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

Optimal. Leaf size=693 \[ \text{result too large to display} \]

[Out]

((-1)^(1/3)*d*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])/(9*a^
(5/3)*b^(1/3)) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) -
((-1)^(2/3)*d*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])/(9*a
^(5/3)*b^(1/3)) + (CosIntegral[d*x]*Sin[c])/a^2 - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/
b^(1/3)])/(3*a^2) - (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) - (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*b*x^3) - Sin[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cos[c]*SinIntegral[d*x])/a^2 + (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) + ((-1)^(1/3)*d*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])/(9*a^(5/3)*b^(1/3)) - (
Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + (d*Sin[c - (a^(1/3)*d)/b^(1/3)]
*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinInt
egral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*S
inIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 1.48497, antiderivative size = 693, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346, 3334} \[ -\frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}-\frac{\sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^2}-\frac{\sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^2}+\frac{\sqrt [3]{-1} d \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{(-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\sqrt [3]{-1} d \sin \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 )}{9 a^{5/3} \sqrt [3]{b}}+\frac{d \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 )}{9 a^{5/3} \sqrt [3]{b}}+\frac{(-1)^{2/3} d \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 )}{9 a^{5/3} \sqrt [3]{b}}+\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^2}-\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^2}-\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^2}+\frac{\sin (c) \text{CosIntegral}(d x)}{a^2}+\frac{\cos (c) \text{Si}(d x)}{a^2}-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\sin (c+d x)}{3 a b x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^3)^2),x]

[Out]

((-1)^(1/3)*d*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])/(9*a^
(5/3)*b^(1/3)) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) -
((-1)^(2/3)*d*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])/(9*a
^(5/3)*b^(1/3)) + (CosIntegral[d*x]*Sin[c])/a^2 - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/
b^(1/3)])/(3*a^2) - (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) - (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*b*x^3) - Sin[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cos[c]*SinIntegral[d*x])/a^2 + (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) + ((-1)^(1/3)*d*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])/(9*a^(5/3)*b^(1/3)) - (
Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + (d*Sin[c - (a^(1/3)*d)/b^(1/3)]
*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinInt
egral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*S
inIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3))

Rule 3343

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(x^(m - n + 1)*(a + b*
x^n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p
+ 1)*Sin[c + d*x], x], x] - Dist[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]

Rule 3345

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

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3346

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

Rule 3334

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{x \left (a+b x^3\right )^2} \, dx &=-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}-\frac{\int \frac{\sin (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{b}+\frac{d \int \frac{\cos (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{3 b}\\ &=-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}-\frac{\int \left (\frac{\sin (c+d x)}{a x^4}-\frac{b \sin (c+d x)}{a^2 x}+\frac{b^2 x^2 \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{b}+\frac{d \int \left (\frac{\cos (c+d x)}{a x^3}-\frac{b \cos (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{3 b}\\ &=-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\int \frac{\sin (c+d x)}{x} \, dx}{a^2}-\frac{\int \frac{\sin (c+d x)}{x^4} \, dx}{a b}-\frac{b \int \frac{x^2 \sin (c+d x)}{a+b x^3} \, dx}{a^2}-\frac{d \int \frac{\cos (c+d x)}{a+b x^3} \, dx}{3 a}+\frac{d \int \frac{\cos (c+d x)}{x^3} \, dx}{3 a b}\\ &=-\frac{d \cos (c+d x)}{6 a b x^2}+\frac{\sin (c+d x)}{3 a b x^3}-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}-\frac{b \int \left (\frac{\sin (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sin (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sin (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a^2}-\frac{d \int \left (-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{3 a}-\frac{d \int \frac{\cos (c+d x)}{x^3} \, dx}{3 a b}-\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{6 a b}+\frac{\cos (c) \int \frac{\sin (d x)}{x} \, dx}{a^2}+\frac{\sin (c) \int \frac{\cos (d x)}{x} \, dx}{a^2}\\ &=\frac{\text{Ci}(d x) \sin (c)}{a^2}+\frac{\sin (c+d x)}{3 a b x^3}+\frac{d^2 \sin (c+d x)}{6 a b x}-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^2}-\frac{\sqrt [3]{b} \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}-\frac{\sqrt [3]{b} \int \frac{\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}-\frac{\sqrt [3]{b} \int \frac{\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{5/3}}+\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{5/3}}+\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{5/3}}+\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{6 a b}-\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{6 a b}\\ &=\frac{\text{Ci}(d x) \sin (c)}{a^2}+\frac{\sin (c+d x)}{3 a b x^3}-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^2}+\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{6 a b}-\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{6 a b}-\frac{\left (\sqrt [3]{b} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{\left (d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{5/3}}+\frac{\left (\sqrt [3]{b} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{\left (d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{5/3}}-\frac{\left (\sqrt [3]{b} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{\left (d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{5/3}}+\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{6 a b}-\frac{\left (\sqrt [3]{b} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}-\frac{\left (d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{5/3}}-\frac{\left (\sqrt [3]{b} \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac{\left (d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{5/3}}-\frac{\left (\sqrt [3]{b} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}-\frac{\left (d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{5/3}}\\ &=-\frac{d^3 \cos (c) \text{Ci}(d x)}{6 a b}+\frac{\sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{(-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^2}-\frac{\text{Ci}\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^2}-\frac{\text{Ci}\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{\text{Ci}\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^2}+\frac{\sin (c+d x)}{3 a b x^3}-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^2}+\frac{d^3 \sin (c) \text{Si}(d x)}{6 a b}+\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^2}+\frac{\sqrt [3]{-1} d \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 )}{9 a^{5/3} \sqrt [3]{b}}-\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^2}+\frac{d \sin \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 )}{9 a^{5/3} \sqrt [3]{b}}-\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^2}+\frac{(-1)^{2/3} d \sin \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 )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{6 a b}-\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{6 a b}\\ &=\frac{\sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{(-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^2}-\frac{\text{Ci}\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^2}-\frac{\text{Ci}\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{\text{Ci}\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^2}+\frac{\sin (c+d x)}{3 a b x^3}-\frac{\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^2}+\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^2}+\frac{\sqrt [3]{-1} d \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 )}{9 a^{5/3} \sqrt [3]{b}}-\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^2}+\frac{d \sin \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 )}{9 a^{5/3} \sqrt [3]{b}}-\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^2}+\frac{(-1)^{2/3} d \sin \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 )}{9 a^{5/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [B]  time = 8.79111, size = 1819, normalized size = 2.62 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sin[c]*(CosIntegral[d*x]/a^2 - ((3*b^(1/3) - 2*(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/3)*b^(1/3))*(Cos[((-1)^(1/3)*a^(
1/3)*d)/b^(1/3)]*CosIntegral[-(((-1)^(1/3)*a^(1/3)*d)/b^(1/3)) + d*x] + Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*Si
nIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]))/((1 + (-1)^(1/3))^2*a^2*b^(1/3)) + ((21 - 22*(-1)^(1/3) + 21
*(-1)^(2/3))*b^(1/3)*(-(Cos[d*x]/(b^(1/3)*(-((-1)^(1/3)*a^(1/3)) + b^(1/3)*x))) + (d*(-(CosIntegral[-(((-1)^(1
/3)*a^(1/3)*d)/b^(1/3)) + d*x]*Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]) + Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinI
ntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]))/b^(2/3)))/(3*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(5/3)) - (
(2*b^(1/3) - 3*(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/3)*b^(1/3))*(Cos[(a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^
(1/3) + d*x] + Sin[(a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x]))/((-1 + (-1)^(1/3))*(1 + (-1)^
(1/3))^2*a^2*b^(1/3)) + ((22 - 21*(-1)^(1/3) + 21*(-1)^(2/3))*b^(1/3)*(-(Cos[d*x]/(b^(1/3)*(a^(1/3) + b^(1/3)*
x))) + (d*(CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[(a^(1/3)*d)/b^(1/3)] - Cos[(a^(1/3)*d)/b^(1/3)]*SinInteg
ral[(a^(1/3)*d)/b^(1/3) + d*x]))/b^(2/3)))/(3*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(5/3)) - ((2*b^(1/3) - 3*
(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/3)*b^(1/3))*(Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3
)*d)/b^(1/3) + d*x] + Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]))/
((-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^2*b^(1/3)) + ((22*b^(1/3) - 21*(-1)^(1/3)*b^(1/3) + 21*(-1)^(2/3)*b^(1
/3))*(-(Cos[d*x]/(b^(1/3)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))) + (d*(CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)
+ d*x]*Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)] - Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/
3)*d)/b^(1/3) + d*x]))/b^(2/3)))/(3*(1 + (-1)^(1/3))^2*a^(5/3))) + Cos[c]*(SinIntegral[d*x]/a^2 - ((3*b^(1/3)
- 2*(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/3)*b^(1/3))*(CosIntegral[-(((-1)^(1/3)*a^(1/3)*d)/b^(1/3)) + d*x]*Sin[((-1)
^(1/3)*a^(1/3)*d)/b^(1/3)] - Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) -
d*x]))/((1 + (-1)^(1/3))^2*a^2*b^(1/3)) + ((21 - 22*(-1)^(1/3) + 21*(-1)^(2/3))*b^(1/3)*(-(Sin[d*x]/(b^(1/3)*(
-((-1)^(1/3)*a^(1/3)) + b^(1/3)*x))) + (d*(Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[-(((-1)^(1/3)*a^(1/
3)*d)/b^(1/3)) + d*x] + Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])
)/b^(2/3)))/(3*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(5/3)) - ((2*b^(1/3) - 3*(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/
3)*b^(1/3))*(-(CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[(a^(1/3)*d)/b^(1/3)]) + Cos[(a^(1/3)*d)/b^(1/3)]*Sin
Integral[(a^(1/3)*d)/b^(1/3) + d*x]))/((-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^2*b^(1/3)) + ((22 - 21*(-1)^(1/3
) + 21*(-1)^(2/3))*b^(1/3)*(-(Sin[d*x]/(b^(1/3)*(a^(1/3) + b^(1/3)*x))) + (d*(Cos[(a^(1/3)*d)/b^(1/3)]*CosInte
gral[(a^(1/3)*d)/b^(1/3) + d*x] + Sin[(a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x]))/b^(2/3)))/
(3*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(5/3)) - ((2*b^(1/3) - 3*(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/3)*b^(1/3))*
(-(CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]) + Cos[((-1)^(2/3)*a^
(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]))/((-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^
2*b^(1/3)) + ((22*b^(1/3) - 21*(-1)^(1/3)*b^(1/3) + 21*(-1)^(2/3)*b^(1/3))*(-(Sin[d*x]/(b^(1/3)*((-1)^(2/3)*a^
(1/3) + b^(1/3)*x))) + (d*(Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*
x] + Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]))/b^(2/3)))/(3*(1 +
(-1)^(1/3))^2*a^(5/3)))

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Maple [C]  time = 0.031, size = 233, normalized size = 0.3 \begin{align*}{\frac{\sin \left ( dx+c \right ){d}^{3}}{3\,a \left ( \left ( dx+c \right ) ^{3}b-3\,c \left ( dx+c \right ) ^{2}b+3\, \left ( dx+c \right ) b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }}-{\frac{\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }-{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) }{3\,{a}^{2}}}+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{{a}^{2}}}-{\frac{{d}^{3}}{9\,ab}\sum _{{\it \_RR1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{{\it Si} \left ( -dx+{\it \_RR1}-c \right ) \sin \left ({\it \_RR1} \right ) +{\it Ci} \left ( dx-{\it \_RR1}+c \right ) \cos \left ({\it \_RR1} \right ) }{{{\it \_RR1}}^{2}-2\,{\it \_RR1}\,c+{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sin(d*x+c)*d^3/a/((d*x+c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+c)*b*c^2+a*d^3-c^3*b)-1/3/a^2*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))+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=R
ootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.57534, size = 1493, normalized size = 2.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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) + (-18*I*b*x^3 - 18*I*a)*Ei(I*d*x)*e^(
I*c) + (18*I*b*x^3 + 18*I*a)*Ei(-I*d*x)*e^(-I*c) + (6*I*b*x^3 + (2*I*b*x^3 + 2*I*a)*(-I*a*d^3/b)^(1/3) + 6*I*a
)*Ei(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) + (-6*I*b*x^3 + (-2*I*b*x^3 - 2*I*a)*(I*a*d^3/b)
^(1/3) - 6*I*a)*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) + 12*a*sin(d*x + c))/(a^2*b*x^3 +
a^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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