3.3.47 \(\int \frac {\sin (a+\frac {b}{\sqrt [3]{c+d x}})}{(c e+d e x)^{7/3}} \, dx\) [247]

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

[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 = 0.10, 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 = {3512, 15, 3377, 2717} \begin {gather*} \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)}} \end {gather*}

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 2717

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

Rule 3377

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 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 )}{(c e+d e x)^{7/3}} \, dx &=-\frac {3 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.11, size = 107, normalized size = 0.62 \begin {gather*} -\frac {3 \left (\left (-b^3 \sqrt [3]{c+d x}+6 b (c+d x)\right ) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+3 \sqrt [3]{c+d x} \left (-2 c-2 d x+b^2 \sqrt [3]{c+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}} \end {gather*}

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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Maple [F]
time = 0.02, 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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.39, size = 1400, normalized size = 8.14 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-3/16*(2*(cos(a)^2 + sin(a)^2)*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2*(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)) + 2*(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)) + ((((-I*gamma(5,
I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) - I*gamma(5, I*b/(d*x + c)^(1/
3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^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)
+ (-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) - I*gamma(5, I
*b/(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 - (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 + ((-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conjugate((d*x + c)^(
-1/3))) - I*gamma(5, I*b/(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^3 - (gamma(5, I*b*conjuga
te((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) + (-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conju
gate((d*x + c)^(-1/3))) - I*gamma(5, I*b/(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2
- (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 + ((
(-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) - I*gamma(5, I*b/
(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^3 - (gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + g
amma(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)))*co
s(a)^2*sin(a) + (-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conjugate((d*x + c)^(-1/3))) -
I*gamma(5, I*b/(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 - (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 + ((-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(5, -I*b*conjugat
e((d*x + c)^(-1/3))) - I*gamma(5, I*b/(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos(a)^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) + (-I*gamma(5, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(
5, -I*b*conjugate((d*x + c)^(-1/3))) - I*gamma(5, I*b/(d*x + c)^(1/3)) + I*gamma(5, -I*b/(d*x + c)^(1/3)))*cos
(a)*sin(a)^2 - (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))*e^(-1/3)/((((cos(a)^2*e^2 + e^2*sin(a)^2)*b^4*d^2*x + (cos(a)^2*e^2 + e^2*sin(a)^2)
*b^4*c*d)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 + ((cos(a)^2*e^2 + e^2*sin(a)^2)*b^4*d^2*x + (cos(a)^
2*e^2 + e^2*sin(a)^2)*b^4*c*d)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*(d*x + c)^(1/3))

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Fricas [A]
time = 0.86, size = 151, normalized size = 0.88 \begin {gather*} \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{3} - 6 \, b d x - 6 \, b c\right )} {\left (d x + c\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 ) e^{\frac {2}{3}} - 3 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} - 2 \, {\left (d x + c\right )}^{\frac {4}{3}}\right )} {\left (d x + c\right )}^{\frac {2}{3}} e^{\frac {2}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )\right )} e^{\left (-3\right )}}{b^{4} d^{3} x^{2} + 2 \, b^{4} c d^{2} x + b^{4} c^{2} d} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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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))/(d*e*x+c*e)^(7/3),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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