3.3.11 \(\int \frac {\sin (a+b \sqrt [3]{c+d x})}{(e+f x)^2} \, dx\) [211]

Optimal. Leaf size=555 \[ -\frac {\sqrt [3]{-1} b d \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac {b d \cos \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Ci}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac {(-1)^{2/3} b d \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}-\frac {\sqrt [3]{-1} b d \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {b d \sin \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {(-1)^{2/3} b d \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}} \]

[Out]

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

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Rubi [A]
time = 1.45, antiderivative size = 555, 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 {\sqrt [3]{-1} b d \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac {b d \cos \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {CosIntegral}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac {(-1)^{2/3} b d \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {\sqrt [3]{-1} b d \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {b d \sin \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {(-1)^{2/3} b d \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.70, size = 180, normalized size = 0.32 \begin {gather*} \frac {\frac {3 i e^{-i \left (a+b \sqrt [3]{c+d x}\right )} \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) f}{e+f x}+b d \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {e^{-i a-i b \text {$\#$1}} \text {Ei}\left (-i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]+b d \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {e^{i a+i b \text {$\#$1}} \text {Ei}\left (i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]}{6 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*f)/(E^(I*(a + b*(c + d*x)^(1/3)))*(e + f*x)) + b*d*RootSum[d*
e - c*f + f*#1^3 & , (E^((-I)*a - I*b*#1)*ExpIntegralEi[(-I)*b*((c + d*x)^(1/3) - #1)])/#1^2 & ] + b*d*RootSum
[d*e - c*f + f*#1^3 & , (E^(I*a + I*b*#1)*ExpIntegralEi[I*b*((c + d*x)^(1/3) - #1)])/#1^2 & ])/(6*f^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.07, size = 1176, normalized size = 2.12

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1176\)
default	\(\text {Expression too large to display}\)	\(1176\)

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*(b^6*a^2*(sin(a+b*(d*x+c)^(1/3))*(1/3/b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))-1/3*a/b^3/(c*f-d*e))/(b^3*c*f-
b^3*d*e+a^3*f-3*a^2*f*(a+b*(d*x+c)^(1/3))+3*a*f*(a+b*(d*x+c)^(1/3))^2-f*(a+b*(d*x+c)^(1/3))^3)-2/9/b^3/f*sum(1
/(c*f-d*e)/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=R
ootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))-1/9/b^3/f*sum(1/(-_RR1+a)/(c*f-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(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+
3*_Z*a^2*f-a^3*f)))+sin(a+b*(d*x+c)^(1/3))*(-2/3*a*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))^2+2/3*a^2*b^3/(c*f-d*e)*(
a+b*(d*x+c)^(1/3)))/(b^3*c*f-b^3*d*e+a^3*f-3*a^2*f*(a+b*(d*x+c)^(1/3))+3*a*f*(a+b*(d*x+c)^(1/3))^2-f*(a+b*(d*x
+c)^(1/3))^3)+2/9*a*b^3/f*sum((_R1+a)/(c*f-d*e)/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b
*(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))+2/9*a*b^3/f*s
um(_RR1/(-_RR1+a)/(c*f-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(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))+sin(a+b*(d*x+c)^(1/3))*(2/3*a*b^3/(c*f-d*e)*(a+b*
(d*x+c)^(1/3))^2-a^2*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))+1/3*b^3*(b^3*c*f-b^3*d*e+a^3*f)/f/(c*f-d*e))/(b^3*c*f-b
^3*d*e+a^3*f-3*a^2*f*(a+b*(d*x+c)^(1/3))+3*a*f*(a+b*(d*x+c)^(1/3))^2-f*(a+b*(d*x+c)^(1/3))^3)-2/9*a*b^3/f*sum(
_R1/(c*f-d*e)/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R
1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))+1/9*b^3/f^2*sum((b^3*c*f-b^3*d*e+2*_RR1^2*a*f-3
*_RR1*a^2*f+a^3*f)/(c*f-d*e)/(_RR1^2-2*_RR1*a+a^2)*(Si(-b*(d*x+c)^(1/3)+_RR1-a)*sin(_RR1)+Ci(b*(d*x+c)^(1/3)-_
RR1+a)*cos(_RR1)),_RR1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f)))

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.43, size = 760, normalized size = 1.37 \begin {gather*} -\frac {{\left (i \, d f x + i \, d e - \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}} + i \, a\right )} + {\left (i \, d f x + i \, d e + \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}} + i \, a\right )} + {\left (-i \, d f x - i \, d e + \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}} - i \, a\right )} + {\left (-i \, d f x - i \, d e - \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}} - i \, a\right )} - 2 \, {\left (i \, d f x + i \, d e\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (i \, a - \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right )} - 2 \, {\left (-i \, d f x - i \, d e\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, a - \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right )} + 12 \, {\left (c f - d e\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\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*d*f*x + I*d*e - sqrt(3)*(d*f*x + d*e))*((I*b^3*c*f - I*b^3*d*e)/f)^(1/3)*Ei(I*(d*x + c)^(1/3)*b + 1/
2*(-I*sqrt(3) - 1)*((I*b^3*c*f - I*b^3*d*e)/f)^(1/3))*e^(1/2*(I*sqrt(3) + 1)*((I*b^3*c*f - I*b^3*d*e)/f)^(1/3)
+ I*a) + (I*d*f*x + I*d*e + sqrt(3)*(d*f*x + d*e))*((I*b^3*c*f - I*b^3*d*e)/f)^(1/3)*Ei(I*(d*x + c)^(1/3)*b +
1/2*(I*sqrt(3) - 1)*((I*b^3*c*f - I*b^3*d*e)/f)^(1/3))*e^(1/2*(-I*sqrt(3) + 1)*((I*b^3*c*f - I*b^3*d*e)/f)^(1
/3) + I*a) + (-I*d*f*x - I*d*e + sqrt(3)*(d*f*x + d*e))*((-I*b^3*c*f + I*b^3*d*e)/f)^(1/3)*Ei(-I*(d*x + c)^(1/
3)*b + 1/2*(-I*sqrt(3) - 1)*((-I*b^3*c*f + I*b^3*d*e)/f)^(1/3))*e^(1/2*(I*sqrt(3) + 1)*((-I*b^3*c*f + I*b^3*d*
e)/f)^(1/3) - I*a) + (-I*d*f*x - I*d*e - sqrt(3)*(d*f*x + d*e))*((-I*b^3*c*f + I*b^3*d*e)/f)^(1/3)*Ei(-I*(d*x
+ c)^(1/3)*b + 1/2*(I*sqrt(3) - 1)*((-I*b^3*c*f + I*b^3*d*e)/f)^(1/3))*e^(1/2*(-I*sqrt(3) + 1)*((-I*b^3*c*f +
I*b^3*d*e)/f)^(1/3) - I*a) - 2*(I*d*f*x + I*d*e)*((I*b^3*c*f - I*b^3*d*e)/f)^(1/3)*Ei(I*(d*x + c)^(1/3)*b + ((
I*b^3*c*f - I*b^3*d*e)/f)^(1/3))*e^(I*a - ((I*b^3*c*f - I*b^3*d*e)/f)^(1/3)) - 2*(-I*d*f*x - I*d*e)*((-I*b^3*c
*f + I*b^3*d*e)/f)^(1/3)*Ei(-I*(d*x + c)^(1/3)*b + ((-I*b^3*c*f + I*b^3*d*e)/f)^(1/3))*e^(-I*a - ((-I*b^3*c*f
+ I*b^3*d*e)/f)^(1/3)) + 12*(c*f - d*e)*sin((d*x + c)^(1/3)*b + a))/(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 + 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((d*x + c)^(1/3)*b + a)/(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+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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