (I*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[
-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (I*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*
Sqrt[-a^2 + b^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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& 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]

Rubi steps

\begin {align*} \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b \csc (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {x^2}{a}-\frac {b x^2}{a (b+a \sin (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac {x^6}{6 a}-\frac {b \text {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac {x^6}{6 a}-\frac {b \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a}\\ &=\frac {x^6}{6 a}+\frac {(i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt {-a^2+b^2}}-\frac {(i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt {-a^2+b^2}}\\ &=\frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {(i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d}+\frac {(i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d}\\ &=\frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {b \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d^2}\\ &=\frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a \sqrt {-a^2+b^2} d^3}\\ &=\frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {i b \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {i b \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}\\ \end {align*}

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Mathematica [A]
time = 1.28, size = 488, normalized size = 1.23 \begin {gather*} \frac {d^3 \sqrt {\left (a^2-b^2\right ) e^{2 i c}} x^6-3 b d^2 e^{i c} x^4 \log \left (1+\frac {a e^{i \left (2 c+d x^2\right )}}{i b e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+3 b d^2 e^{i c} x^4 \log \left (1+\frac {a e^{i \left (2 c+d x^2\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+6 i b d e^{i c} x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (2 c+d x^2\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-6 i b d e^{i c} x^2 \text {PolyLog}\left (2,-\frac {a e^{i \left (2 c+d x^2\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-6 b e^{i c} \text {PolyLog}\left (3,\frac {i a e^{i \left (2 c+d x^2\right )}}{b e^{i c}+i \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+6 b e^{i c} \text {PolyLog}\left (3,-\frac {a e^{i \left (2 c+d x^2\right )}}{i b e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )}{6 a d^3 \sqrt {\left (a^2-b^2\right ) e^{2 i c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^6 - 3*b*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sq
rt[(a^2 - b^2)*E^((2*I)*c)])] + 3*b*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 -
b^2)*E^((2*I)*c)])] + (6*I)*b*d*E^(I*c)*x^2*PolyLog[2, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b
^2)*E^((2*I)*c)])] - (6*I)*b*d*E^(I*c)*x^2*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2
)*E^((2*I)*c)]))] - 6*b*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*
c)])] + 6*b*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])/(6*a
*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)])

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{5}}{a +b \csc \left (d \,x^{2}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(x^6 - 12*a*b*integrate((2*b*x^5*cos(d*x^2 + c)^2 + a*x^5*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) - a*x^5*cos(2*
d*x^2 + 2*c)*sin(d*x^2 + c) + 2*b*x^5*sin(d*x^2 + c)^2 + a*x^5*sin(d*x^2 + c))/(a^3*cos(2*d*x^2 + 2*c)^2 + 4*a
*b^2*cos(d*x^2 + c)^2 + 4*a^2*b*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + a^3*sin(2*d*x^2 + 2*c)^2 + 4*a*b^2*sin(d*x
^2 + c)^2 + 4*a^2*b*sin(d*x^2 + c) + a^3 - 2*(2*a^2*b*sin(d*x^2 + c) + a^3)*cos(2*d*x^2 + 2*c)), x))/a

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (332) = 664\).
time = 3.70, size = 1445, normalized size = 3.65 \begin {gather*} \frac {2 \, {\left (a^{2} - b^{2}\right )} d^{3} x^{6} + 6 i \, a b d x^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) + {\left (a \cos \left (d x^{2} + c\right ) + i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a} + 1\right ) - 6 i \, a b d x^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) - {\left (a \cos \left (d x^{2} + c\right ) + i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a} + 1\right ) - 6 i \, a b d x^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {-i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) + {\left (a \cos \left (d x^{2} + c\right ) - i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a} + 1\right ) + 6 i \, a b d x^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {-i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) - {\left (a \cos \left (d x^{2} + c\right ) - i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a} + 1\right ) + 3 \, a b c^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cos \left (d x^{2} + c\right ) + 2 i \, a \sin \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + 2 i \, b\right ) + 3 \, a b c^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cos \left (d x^{2} + c\right ) - 2 i \, a \sin \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - 2 i \, b\right ) - 3 \, a b c^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (-2 \, a \cos \left (d x^{2} + c\right ) + 2 i \, a \sin \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + 2 i \, b\right ) - 3 \, a b c^{2} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (-2 \, a \cos \left (d x^{2} + c\right ) - 2 i \, a \sin \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - 2 i \, b\right ) + 6 \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm polylog}\left (3, -\frac {i \, b \cos \left (d x^{2} + c\right ) + b \sin \left (d x^{2} + c\right ) + {\left (a \cos \left (d x^{2} + c\right ) - i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}}}{a}\right ) - 6 \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm polylog}\left (3, -\frac {i \, b \cos \left (d x^{2} + c\right ) + b \sin \left (d x^{2} + c\right ) - {\left (a \cos \left (d x^{2} + c\right ) - i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}}}{a}\right ) + 6 \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm polylog}\left (3, -\frac {-i \, b \cos \left (d x^{2} + c\right ) + b \sin \left (d x^{2} + c\right ) + {\left (a \cos \left (d x^{2} + c\right ) + i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}}}{a}\right ) - 6 \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm polylog}\left (3, -\frac {-i \, b \cos \left (d x^{2} + c\right ) + b \sin \left (d x^{2} + c\right ) - {\left (a \cos \left (d x^{2} + c\right ) + i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}}}{a}\right ) - 3 \, {\left (a b d^{2} x^{4} - a b c^{2}\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (-\frac {i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) + {\left (a \cos \left (d x^{2} + c\right ) + i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a}\right ) + 3 \, {\left (a b d^{2} x^{4} - a b c^{2}\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (-\frac {i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) - {\left (a \cos \left (d x^{2} + c\right ) + i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a}\right ) - 3 \, {\left (a b d^{2} x^{4} - a b c^{2}\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (-\frac {-i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) + {\left (a \cos \left (d x^{2} + c\right ) - i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a}\right ) + 3 \, {\left (a b d^{2} x^{4} - a b c^{2}\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} \log \left (-\frac {-i \, b \cos \left (d x^{2} + c\right ) - b \sin \left (d x^{2} + c\right ) - {\left (a \cos \left (d x^{2} + c\right ) - i \, a \sin \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - a}{a}\right )}{12 \, {\left (a^{3} - a b^{2}\right )} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*(a^2 - b^2)*d^3*x^6 + 6*I*a*b*d*x^2*sqrt((a^2 - b^2)/a^2)*dilog((I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c)
+ (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a + 1) - 6*I*a*b*d*x^2*sqrt((a^2 - b^2)/
a^2)*dilog((I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a
^2) - a)/a + 1) - 6*I*a*b*d*x^2*sqrt((a^2 - b^2)/a^2)*dilog((-I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d
*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a + 1) + 6*I*a*b*d*x^2*sqrt((a^2 - b^2)/a^2)*dilog(
(-I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a
+ 1) + 3*a*b*c^2*sqrt((a^2 - b^2)/a^2)*log(2*a*cos(d*x^2 + c) + 2*I*a*sin(d*x^2 + c) + 2*a*sqrt((a^2 - b^2)/a
^2) + 2*I*b) + 3*a*b*c^2*sqrt((a^2 - b^2)/a^2)*log(2*a*cos(d*x^2 + c) - 2*I*a*sin(d*x^2 + c) + 2*a*sqrt((a^2 -
b^2)/a^2) - 2*I*b) - 3*a*b*c^2*sqrt((a^2 - b^2)/a^2)*log(-2*a*cos(d*x^2 + c) + 2*I*a*sin(d*x^2 + c) + 2*a*sqr
t((a^2 - b^2)/a^2) + 2*I*b) - 3*a*b*c^2*sqrt((a^2 - b^2)/a^2)*log(-2*a*cos(d*x^2 + c) - 2*I*a*sin(d*x^2 + c) +
2*a*sqrt((a^2 - b^2)/a^2) - 2*I*b) + 6*a*b*sqrt((a^2 - b^2)/a^2)*polylog(3, -(I*b*cos(d*x^2 + c) + b*sin(d*x^
2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))/a) - 6*a*b*sqrt((a^2 - b^2)/a^2)*polyl
og(3, -(I*b*cos(d*x^2 + c) + b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))
/a) + 6*a*b*sqrt((a^2 - b^2)/a^2)*polylog(3, -(-I*b*cos(d*x^2 + c) + b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*
a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))/a) - 6*a*b*sqrt((a^2 - b^2)/a^2)*polylog(3, -(-I*b*cos(d*x^2 + c) + b
*sin(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2))/a) - 3*(a*b*d^2*x^4 - a*b*c^2
)*sqrt((a^2 - b^2)/a^2)*log(-(I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*
sqrt((a^2 - b^2)/a^2) - a)/a) + 3*(a*b*d^2*x^4 - a*b*c^2)*sqrt((a^2 - b^2)/a^2)*log(-(I*b*cos(d*x^2 + c) - b*s
in(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a) - 3*(a*b*d^2*x^4 - a*b*c
^2)*sqrt((a^2 - b^2)/a^2)*log(-(-I*b*cos(d*x^2 + c) - b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c
))*sqrt((a^2 - b^2)/a^2) - a)/a) + 3*(a*b*d^2*x^4 - a*b*c^2)*sqrt((a^2 - b^2)/a^2)*log(-(-I*b*cos(d*x^2 + c) -
b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt((a^2 - b^2)/a^2) - a)/a))/((a^3 - a*b^2)*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{a + b \csc {\left (c + d x^{2} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{a+\frac {b}{\sin \left (d\,x^2+c\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b/sin(c + d*x^2)),x)

[Out]

int(x^5/(a + b/sin(c + d*x^2)), x)

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