3.247 \(\int \frac {\sin (a+\frac {b}{\sqrt [3]{c+d x}})}{(c e+d e x)^{7/3}} \, dx\)

Optimal. Leaf size=172 \[ \frac {18 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 \sqrt [3]{e (c+d x)}}-\frac {18 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}-\frac {9 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}} \]

[Out]

-18*cos(a+b/(d*x+c)^(1/3))/b^3/d/e^2/(e*(d*x+c))^(1/3)+3*cos(a+b/(d*x+c)^(1/3))/b/d/e^2/(d*x+c)^(2/3)/(e*(d*x+
c))^(1/3)-9*sin(a+b/(d*x+c)^(1/3))/b^2/d/e^2/(d*x+c)^(1/3)/(e*(d*x+c))^(1/3)+18*(d*x+c)^(1/3)*sin(a+b/(d*x+c)^
(1/3))/b^4/d/e^2/(e*(d*x+c))^(1/3)

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Rubi [A]  time = 0.15, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3431, 15, 3296, 2637} \[ \frac {18 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 \sqrt [3]{e (c+d x)}}-\frac {9 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac {18 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3),x]

[Out]

(-18*Cos[a + b/(c + d*x)^(1/3)])/(b^3*d*e^2*(e*(c + d*x))^(1/3)) + (3*Cos[a + b/(c + d*x)^(1/3)])/(b*d*e^2*(c
+ d*x)^(2/3)*(e*(c + d*x))^(1/3)) - (9*Sin[a + b/(c + d*x)^(1/3)])/(b^2*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1
/3)) + (18*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(b^4*d*e^2*(e*(c + d*x))^(1/3))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3431

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 )}{(c e+d e x)^{7/3}} \, dx &=-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{\left (\frac {e}{x^3}\right )^{7/3} x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac {\left (9 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac {9 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac {\left (18 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac {18 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac {9 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac {\left (18 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac {18 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac {9 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac {18 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 \sqrt [3]{e (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 107, normalized size = 0.62 \[ -\frac {3 \left (\left (6 b (c+d x)-b^3 \sqrt [3]{c+d x}\right ) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+3 \sqrt [3]{c+d x} \left (b^2 \sqrt [3]{c+d x}-2 c-2 d x\right ) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{b^4 d e (e (c+d x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3),x]

[Out]

(-3*((-(b^3*(c + d*x)^(1/3)) + 6*b*(c + d*x))*Cos[a + b/(c + d*x)^(1/3)] + 3*(c + d*x)^(1/3)*(-2*c - 2*d*x + b
^2*(c + d*x)^(1/3))*Sin[a + b/(c + d*x)^(1/3)]))/(b^4*d*e*(e*(c + d*x))^(4/3))

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fricas [A]  time = 1.97, size = 160, normalized size = 0.93 \[ \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{3} - 6 \, b d x - 6 \, b c\right )} {\left (d e x + c e\right )}^{\frac {2}{3}} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - 3 \, {\left (d e x + c e\right )}^{\frac {2}{3}} {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} - 2 \, {\left (d x + c\right )}^{\frac {4}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )\right )}}{b^{4} d^{3} e^{3} x^{2} + 2 \, b^{4} c d^{2} e^{3} x + b^{4} c^{2} d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="fricas")

[Out]

3*(((d*x + c)^(1/3)*b^3 - 6*b*d*x - 6*b*c)*(d*e*x + c*e)^(2/3)*cos((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)
) - 3*(d*e*x + c*e)^(2/3)*((d*x + c)^(2/3)*b^2 - 2*(d*x + c)^(4/3))*sin((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x
+ c)))/(b^4*d^3*e^3*x^2 + 2*b^4*c*d^2*e^3*x + b^4*c^2*d*e^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(1/3))/(d*e*x + c*e)^(7/3), x)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x)

[Out]

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x)

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maxima [C]  time = 2.37, size = 1390, normalized size = 8.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="maxima")

[Out]

-1/16*(6*(cos(a)^2 + sin(a)^2)*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 6*(b^4*cos(((d*x + c)^(1/3)*
a + b)/(d*x + c)^(1/3))^2*sin(a) + b^4*sin(a)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*cos((2*(d*x + c)
^(1/3)*a + b)/(d*x + c)^(1/3)) + 6*(b^4*cos(a)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 + b^4*cos(a)*sin
(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*sin((2*(d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - ((((3*I*gamma(5,
I*b*conjugate((d*x + c)^(-1/3))) - 3*I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + 3*I*gamma(5, I*b/(d*x + c
)^(1/3)) - 3*I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^3 + 3*(gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + gamma
(5, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, I*b/(d*x + c)^(1/3)) + gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)
^2*sin(a) + (3*I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) - 3*I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) +
3*I*gamma(5, I*b/(d*x + c)^(1/3)) - 3*I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 + 3*(gamma(5, I*b*conj
ugate((d*x + c)^(-1/3))) + gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, I*b/(d*x + c)^(1/3)) + gamma(
5, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*d*x + ((3*I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) - 3*I*gamma(5, -I*b*
conjugate((d*x + c)^(-1/3))) + 3*I*gamma(5, I*b/(d*x + c)^(1/3)) - 3*I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^
3 + 3*(gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, I*b/(
d*x + c)^(1/3)) + gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + (3*I*gamma(5, I*b*conjugate((d*x + c)^(-1/
3))) - 3*I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + 3*I*gamma(5, I*b/(d*x + c)^(1/3)) - 3*I*gamma(5, -I*b/
(d*x + c)^(1/3)))*cos(a)*sin(a)^2 + 3*(gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, -I*b*conjugate((d*
x + c)^(-1/3))) + gamma(5, I*b/(d*x + c)^(1/3)) + gamma(5, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*c)*cos(((d*x + c)^
(1/3)*a + b)/(d*x + c)^(1/3))^2 + (((3*I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) - 3*I*gamma(5, -I*b*conjuga
te((d*x + c)^(-1/3))) + 3*I*gamma(5, I*b/(d*x + c)^(1/3)) - 3*I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^3 + 3*(
gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, I*b/(d*x + c
)^(1/3)) + gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + (3*I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) -
3*I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + 3*I*gamma(5, I*b/(d*x + c)^(1/3)) - 3*I*gamma(5, -I*b/(d*x +
c)^(1/3)))*cos(a)*sin(a)^2 + 3*(gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, -I*b*conjugate((d*x + c)^
(-1/3))) + gamma(5, I*b/(d*x + c)^(1/3)) + gamma(5, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*d*x + ((3*I*gamma(5, I*b*
conjugate((d*x + c)^(-1/3))) - 3*I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + 3*I*gamma(5, I*b/(d*x + c)^(1/
3)) - 3*I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^3 + 3*(gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, -
I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, I*b/(d*x + c)^(1/3)) + gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^2*si
n(a) + (3*I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) - 3*I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + 3*I*g
amma(5, I*b/(d*x + c)^(1/3)) - 3*I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 + 3*(gamma(5, I*b*conjugate
((d*x + c)^(-1/3))) + gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(5, I*b/(d*x + c)^(1/3)) + gamma(5, -I
*b/(d*x + c)^(1/3)))*sin(a)^3)*c)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*(d*x + c)^(1/3))/((((cos(a)^
2 + sin(a)^2)*b^4*d^2*e^2*x + (cos(a)^2 + sin(a)^2)*b^4*c*d*e^2)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^
2 + ((cos(a)^2 + sin(a)^2)*b^4*d^2*e^2*x + (cos(a)^2 + sin(a)^2)*b^4*c*d*e^2)*sin(((d*x + c)^(1/3)*a + b)/(d*x
+ c)^(1/3))^2)*(d*x + c)^(1/3)*e^(1/3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{7/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b/(c + d*x)^(1/3))/(c*e + d*e*x)^(7/3),x)

[Out]

int(sin(a + b/(c + d*x)^(1/3))/(c*e + d*e*x)^(7/3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)**(1/3))/(d*e*x+c*e)**(7/3),x)

[Out]

Timed out

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