\(\int \frac {x^3}{a+b \csc (c+d x^2)} \, dx\) [18]

Optimal result

Integrand size = 18, antiderivative size = 271 \[ \int \frac {x^3}{a+b \csc \left (c+d x^2\right )} \, dx=\frac {x^4}{4 a}+\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2} \]

[Out]

1/4*x^4/a+1/2*I*b*x^2*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^2*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*b*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)-1/2*b*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)

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4290, 4276, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {x^3}{a+b \csc \left (c+d x^2\right )} \, dx=\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a d^2 \sqrt {b^2-a^2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{2 a d^2 \sqrt {b^2-a^2}}+\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a d \sqrt {b^2-a^2}}-\frac {i b x^2 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a d \sqrt {b^2-a^2}}+\frac {x^4}{4 a} \]

[In]

Int[x^3/(a + b*Csc[c + d*x^2]),x]

[Out]

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

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(987\) vs. \(2(271)=542\).

Time = 5.40 (sec) , antiderivative size = 987, normalized size of antiderivative = 3.64 \[ \int \frac {x^3}{a+b \csc \left (c+d x^2\right )} \, dx=\frac {\csc \left (c+d x^2\right ) \left (x^4-\frac {2 b \left (\frac {\pi \arctan \left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {2 \left (c-\arccos \left (-\frac {b}{a}\right )\right ) \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-2 c+\pi -2 d x^2\right ) \text {arctanh}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )-\left (\arccos \left (-\frac {b}{a}\right )-2 i \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (a-b-i \sqrt {a^2-b^2}\right ) \left (1+i \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )\right )}\right )+\left (\arccos \left (-\frac {b}{a}\right )+2 i \left (-\text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+\text {arctanh}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {a^2-b^2} e^{-\frac {1}{2} i \left (c+d x^2\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \sin \left (c+d x^2\right )}}\right )+\left (\arccos \left (-\frac {b}{a}\right )+2 i \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )-2 i \text {arctanh}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {a^2-b^2} e^{\frac {1}{2} i \left (c+d x^2\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \sin \left (c+d x^2\right )}}\right )-\left (\arccos \left (-\frac {b}{a}\right )+2 i \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (1+\frac {i \left (i b+\sqrt {a^2-b^2}\right ) \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{4} \left (2 c-\pi +2 d x^2\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (b-i \sqrt {a^2-b^2}\right ) \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{4} \left (2 c-\pi +2 d x^2\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (b+i \sqrt {a^2-b^2}\right ) \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{4} \left (2 c-\pi +2 d x^2\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^2\right )\right )\right )}\right )\right )}{\sqrt {a^2-b^2}}\right )}{d^2}\right ) \left (b+a \sin \left (c+d x^2\right )\right )}{4 a \left (a+b \csc \left (c+d x^2\right )\right )} \]

[In]

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

[Out]

(Csc[c + d*x^2]*(x^4 - (2*b*((Pi*ArcTan[(a + b*Tan[(c + d*x^2)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(c
- ArcCos[-(b/a)])*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] + (-2*c + Pi - 2*d*x^2)*ArcT
anh[((a + b)*Tan[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] - (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a - b)*Cot[(2*c
+ Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 - b^2])*(1 + I*Cot[(2*c + Pi + 2*d*x^2
)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))] + (ArcCos[-(b/a)] + (2*I)*(-ArcTanh[((a - b)
*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] + ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]
]))*Log[((-1)^(1/4)*Sqrt[a^2 - b^2])/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^2))*Sqrt[b + a*Sin[c + d*x^2]])] + (Ar
cCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] - (2*I)*ArcTanh[((a + b)*T
an[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^2)))/(Sqrt[2
]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^2]]))] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4]
)/Sqrt[a^2 - b^2]])*Log[1 + (I*(I*b + Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^2)/4]))/
(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b + S
qrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^2)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))] - Poly
Log[2, ((b + I*Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^2)/4]))/(a*(a + b + Sqrt[a^2 -
b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))]))/Sqrt[a^2 - b^2]))/d^2)*(b + a*Sin[c + d*x^2]))/(4*a*(a + b*Csc[c + d*x^2
]))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (223) = 446\).

Time = 0.36 (sec) , antiderivative size = 1050, normalized size of antiderivative = 3.87 \[ \int \frac {x^3}{a+b \csc \left (c+d x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*((a^2 - b^2)*d^2*x^4 - a*b*c*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) - a*b*c*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) + a*b*c*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) + a*b*c*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) + I*a*b*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) - I*a*b*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) - I*a*b*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) + I*a*b*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) - (a*b*d*x^2 +
a*b*c)*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*b*d*x^2 + a*b*c)*sqrt((a^2 - b^2)/a^2)*log(-(I*b*cos(d*x^2 + c) - b*si
n(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*b*d*x^2 + a*b*c)*sqr
t((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*b*d*x^2 + a*b*c)*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^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/4*(x^4 - 8*a*b*integrate((2*b*x^3*cos(d*x^2 + c)^2 + a*x^3*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) - a*x^3*cos(2*d
*x^2 + 2*c)*sin(d*x^2 + c) + 2*b*x^3*sin(d*x^2 + c)^2 + a*x^3*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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^3/(a + b/sin(c + d*x^2)),x)

[Out]

int(x^3/(a + b/sin(c + d*x^2)), x)