3.3.54 \(\int \frac {\sin (a+\frac {b}{(c+d x)^{2/3}})}{(c e+d e x)^{4/3}} \, dx\) [254]

Optimal. Leaf size=141 \[ -\frac {3 \sqrt {\pi } \sqrt [3]{c+d x} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {\pi } \sqrt [3]{c+d x} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}} \]

[Out]

-3/2*(d*x+c)^(1/3)*cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))*Pi^(1/2)/d/e/(e*(d*x+c))^(1/3)*2^(1
/2)/b^(1/2)-3/2*(d*x+c)^(1/3)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))*sin(a)*Pi^(1/2)/d/e/(e*(d*x+c))
^(1/3)*2^(1/2)/b^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3516, 3498, 3464, 3434, 3433, 3432} \begin {gather*} -\frac {3 \sqrt {\pi } \sin (a) \sqrt [3]{c+d x} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {\pi } \cos (a) \sqrt [3]{c+d x} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

(-3*Sqrt[Pi]*(c + d*x)^(1/3)*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(Sqrt[2]*Sqrt[b]*d*e*(e*(c
+ d*x))^(1/3)) - (3*Sqrt[Pi]*(c + d*x)^(1/3)*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(Sqrt[2]*
Sqrt[b]*d*e*(e*(c + d*x))^(1/3))

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3434

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 3464

Int[(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_)], x_Symbol] :> Dist[2/n, Subst[Int[Sin[a + b*x^2], x], x, x^(n/2)],
x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]

Rule 3498

Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[e^IntPart[m]*((e*x)
^FracPart[m]/x^FracPart[m]), Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && Integ
erQ[p] && FractionQ[n]

Rule 3516

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

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 96, normalized size = 0.68 \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{4/3} \left (\cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)\right )}{\sqrt {b} d (e (c+d x))^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.62, size = 486, normalized size = 3.45 \begin {gather*} \frac {3 \, {\left ({\left ({\left (-i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) - {\left ({\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )}\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {4}{3}\right )}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} d \sqrt {\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(2/3))/(d*e*x+c*e)^(4/3),x, algorithm="maxima")

[Out]

3/8*(((-I*sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(-2/3)))) - 1) + I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3
))) - 1))*cos(1/4*pi + 1/3*arctan2(0, d*x + c)) + (I*sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(-2/3)))) - 1
) - I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/3*arctan2(0, d*x + c)) - (sqrt(pi)*(erf(s
qrt(I*b*conjugate((d*x + c)^(-2/3)))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/3*
arctan2(0, d*x + c)) + (sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(-2/3)))) - 1) + sqrt(pi)*(erf(sqrt(I*b/(d
*x + c)^(2/3))) - 1))*sin(-1/4*pi + 1/3*arctan2(0, d*x + c)))*cos(a) - ((sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x
+ c)^(-2/3)))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(1/4*pi + 1/3*arctan2(0, d*x + c)) +
(sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(-2/3)))) - 1) + sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1))*
cos(-1/4*pi + 1/3*arctan2(0, d*x + c)) - (I*sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(-2/3)))) - 1) - I*sqrt
(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/3*arctan2(0, d*x + c)) - (I*sqrt(pi)*(erf(sqrt(-I*b
*conjugate((d*x + c)^(-2/3)))) - 1) - I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4*pi + 1/3*arcta
n2(0, d*x + c)))*sin(a))*e^(-4/3)/((d*x + c)^(1/3)*d*sqrt(b/(d*x + c)^(2/3)))

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Fricas [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)^(2/3))/(d*e*x+c*e)^(4/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {4}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)**(2/3))/(d*e*x+c*e)**(4/3),x)

[Out]

Integral(sin(a + b/(c + d*x)**(2/3))/(e*(c + d*x))**(4/3), 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)^(2/3))/(d*e*x+c*e)^(4/3),x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(2/3))/(d*x*e + c*e)^(4/3), 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 )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{4/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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