3.1.36 \(\int \frac {x^3}{a+b \sin (c+d x^2)} \, dx\) [36]

3.1.36.1 Optimal result

Integrand size = 18, antiderivative size = 245 \[ \int \frac {x^3}{a+b \sin \left (c+d x^2\right )} \, dx=-\frac {i x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}-\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d^2}+\frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d^2} \]

output

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

3.1.36.2 Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{a+b \sin \left (c+d x^2\right )} \, dx=\frac {-i d x^2 \left (\log \left (1+\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )-\log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )+\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d^2} \]

input

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

output

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

3.1.36.3 Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3860, 3042, 3804, 2694, 27, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3}{a+b \sin \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 3860

\(\displaystyle \frac {1}{2} \int \frac {x^2}{a+b \sin \left (d x^2+c\right )}dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {x^2}{a+b \sin \left (d x^2+c\right )}dx^2\)

\(\Big \downarrow \) 3804

\(\displaystyle \int \frac {x^2 e^{i \left (c+d x^2\right )}}{2 a e^{i \left (c+d x^2\right )}-i b e^{2 i \left (c+d x^2\right )}+i b}dx^2\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^2}{2 \left (a-i b e^{i \left (d x^2+c\right )}+\sqrt {a^2-b^2}\right )}dx^2}{\sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^2}{2 \left (a-i b e^{i \left (d x^2+c\right )}-\sqrt {a^2-b^2}\right )}dx^2}{\sqrt {a^2-b^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^2}{a-i b e^{i \left (d x^2+c\right )}+\sqrt {a^2-b^2}}dx^2}{2 \sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^2}{a-i b e^{i \left (d x^2+c\right )}-\sqrt {a^2-b^2}}dx^2}{2 \sqrt {a^2-b^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {i b \left (\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {\int \log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )dx^2}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\int \log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )dx^2}{b d}\right )}{2 \sqrt {a^2-b^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {i b \left (\frac {i \int \frac {\log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{b d^2}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {i \int \frac {\log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{b d^2}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {i b \left (\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}\right )}{2 \sqrt {a^2-b^2}}\)

input

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

output

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

3.1.36.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2694

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]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[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 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

3.1.36.4 Maple [F]

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

input

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

output

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

3.1.36.5 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. 1041 vs. \(2 (199) = 398\).

Time = 0.42 (sec) , antiderivative size = 1041, normalized size of antiderivative = 4.25 \[ \int \frac {x^3}{a+b \sin \left (c+d x^2\right )} \, dx=\text {Too large to display} \]

input

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

output

-1/4*(b*c*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x^2 + c) + 2*I*b*sin(d*x^2 
+ c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) + b*c*sqrt(-(a^2 - b^2)/b^2)*lo 
g(2*b*cos(d*x^2 + c) - 2*I*b*sin(d*x^2 + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 
 2*I*a) - b*c*sqrt(-(a^2 - b^2)/b^2)*log(-2*b*cos(d*x^2 + c) + 2*I*b*sin(d 
*x^2 + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - b*c*sqrt(-(a^2 - b^2)/b^ 
2)*log(-2*b*cos(d*x^2 + c) - 2*I*b*sin(d*x^2 + c) + 2*b*sqrt(-(a^2 - b^2)/ 
b^2) - 2*I*a) - I*b*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x^2 + c) - a*s 
in(d*x^2 + c) + (b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/ 
b^2) - b)/b + 1) + I*b*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x^2 + c) - 
a*sin(d*x^2 + c) - (b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c))*sqrt(-(a^2 - b^ 
2)/b^2) - b)/b + 1) + I*b*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x^2 + c 
) - a*sin(d*x^2 + c) + (b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c))*sqrt(-(a^2 
- b^2)/b^2) - b)/b + 1) - I*b*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x^2 
 + c) - a*sin(d*x^2 + c) - (b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c))*sqrt(-( 
a^2 - b^2)/b^2) - b)/b + 1) + (b*d*x^2 + b*c)*sqrt(-(a^2 - b^2)/b^2)*log(- 
(I*a*cos(d*x^2 + c) - a*sin(d*x^2 + c) + (b*cos(d*x^2 + c) + I*b*sin(d*x^2 
 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) - (b*d*x^2 + b*c)*sqrt(-(a^2 - b^2)/ 
b^2)*log(-(I*a*cos(d*x^2 + c) - a*sin(d*x^2 + c) - (b*cos(d*x^2 + c) + I*b 
*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) + (b*d*x^2 + b*c)*sqrt(-(a 
^2 - b^2)/b^2)*log(-(-I*a*cos(d*x^2 + c) - a*sin(d*x^2 + c) + (b*cos(d*...
 

3.1.36.6 Sympy [F]

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

input

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

output

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

3.1.36.7 Maxima [F]

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

input

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

output

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

3.1.36.8 Giac [F]

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

input

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

output

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

3.1.36.9 Mupad [F(-1)]

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

input

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

output

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