Integrand size = 19, antiderivative size = 693 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Output:
1/9*(-1)^(1/3)*d*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Ci((-1)^(1/3)*a^(1/3) *d/b^(1/3)-d*x)/a^(5/3)/b^(1/3)-1/9*d*cos(c-a^(1/3)*d/b^(1/3))*Ci(a^(1/3)* d/b^(1/3)+d*x)/a^(5/3)/b^(1/3)-1/9*(-1)^(2/3)*d*cos(c-(-1)^(2/3)*a^(1/3)*d /b^(1/3))*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(1/3)+Ci(d*x)*sin (c)/a^2-1/3*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^2-1/3*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^2- 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^2+1/3*sin(d*x+c)/a/b/x^3-1/3*sin(d*x+c)/b/x^3/(b*x^3+a)+cos(c)*Si(d*x )/a^2-1/3*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^2-1/9*(-1)^(1/3)*d*sin(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^(5/3)/b^(1/3)-1/3*cos(c-a^(1/3)*d/b^(1/3) )*Si(a^(1/3)*d/b^(1/3)+d*x)/a^2+1/9*d*sin(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)* d/b^(1/3)+d*x)/a^(5/3)/b^(1/3)-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^2+1/9*(-1)^(2/3)*d*sin(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^(5/3)/b^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.63 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.64 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\frac {-\frac {1}{2} i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-i \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))\&\right ]+\frac {1}{2} i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))+i \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))\&\right ]-\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-i \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{6 b}-\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))+i \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{6 b}+\frac {a \cos (d x) \sin (c)}{a+b x^3}+3 \operatorname {CosIntegral}(d x) \sin (c)+\frac {a \cos (c) \sin (d x)}{a+b x^3}+3 \cos (c) \text {Si}(d x)}{3 a^2} \] Input:
Integrate[Sin[c + d*x]/(x*(a + b*x^3)^2),x]
Output:
((-1/2*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]*SinIntegral[d*(x - #1)] & ] + (I/2)*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]*Si nIntegral[d*(x - #1)] & ] - (a*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*Cos Integral[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]*SinIntegral[d*(x - #1)])/#1 ^2 & ])/(6*b) - (a*d*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]*SinIn tegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ])/(6*b ) + (a*Cos[d*x]*Sin[c])/(a + b*x^3) + 3*CosIntegral[d*x]*Sin[c] + (a*Cos[c ]*Sin[d*x])/(a + b*x^3) + 3*Cos[c]*SinIntegral[d*x])/(3*a^2)
Time = 1.98 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3824, 3826, 2009, 3827, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\int \frac {\sin (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle -\frac {\int \left (\frac {b^2 \sin (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \sin (c+d x)}{a^2 x}+\frac {\sin (c+d x)}{a x^4}\right )dx}{b}+\frac {d \int \frac {\cos (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {-\frac {b \sin (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {d^3 \cos (c) \operatorname {CosIntegral}(d x)}{6 a}+\frac {d^3 \sin (c) \text {Si}(d x)}{6 a}+\frac {d^2 \sin (c+d x)}{6 a x}-\frac {\sin (c+d x)}{3 a x^3}-\frac {d \cos (c+d x)}{6 a x^2}}{b}-\frac {\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 3827 |
| \(\displaystyle \frac {d \int \left (\frac {\cos (c+d x)}{a x^3}-\frac {b \cos (c+d x)}{a \left (b x^3+a\right )}\right )dx}{3 b}-\frac {-\frac {b \sin (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {b \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {d^3 \cos (c) \operatorname {CosIntegral}(d x)}{6 a}+\frac {d^3 \sin (c) \text {Si}(d x)}{6 a}+\frac {d^2 \sin (c+d x)}{6 a x}-\frac {\sin (c+d x)}{3 a x^3}-\frac {d \cos (c+d x)}{6 a x^2}}{b}-\frac {\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {-\frac {\cos (c) \operatorname {CosIntegral}(d x) d^3}{6 a}+\frac {\sin (c) \text {Si}(d x) d^3}{6 a}+\frac {\sin (c+d x) d^2}{6 a x}-\frac {\cos (c+d x) d}{6 a x^2}-\frac {b \operatorname {CosIntegral}(d x) \sin (c)}{a^2}+\frac {b \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {b \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^2}+\frac {b \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\sin (c+d x)}{3 a x^3}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \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^2}+\frac {b \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^2}+\frac {b \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^2}}{b}+\frac {d \left (-\frac {\cos (c) \operatorname {CosIntegral}(d x) d^2}{2 a}+\frac {\sin (c) \text {Si}(d x) d^2}{2 a}+\frac {\sin (c+d x) d}{2 a x}-\frac {\cos (c+d x)}{2 a x^2}+\frac {\sqrt [3]{-1} b^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \cos \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^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cos \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^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \sin \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^{5/3}}+\frac {b^{2/3} \sin \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^{5/3}}+\frac {(-1)^{2/3} b^{2/3} \sin \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^{5/3}}\right )}{3 b}\) |
Input:
Int[Sin[c + d*x]/(x*(a + b*x^3)^2),x]
Output:
-1/3*Sin[c + d*x]/(b*x^3*(a + b*x^3)) - (-1/6*(d*Cos[c + d*x])/(a*x^2) - ( d^3*Cos[c]*CosIntegral[d*x])/(6*a) - (b*CosIntegral[d*x]*Sin[c])/a^2 + (b* CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^ 2) + (b*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^2) + (b*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^2) - Sin[c + d *x]/(3*a*x^3) + (d^2*Sin[c + d*x])/(6*a*x) - (b*Cos[c]*SinIntegral[d*x])/a ^2 + (d^3*Sin[c]*SinIntegral[d*x])/(6*a) - (b*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^2) + ( b*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3* a^2) + (b*Cos[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^2))/b + (d*(-1/2*Cos[c + d*x]/(a*x^2) - (d ^2*Cos[c]*CosIntegral[d*x])/(2*a) + ((-1)^(1/3)*b^(2/3)*Cos[c + ((-1)^(1/3 )*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/( 3*a^(5/3)) - (b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d) /b^(1/3) + d*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Cos[c - ((-1)^(2/3)*a^( 1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^( 5/3)) + (d*Sin[c + d*x])/(2*a*x) + (d^2*Sin[c]*SinIntegral[d*x])/(2*a) + ( (-1)^(1/3)*b^(2/3)*Sin[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^(5/3)) + (b^(2/3)*Sin[c - (a^(...
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))) , x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1) *Sin[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b*x^n )^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
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]
Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sym bol] :> Int[ExpandIntegrand[Cos[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.30 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.34
| method | result | size |
| derivativedivides | \(\frac {\sin \left (d x +c \right ) d^{3}}{3 a \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\left (-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{3 a^{2}}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {d^{3} \left (\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{9 a b}\) | \(233\) |
| default | \(\frac {\sin \left (d x +c \right ) d^{3}}{3 a \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\left (-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{3 a^{2}}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {d^{3} \left (\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{9 a b}\) | \(233\) |
| risch | \(\frac {i {\mathrm e}^{i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a^{2}}-\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 \textit {\_Z} b \,c^{2}\right )}{\sum }\frac {\left (-i d^{3} a -6 i \textit {\_R1} b c +3 \textit {\_R1}^{2} b -3 b \,c^{2}\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2} b}-\frac {{\mathrm e}^{-i c} \pi \,\operatorname {csgn}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-i c} \operatorname {Si}\left (d x \right )}{a^{2}}-\frac {i {\mathrm e}^{-i c} \operatorname {expIntegral}_{1}\left (-i d x \right )}{2 a^{2}}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 \textit {\_Z} b \,c^{2}\right )}{\sum }\frac {\left (i d^{3} a -6 i \textit {\_R1} b c +3 \textit {\_R1}^{2} b -3 b \,c^{2}\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2} b}+\frac {d^{3} \sin \left (d x +c \right )}{3 a \left (d^{3} x^{3} b +a \,d^{3}\right )}\) | \(312\) |
Input:
int(sin(d*x+c)/x/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
1/3*sin(d*x+c)*d^3/a/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c )^3)-1/3/a^2*sum(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1),_R1=RootO f(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/a^2*(Si(d*x)*cos(c)+Ci(d*x) *sin(c))-1/9*d^3/a/b*sum(1/(_RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_RR1-c)*sin(_RR1 )+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3 -b*c^3))
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.83 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(sin(d*x+c)/x/(b*x^3+a)^2,x, algorithm="fricas")
Output:
1/36*((-6*I*b*x^3 + (I*b*x^3 - sqrt(3)*(b*x^3 + a) + I*a)*(I*a*d^3/b)^(1/3 ) - 6*I*a)*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) + (6*I*b*x^3 + (-I*b*x^3 + sqrt(3)*(b *x^3 + a) - I*a)*(-I*a*d^3/b)^(1/3) + 6*I*a)*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) + (-6*I*b*x^3 + (I*b*x^3 + sqrt(3)*(b*x^3 + a) + I*a)*(I*a*d^3/b)^(1/3) - 6* I*a)*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) + (6*I*b*x^3 + (-I*b*x^3 - sqrt(3)*(b*x^3 + a) - I*a)*(-I*a*d^3/b)^(1/3) + 6*I*a)*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) - 2*(-3* I*b*x^3 + (-I*b*x^3 - I*a)*(-I*a*d^3/b)^(1/3) - 3*I*a)*Ei(I*d*x + (-I*a*d^ 3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) - 2*(3*I*b*x^3 + (I*b*x^3 + I*a)* (I*a*d^3/b)^(1/3) + 3*I*a)*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d ^3/b)^(1/3)) + 36*(b*x^3 + a)*cos_integral(d*x)*sin(c) + 36*(b*x^3 + a)*co s(c)*sin_integral(d*x) + 12*a*sin(d*x + c))/(a^2*b*x^3 + a^3)
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x^{3}\right )^{2}}\, dx \] Input:
integrate(sin(d*x+c)/x/(b*x**3+a)**2,x)
Output:
Integral(sin(c + d*x)/(x*(a + b*x**3)**2), x)
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x} \,d x } \] Input:
integrate(sin(d*x+c)/x/(b*x^3+a)^2,x, algorithm="maxima")
Output:
integrate(sin(d*x + c)/((b*x^3 + a)^2*x), x)
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x} \,d x } \] Input:
integrate(sin(d*x+c)/x/(b*x^3+a)^2,x, algorithm="giac")
Output:
integrate(sin(d*x + c)/((b*x^3 + a)^2*x), x)
Timed out. \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x\,{\left (b\,x^3+a\right )}^2} \,d x \] Input:
int(sin(c + d*x)/(x*(a + b*x^3)^2),x)
Output:
int(sin(c + d*x)/(x*(a + b*x^3)^2), x)
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx=\int \frac {\sin \left (d x +c \right )}{b^{2} x^{7}+2 a b \,x^{4}+a^{2} x}d x \] Input:
int(sin(d*x+c)/x/(b*x^3+a)^2,x)
Output:
int(sin(c + d*x)/(a**2*x + 2*a*b*x**4 + b**2*x**7),x)