3.3.21 \(\int \frac {\sin (a+\frac {b}{\sqrt [3]{c+d x}})}{(e+f x)^2} \, dx\) [221]

Optimal. Leaf size=566 \[ -\frac {b d \cos \left (a+\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Ci}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac {(-1)^{2/3} b d \cos \left (a+\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}+\frac {\sqrt [3]{-1} b d \cos \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac {b d \sin \left (a+\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac {(-1)^{2/3} b d \sin \left (a+\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac {\sqrt [3]{-1} b d \sin \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}} \]

[Out]

-1/3*b*d*Ci(b*f^(1/3)/(c*f-d*e)^(1/3)-b/(d*x+c)^(1/3))*cos(a+b*f^(1/3)/(c*f-d*e)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3
)+1/3*(-1)^(1/3)*b*d*Ci((-1)^(1/3)*b*f^(1/3)/(c*f-d*e)^(1/3)+b/(d*x+c)^(1/3))*cos(a-(-1)^(1/3)*b*f^(1/3)/(c*f-
d*e)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*(-1)^(2/3)*b*d*Ci((-1)^(2/3)*b*f^(1/3)/(c*f-d*e)^(1/3)-b/(d*x+c)^(1/3)
)*cos(a+(-1)^(2/3)*b*f^(1/3)/(c*f-d*e)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*b*d*Si(b*f^(1/3)/(c*f-d*e)^(1/3)-b/(
d*x+c)^(1/3))*sin(a+b*f^(1/3)/(c*f-d*e)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*(-1)^(1/3)*b*d*Si((-1)^(1/3)*b*f^(1
/3)/(c*f-d*e)^(1/3)+b/(d*x+c)^(1/3))*sin(a-(-1)^(1/3)*b*f^(1/3)/(c*f-d*e)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*(
-1)^(2/3)*b*d*Si((-1)^(2/3)*b*f^(1/3)/(c*f-d*e)^(1/3)-b/(d*x+c)^(1/3))*sin(a+(-1)^(2/3)*b*f^(1/3)/(c*f-d*e)^(1
/3))/f^(2/3)/(c*f-d*e)^(4/3)+(d*x+c)*sin(a+b/(d*x+c)^(1/3))/(-c*f+d*e)/(f*x+e)

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Rubi [A]
time = 1.66, antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3512, 3422, 3415, 3384, 3380, 3383} \begin {gather*} -\frac {b d \cos \left (a+\frac {b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text {CosIntegral}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac {(-1)^{2/3} b d \cos \left (a+\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}+\frac {\sqrt [3]{-1} b d \cos \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac {b d \sin \left (a+\frac {b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text {Si}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac {(-1)^{2/3} b d \sin \left (a+\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}-\frac {\sqrt [3]{-1} b d \sin \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{c f-d e}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{f} b}{\sqrt [3]{c f-d e}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (c f-d e)^{4/3}}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(1/3)]/(e + f*x)^2,x]

[Out]

-1/3*(b*d*Cos[a + (b*f^(1/3))/(-(d*e) + c*f)^(1/3)]*CosIntegral[(b*f^(1/3))/(-(d*e) + c*f)^(1/3) - b/(c + d*x)
^(1/3)])/(f^(2/3)*(-(d*e) + c*f)^(4/3)) - ((-1)^(2/3)*b*d*Cos[a + ((-1)^(2/3)*b*f^(1/3))/(-(d*e) + c*f)^(1/3)]
*CosIntegral[((-1)^(2/3)*b*f^(1/3))/(-(d*e) + c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*f^(2/3)*(-(d*e) + c*f)^(4/3)
) + ((-1)^(1/3)*b*d*Cos[a - ((-1)^(1/3)*b*f^(1/3))/(-(d*e) + c*f)^(1/3)]*CosIntegral[((-1)^(1/3)*b*f^(1/3))/(-
(d*e) + c*f)^(1/3) + b/(c + d*x)^(1/3)])/(3*f^(2/3)*(-(d*e) + c*f)^(4/3)) + ((c + d*x)*Sin[a + b/(c + d*x)^(1/
3)])/((d*e - c*f)*(e + f*x)) - (b*d*Sin[a + (b*f^(1/3))/(-(d*e) + c*f)^(1/3)]*SinIntegral[(b*f^(1/3))/(-(d*e)
+ c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*f^(2/3)*(-(d*e) + c*f)^(4/3)) - ((-1)^(2/3)*b*d*Sin[a + ((-1)^(2/3)*b*f^
(1/3))/(-(d*e) + c*f)^(1/3)]*SinIntegral[((-1)^(2/3)*b*f^(1/3))/(-(d*e) + c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*
f^(2/3)*(-(d*e) + c*f)^(4/3)) - ((-1)^(1/3)*b*d*Sin[a - ((-1)^(1/3)*b*f^(1/3))/(-(d*e) + c*f)^(1/3)]*SinIntegr
al[((-1)^(1/3)*b*f^(1/3))/(-(d*e) + c*f)^(1/3) + b/(c + d*x)^(1/3)])/(3*f^(2/3)*(-(d*e) + c*f)^(4/3))

Rule 3380

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 3383

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 3384

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 3415

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])

Rule 3422

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

Rule 3512

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.76, size = 313, normalized size = 0.55 \begin {gather*} \frac {(\cos (a)+i \sin (a)) \left (b d (e+f x) \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {\text {Ei}\left (\frac {i b}{\sqrt [3]{c+d x}}\right )-e^{\frac {i b}{\text {$\#$1}}} \text {Ei}\left (i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )}{\text {$\#$1}}\&\right ]+(c+d x) \left (3 i f \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right )-3 f \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )\right )+i \left (-3 c f-3 d f x+b d (e+f x) \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {\text {Ei}\left (-\frac {i b}{\sqrt [3]{c+d x}}\right )-e^{-\frac {i b}{\text {$\#$1}}} \text {Ei}\left (-i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )}{\text {$\#$1}}\&\right ] \left (-i \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right )+\sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )\right ) \left (\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )-i \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{6 f (-d e+c f) (e+f x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(1/3)]/(e + f*x)^2,x]

[Out]

((Cos[a] + I*Sin[a])*(b*d*(e + f*x)*RootSum[d*e - c*f + f*#1^3 & , (ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - E^(
(I*b)/#1)*ExpIntegralEi[I*b*((c + d*x)^(-1/3) - #1^(-1))])/#1 & ] + (c + d*x)*((3*I)*f*Cos[b/(c + d*x)^(1/3)]
- 3*f*Sin[b/(c + d*x)^(1/3)])) + I*(-3*c*f - 3*d*f*x + b*d*(e + f*x)*RootSum[d*e - c*f + f*#1^3 & , (ExpIntegr
alEi[((-I)*b)/(c + d*x)^(1/3)] - ExpIntegralEi[(-I)*b*((c + d*x)^(-1/3) - #1^(-1))]/E^((I*b)/#1))/#1 & ]*((-I)
*Cos[b/(c + d*x)^(1/3)] + Sin[b/(c + d*x)^(1/3)]))*(Cos[a + b/(c + d*x)^(1/3)] - I*Sin[a + b/(c + d*x)^(1/3)])
)/(6*f*(-(d*e) + c*f)*(e + f*x))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 1554, normalized size = 2.75

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1554\)
default	\(\text {Expression too large to display}\)	\(1554\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/(d*x+c)^(1/3))/(f*x+e)^2,x,method=_RETURNVERBOSE)

[Out]

-3*d*b^3*(a^2*(sin(a+b/(d*x+c)^(1/3))*(1/3/f/b^3*(a+b/(d*x+c)^(1/3))-1/3*a/f/b^3)/(a^3*c*f-a^3*d*e-3*a^2*c*f*(
a+b/(d*x+c)^(1/3))+3*a^2*d*e*(a+b/(d*x+c)^(1/3))+3*a*c*f*(a+b/(d*x+c)^(1/3))^2-3*a*d*e*(a+b/(d*x+c)^(1/3))^2-c
*f*(a+b/(d*x+c)^(1/3))^3+d*e*(a+b/(d*x+c)^(1/3))^3+f*b^3)-2/9/f/b^3*sum(1/(_R1^2*c*f-_R1^2*d*e-2*_R1*a*c*f+2*_
R1*a*d*e+a^2*c*f-a^2*d*e)*(-Si(-b/(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b/(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf
((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a^2*d*e)*_Z-a^3*c*f+a^3*d*e-f*b^3))-1/9/f/b^3*sum(1/(-_RR
1*c*f+_RR1*d*e+a*c*f-a*d*e)*(Si(-b/(d*x+c)^(1/3)+_RR1-a)*sin(_RR1)+Ci(b/(d*x+c)^(1/3)-_RR1+a)*cos(_RR1)),_RR1=
RootOf((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a^2*d*e)*_Z-a^3*c*f+a^3*d*e-f*b^3)))+sin(a+b/(d*x+c
)^(1/3))*(-2/3*a/f/b^3*(a+b/(d*x+c)^(1/3))^2+2/3*a^2/f/b^3*(a+b/(d*x+c)^(1/3)))/(a^3*c*f-a^3*d*e-3*a^2*c*f*(a+
b/(d*x+c)^(1/3))+3*a^2*d*e*(a+b/(d*x+c)^(1/3))+3*a*c*f*(a+b/(d*x+c)^(1/3))^2-3*a*d*e*(a+b/(d*x+c)^(1/3))^2-c*f
*(a+b/(d*x+c)^(1/3))^3+d*e*(a+b/(d*x+c)^(1/3))^3+f*b^3)+2/9*a/f/b^3*sum((_R1+a)/(_R1^2*c*f-_R1^2*d*e-2*_R1*a*c
*f+2*_R1*a*d*e+a^2*c*f-a^2*d*e)*(-Si(-b/(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b/(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=
RootOf((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a^2*d*e)*_Z-a^3*c*f+a^3*d*e-f*b^3))+2/9*a/f/b^3*sum
(_RR1/(-_RR1*c*f+_RR1*d*e+a*c*f-a*d*e)*(Si(-b/(d*x+c)^(1/3)+_RR1-a)*sin(_RR1)+Ci(b/(d*x+c)^(1/3)-_RR1+a)*cos(_
RR1)),_RR1=RootOf((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a^2*d*e)*_Z-a^3*c*f+a^3*d*e-f*b^3))+sin(
a+b/(d*x+c)^(1/3))*(2/3*a/f/b^3*(a+b/(d*x+c)^(1/3))^2-a^2/f/b^3*(a+b/(d*x+c)^(1/3))+1/3*(a^3*c*f-a^3*d*e+b^3*f
)/f/b^3/(c*f-d*e))/(a^3*c*f-a^3*d*e-3*a^2*c*f*(a+b/(d*x+c)^(1/3))+3*a^2*d*e*(a+b/(d*x+c)^(1/3))+3*a*c*f*(a+b/(
d*x+c)^(1/3))^2-3*a*d*e*(a+b/(d*x+c)^(1/3))^2-c*f*(a+b/(d*x+c)^(1/3))^3+d*e*(a+b/(d*x+c)^(1/3))^3+f*b^3)-2/9*a
/f/b^3*sum(_R1/(_R1^2*c*f-_R1^2*d*e-2*_R1*a*c*f+2*_R1*a*d*e+a^2*c*f-a^2*d*e)*(-Si(-b/(d*x+c)^(1/3)+_R1-a)*cos(
_R1)+Ci(b/(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a^2*d*
e)*_Z-a^3*c*f+a^3*d*e-f*b^3))+1/9/f/b^3*sum((2*_RR1^2*a*c*f-2*_RR1^2*a*d*e-3*_RR1*a^2*c*f+3*_RR1*a^2*d*e+a^3*c
*f-a^3*d*e+b^3*f)/(c*f-d*e)/(_RR1^2*c*f-_RR1^2*d*e-2*_RR1*a*c*f+2*_RR1*a*d*e+a^2*c*f-a^2*d*e)*(Si(-b/(d*x+c)^(
1/3)+_RR1-a)*sin(_RR1)+Ci(b/(d*x+c)^(1/3)-_RR1+a)*cos(_RR1)),_RR1=RootOf((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^
2+(3*a^2*c*f-3*a^2*d*e)*_Z-a^3*c*f+a^3*d*e-f*b^3)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate(sin(a + b/(d*x + c)^(1/3))/(f*x + e)^2, x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.44, size = 827, normalized size = 1.46 \begin {gather*} -\frac {\left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, d f x - i \, d e + \sqrt {3} {\left (d f x + d e\right )}\right )} {\rm Ei}\left (\frac {2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (-i \, d x - i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, a\right )} + \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, d f x + i \, d e - \sqrt {3} {\left (d f x + d e\right )}\right )} {\rm Ei}\left (\frac {-2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (-i \, d x - i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, a\right )} + \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, d f x - i \, d e - \sqrt {3} {\left (d f x + d e\right )}\right )} {\rm Ei}\left (\frac {2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (i \, d x + i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, a\right )} + \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, d f x + i \, d e + \sqrt {3} {\left (d f x + d e\right )}\right )} {\rm Ei}\left (\frac {-2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (i \, d x + i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, a\right )} - 2 \, \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, d f x - i \, d e\right )} {\rm Ei}\left (\frac {i \, {\left (d x + c\right )}^{\frac {2}{3}} b + \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x + c\right )}}{d x + c}\right ) e^{\left (i \, a - \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}}\right )} - 2 \, \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, d f x + i \, d e\right )} {\rm Ei}\left (\frac {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b + \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x + c\right )}}{d x + c}\right ) e^{\left (-i \, a - \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}}\right )} + 12 \, {\left (d f x + c f\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )}{12 \, {\left (c f^{3} x - d f e^{2} - {\left (d f^{2} x - c f^{2}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(f*x+e)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}}{\left (e + f x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)**(1/3))/(f*x+e)**2,x)

[Out]

Integral(sin(a + b/(c + d*x)**(1/3))/(e + f*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(f*x+e)^2,x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(1/3))/(f*x + e)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b/(c + d*x)^(1/3))/(e + f*x)^2,x)

[Out]

int(sin(a + b/(c + d*x)^(1/3))/(e + f*x)^2, x)

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