\(\int \frac {\sin (c+d x)}{x (a+b x^3)} \, dx\) [99]

Optimal result

Integrand size = 19, antiderivative size = 301 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )} \, dx=\frac {\operatorname {CosIntegral}(d x) \sin (c)}{a}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\cos (c) \text {Si}(d x)}{a}+\frac {\cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a}-\frac {\cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a} \] Output:

Ci(d*x)*sin(c)/a-1/3*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a- 
1/3*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3 
))/a-1/3*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b 
^(1/3))/a+cos(c)*Si(d*x)/a-1/3*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Si(-(-1 
)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a-1/3*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d 
/b^(1/3)+d*x)/a-1/3*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1 
/3)*d/b^(1/3)+d*x)/a
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.68 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )} \, dx=\frac {-i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-i \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))\&\right ]+i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))+i \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))\&\right ]+6 \operatorname {CosIntegral}(d x) \sin (c)+6 \cos (c) \text {Si}(d x)}{6 a} \] Input:

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

Output:

((-I)*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*Cos 
Integral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1 
)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] + I*RootSum[a + b*#1^3 & , 
Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + 
d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegra 
l[d*(x - #1)] & ] + 6*CosIntegral[d*x]*Sin[c] + 6*Cos[c]*SinIntegral[d*x]) 
/(6*a)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3826, 2009}

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.

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

\(\Big \downarrow \) 3826

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\sin (c) \operatorname {CosIntegral}(d x)}{a}+\frac {\cos (c) \text {Si}(d x)}{a}\)

Input:

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

Output:

(CosIntegral[d*x]*Sin[c])/a - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[ 
c - (a^(1/3)*d)/b^(1/3)])/(3*a) - (CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1 
/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a) - (CosIntegral[( 
(-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3 
)])/(3*a) + (Cos[c]*SinIntegral[d*x])/a + (Cos[c + ((-1)^(1/3)*a^(1/3)*d)/ 
b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a) - (Cos[c 
 - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a) - (C 
os[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/ 
b^(1/3) + d*x])/(3*a)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3826

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym 
bol] :> Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Free 
Q[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, - 
1]) && IntegerQ[m]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.96 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.29

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\left (-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{3 a}\)	\(88\)
default	\(\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\left (-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{3 a}\)	\(88\)
risch	\(\frac {i {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a}-\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 \textit {\_Z} b \,c^{2}\right )}{\sum }{\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-i d x -i c +\textit {\_R1} \right )\right )}{6 a}-\frac {{\mathrm e}^{-i c} \pi \,\operatorname {csgn}\left (d x \right )}{2 a}+\frac {{\mathrm e}^{-i c} \operatorname {Si}\left (d x \right )}{a}-\frac {i {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 \textit {\_Z} b \,c^{2}\right )}{\sum }{\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (i d x +i c -\textit {\_R1} \right )\right )}{6 a}\)	\(192\)

Input:

int(sin(d*x+c)/x/(b*x^3+a),x,method=_RETURNVERBOSE)
 

Output:

1/a*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))-1/3/a*sum(-Si(-d*x+_R1-c)*cos(_R1)+Ci( 
d*x-_R1+c)*sin(_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.02 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )} \, dx=\frac {-i \, {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + i \, {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} - i \, {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + i \, {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} + i \, {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - i \, {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} + 6 \, \operatorname {Ci}\left (d x\right ) \sin \left (c\right ) + 6 \, \cos \left (c\right ) \operatorname {Si}\left (d x\right )}{6 \, a} \] Input:

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

Output:

1/6*(-I*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(I*a*d^ 
3/b)^(1/3)*(I*sqrt(3) + 1) - I*c) + I*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(- 
I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) + I*c) - I*Ei(-I 
*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(I*a*d^3/b)^(1/3)*(-I 
*sqrt(3) + 1) - I*c) + I*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1) 
)*e^(1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) + I*c) + I*Ei(I*d*x + (-I*a*d 
^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) - I*Ei(-I*d*x + (I*a*d^3/b)^(1/3 
))*e^(-I*c - (I*a*d^3/b)^(1/3)) + 6*cos_integral(d*x)*sin(c) + 6*cos(c)*si 
n_integral(d*x))/a
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )} \, dx=\int \frac {\sin \left (d x +c \right )}{b \,x^{4}+a x}d x \] Input:

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

Output:

int(sin(c + d*x)/(a*x + b*x**4),x)