Integrand size = 16, antiderivative size = 735 \[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Output:
1/9*(-1)^(2/3)*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^(4/3)/b^(2/3)+1/9*d*cos(c-a^(1/3)*d/b^(1/3))*Ci(a^(1/3)* d/b^(1/3)+d*x)/a^(4/3)/b^(2/3)-1/9*(-1)^(1/3)*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^(4/3)/b^(2/3)+2/9*Ci(a^(1 /3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(1/3)-2/9*(-1)^(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^(5/3)/b^(1/3)+2/9*(-1)^(2/3)*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^(5/3)/b^(1/3)+1/3*sin(d*x+c)/a/b/x^2-1/3*s in(d*x+c)/b/x^2/(b*x^3+a)-2/9*(-1)^(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^(5/3)/b^(1/3)-1/9*(-1)^(2/3)*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^(4/3)/b^(2/3)+2/9*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^( 5/3)/b^(1/3)-1/9*d*sin(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(4 /3)/b^(2/3)+2/9*(-1)^(2/3)*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^(5/3)/b^(1/3)+1/9*(-1)^(1/3)*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^(4/3)/b^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.26 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.55 \[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx=-\frac {\left (a+b x^3\right ) \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-2 i \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-2 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+2 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}-i d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+\left (a+b x^3\right ) \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {2 i \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-2 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-2 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}+i d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}+i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-6 b x \sin (c+d x)}{18 a b \left (a+b x^3\right )} \] Input:
Integrate[Sin[c + d*x]/(a + b*x^3)^2,x]
Output:
-1/18*((a + b*x^3)*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegra l[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]* SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d* Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin [c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d *#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + (a + b*x^3)*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - # 1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]* #1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinI ntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - 6*b*x*Sin[c + d*x])/(a*b*(a + b*x^3))
Time = 1.87 (sec) , antiderivative size = 830, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3812, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 3812 |
| \(\displaystyle -\frac {2 \int \frac {\sin (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^2 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle -\frac {2 \int \left (\frac {\sin (c+d x)}{a x^3}-\frac {b \sin (c+d x)}{a \left (b x^3+a\right )}\right )dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^2 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x^2 \left (b x^3+a\right )}dx}{3 b}-\frac {2 \left (-\frac {b^{2/3} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {\sqrt [3]{-1} b^{2/3} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x}\right )}{3 b}-\frac {\sin (c+d x)}{3 b x^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 3827 |
| \(\displaystyle \frac {d \int \left (\frac {\cos (c+d x)}{a x^2}-\frac {b x \cos (c+d x)}{a \left (b x^3+a\right )}\right )dx}{3 b}-\frac {2 \left (-\frac {b^{2/3} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {\sqrt [3]{-1} b^{2/3} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac {b^{2/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac {d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x}\right )}{3 b}-\frac {\sin (c+d x)}{3 b x^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{3 b x^2 \left (b x^3+a\right )}-\frac {2 \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^2}{2 a}-\frac {\cos (c) \text {Si}(d x) d^2}{2 a}-\frac {\cos (c+d x) d}{2 a x}-\frac {b^{2/3} \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^{5/3}}+\frac {\sqrt [3]{-1} b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}-\frac {\sin (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 ) \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} \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^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}\right )}{3 b}+\frac {d \left (-\frac {\cos (c+d x)}{a x}+\frac {(-1)^{2/3} \sqrt [3]{b} \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^{4/3}}+\frac {\sqrt [3]{b} \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^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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^{4/3}}-\frac {d \operatorname {CosIntegral}(d x) \sin (c)}{a}-\frac {d \cos (c) \text {Si}(d x)}{a}+\frac {(-1)^{2/3} \sqrt [3]{b} \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^{4/3}}-\frac {\sqrt [3]{b} \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^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} \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^{4/3}}\right )}{3 b}\) |
Input:
Int[Sin[c + d*x]/(a + b*x^3)^2,x]
Output:
-1/3*Sin[c + d*x]/(b*x^2*(a + b*x^3)) - (2*(-1/2*(d*Cos[c + d*x])/(a*x) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) - (b^(2/3)*CosIntegral[(a^(1/3)*d)/b^( 1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3 )*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^(5/3)) - ((-1)^(2/3)*b^(2/3)*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 ^(5/3)) - Sin[c + d*x]/(2*a*x^2) - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) - ( (-1)^(1/3)*b^(2/3)*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^(5/3)) - (b^(2/3)*Cos[c - (a^(1/3 )*d)/b^(1/3)]*SinIntegral[(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)]*SinIntegral[((-1)^(2 /3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3))))/(3*b) + (d*(-(Cos[c + d*x]/(a *x)) + ((-1)^(2/3)*b^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosInte gral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(4/3)) + (b^(1/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*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^(4/3)) - (d*CosIntegral[d*x]*S in[c])/a - (d*Cos[c]*SinIntegral[d*x])/a + ((-1)^(2/3)*b^(1/3)*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^(4/3)) - (b^(1/3)*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(...
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Sim p[x^(-n + 1)*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), x] + (-Simp[ (-n + 1)/(b*n*(p + 1)) Int[((a + b*x^n)^(p + 1)*Sin[c + d*x])/x^n, x], x] - Simp[d/(b*n*(p + 1)) Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]
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 = 1.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.34
| method | result | size |
| derivativedivides | \(d^{5} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}+\frac {\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a \,d^{3} b}\right )\) | \(248\) |
| default | \(d^{5} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}+\frac {\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a \,d^{3} b}\right )\) | \(248\) |
| risch | \(-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 \textit {\_Z} b \,c^{2}\right )}{\sum }\frac {\left (i \textit {\_R1} +c -2 i\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {expIntegral}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{18 b a}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 \textit {\_Z} b \,c^{2}\right )}{\sum }\frac {\left (i \textit {\_R1} +c +2 i\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (i d x +i c -\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{18 b a}+\frac {d^{2} \left (c +i \left (i d x +i c \right )\right ) \sin \left (d x +c \right )}{3 \left (-3 \left (i d x +i c \right )^{2} b c -a \,d^{3}+b \,c^{3}-i b \left (i d x +i c \right )^{3}+3 i b \left (i d x +i c \right ) c^{2}\right ) a}\) | \(278\) |
Input:
int(sin(d*x+c)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
d^5*(sin(d*x+c)*(1/3/a/d^3*(d*x+c)-1/3*c/a/d^3)/(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)+2/9/a/d^3/b*sum(1/(_R1^2-2*_R1*c+c^2)*(-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))+1/9/a/d^3/b*sum(1/(-_RR1+c)*(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 = 0.11 (sec) , antiderivative size = 669, normalized size of antiderivative = 0.91 \[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(sin(d*x+c)/(b*x^3+a)^2,x, algorithm="fricas")
Output:
1/36*(12*a*d*x*sin(d*x + c) + ((b*x^3 + sqrt(3)*(-I*b*x^3 - I*a) + a)*(I*a *d^3/b)^(2/3) + 2*(b*x^3 - sqrt(3)*(-I*b*x^3 - I*a) + a)*(I*a*d^3/b)^(1/3) )*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) + ((b*x^3 + sqrt(3)*(-I*b*x^3 - I*a) + a)*(-I* a*d^3/b)^(2/3) + 2*(b*x^3 - sqrt(3)*(-I*b*x^3 - I*a) + a)*(-I*a*d^3/b)^(1/ 3))*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) + ((b*x^3 + sqrt(3)*(I*b*x^3 + I*a) + a)*(I *a*d^3/b)^(2/3) + 2*(b*x^3 - sqrt(3)*(I*b*x^3 + I*a) + a)*(I*a*d^3/b)^(1/3 ))*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) + ((b*x^3 + sqrt(3)*(I*b*x^3 + I*a) + a)*(-I* a*d^3/b)^(2/3) + 2*(b*x^3 - sqrt(3)*(I*b*x^3 + I*a) + a)*(-I*a*d^3/b)^(1/3 ))*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*((b*x^3 + a)*(-I*a*d^3/b)^(2/3) + 2*(b*x ^3 + a)*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a* d^3/b)^(1/3)) - 2*((b*x^3 + a)*(I*a*d^3/b)^(2/3) + 2*(b*x^3 + a)*(I*a*d^3/ b)^(1/3))*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)))/(a^ 2*b*d*x^3 + a^3*d)
Timed out. \[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sin(d*x+c)/(b*x**3+a)**2,x)
Output:
Timed out
\[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(sin(d*x+c)/(b*x^3+a)^2,x, algorithm="maxima")
Output:
integrate(sin(d*x + c)/(b*x^3 + a)^2, x)
\[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(sin(d*x+c)/(b*x^3+a)^2,x, algorithm="giac")
Output:
integrate(sin(d*x + c)/(b*x^3 + a)^2, x)
Timed out. \[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^2} \,d x \] Input:
int(sin(c + d*x)/(a + b*x^3)^2,x)
Output:
int(sin(c + d*x)/(a + b*x^3)^2, x)
\[ \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx=\int \frac {\sin \left (d x +c \right )}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \] Input:
int(sin(d*x+c)/(b*x^3+a)^2,x)
Output:
int(sin(c + d*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)