3.3.35 \(\int (c e+d e x)^{4/3} \sin (a+b (c+d x)^{2/3}) \, dx\) [235]

Optimal. Leaf size=267 \[ \frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d} \]

[Out]

45/8*e*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(2/3))/b^3/d-3/2*e*(d*x+c)^(4/3)*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(2
/3))/b/d+15/4*e*(d*x+c)^(2/3)*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^(2/3))/b^2/d-45/16*e*(e*(d*x+c))^(1/3)*cos(a)*
FresnelC((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(7/2)/d/(d*x+c)^(1/3)*2^(1/2)+45/16*e*(e*(d*x+c))^
(1/3)*FresnelS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*Pi^(1/2)/b^(7/2)/d/(d*x+c)^(1/3)*2^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3516, 3498, 3496, 3466, 3467, 3435, 3433, 3432} \begin {gather*} -\frac {45 \sqrt {\pi } e \cos (a) \sqrt [3]{e (c+d x)} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 \sqrt {\pi } e \sin (a) \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(45*e*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(8*b^3*d) - (3*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Cos
[a + b*(c + d*x)^(2/3)])/(2*b*d) - (45*e*Sqrt[Pi]*(e*(c + d*x))^(1/3)*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c +
d*x)^(1/3)])/(8*Sqrt[2]*b^(7/2)*d*(c + d*x)^(1/3)) + (45*e*Sqrt[Pi]*(e*(c + d*x))^(1/3)*FresnelS[Sqrt[b]*Sqrt[
2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(8*Sqrt[2]*b^(7/2)*d*(c + d*x)^(1/3)) + (15*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/
3)*Sin[a + b*(c + d*x)^(2/3)])/(4*b^2*d)

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 3435

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

Rule 3466

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^(n - 1))*(e*x)^(m - n + 1)*(Cos[c +
d*x^n]/(d*n)), x] + Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3467

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*
x^n]/(d*n)), x] - Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3496

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Module[{k = Denominator[n]}, D
ist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}
, x] && IntegerQ[p] && FractionQ[n]

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 (c e+d e x)^{4/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {\text {Subst}\left (\int (e x)^{4/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^{4/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=\frac {\left (3 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^6 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {\left (15 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^4 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)} \cos (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}+\frac {\left (45 e \sqrt [3]{e (c+d x)} \sin (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 175, normalized size = 0.66 \begin {gather*} -\frac {3 (e (c+d x))^{4/3} \left (15 \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )-15 \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+2 \sqrt {b} \left (\sqrt [3]{c+d x} \left (-15+4 b^2 (c+d x)^{4/3}\right ) \cos \left (a+b (c+d x)^{2/3}\right )-10 b (c+d x) \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{16 b^{7/2} d (c+d x)^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(e*(c + d*x))^(4/3)*(15*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)] - 15*Sqrt[2*Pi]*Fre
snelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a] + 2*Sqrt[b]*((c + d*x)^(1/3)*(-15 + 4*b^2*(c + d*x)^(4/3))*Co
s[a + b*(c + d*x)^(2/3)] - 10*b*(c + d*x)*Sin[a + b*(c + d*x)^(2/3)])))/(16*b^(7/2)*d*(c + d*x)^(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.70, size = 425, normalized size = 1.59 \begin {gather*} \frac {3 \, {\left ({\left ({\left (i \, e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) - {\left ({\left (e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (-i \, e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (-i \, e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\frac {1}{3}}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{4} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(((I*e*gamma(7/2, -I*b*conjugate((d*x + c)^(2/3))) - I*e*gamma(7/2, I*(d*x + c)^(2/3)*b))*cos(7/4*pi + 7/3
*arctan2(0, d*x + c)) + (-I*e*gamma(7/2, I*b*conjugate((d*x + c)^(2/3))) + I*e*gamma(7/2, -I*(d*x + c)^(2/3)*b
))*cos(-7/4*pi + 7/3*arctan2(0, d*x + c)) - (e*gamma(7/2, -I*b*conjugate((d*x + c)^(2/3))) + e*gamma(7/2, I*(d
*x + c)^(2/3)*b))*sin(7/4*pi + 7/3*arctan2(0, d*x + c)) + (e*gamma(7/2, I*b*conjugate((d*x + c)^(2/3))) + e*ga
mma(7/2, -I*(d*x + c)^(2/3)*b))*sin(-7/4*pi + 7/3*arctan2(0, d*x + c)))*cos(a) - ((e*gamma(7/2, -I*b*conjugate
((d*x + c)^(2/3))) + e*gamma(7/2, I*(d*x + c)^(2/3)*b))*cos(7/4*pi + 7/3*arctan2(0, d*x + c)) + (e*gamma(7/2,
I*b*conjugate((d*x + c)^(2/3))) + e*gamma(7/2, -I*(d*x + c)^(2/3)*b))*cos(-7/4*pi + 7/3*arctan2(0, d*x + c)) -
(-I*e*gamma(7/2, -I*b*conjugate((d*x + c)^(2/3))) + I*e*gamma(7/2, I*(d*x + c)^(2/3)*b))*sin(7/4*pi + 7/3*arc
tan2(0, d*x + c)) - (-I*e*gamma(7/2, I*b*conjugate((d*x + c)^(2/3))) + I*e*gamma(7/2, -I*(d*x + c)^(2/3)*b))*s
in(-7/4*pi + 7/3*arctan2(0, d*x + c)))*sin(a))*sqrt((d*x + c)^(2/3)*b)*e^(1/3)/((d*x + c)^(1/3)*b^4*d)

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

[Out]

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

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3003 deep

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Giac [C] Result contains complex when optimal does not.
time = 3.49, size = 713, normalized size = 2.67 \begin {gather*} -\frac {3 \, {\left (8 \, {\left (-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}}\right )} c^{2} e + {\left (-\frac {8 i \, \sqrt {\pi } c^{2} \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}} d^{2}} + \frac {8 i \, \sqrt {\pi } c^{2} \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}} d^{2}} + \frac {-\frac {2 i \, {\left (4 i \, {\left (d x e + c e\right )}^{\frac {5}{3}} b^{2} e^{\left (-\frac {4}{3}\right )} - 10 \, {\left (d x e + c e\right )} b e^{\left (-\frac {2}{3}\right )} - 15 i \, {\left (d x e + c e\right )}^{\frac {1}{3}}\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a\right )}}{b^{3}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}} b^{3}}}{d^{2}} + \frac {-\frac {2 i \, {\left (4 i \, {\left (d x e + c e\right )}^{\frac {5}{3}} b^{2} e^{\left (-\frac {4}{3}\right )} + 10 \, {\left (d x e + c e\right )} b e^{\left (-\frac {2}{3}\right )} - 15 i \, {\left (d x e + c e\right )}^{\frac {1}{3}}\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a\right )}}{b^{3}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}} b^{3}}}{d^{2}} - \frac {16 i \, {\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b c e^{\left (-\frac {2}{3}\right )} + c\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {1}{3}\right )}}{b^{2} d^{2}} - \frac {16 i \, {\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b c e^{\left (-\frac {2}{3}\right )} - c\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {1}{3}\right )}}{b^{2} d^{2}}\right )} d^{2} e + 16 \, {\left (\frac {i \, \sqrt {\pi } c \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a + 1\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} - \frac {i \, \sqrt {\pi } c \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a + 1\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}} - \frac {i \, {\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - 1\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {4}{3}\right )}}{b^{2}} - \frac {i \, {\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + 1\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {4}{3}\right )}}{b^{2}}\right )} c\right )}}{32 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*(8*(-I*sqrt(pi)*erf(-(d*x*e + c*e)^(1/3)*sqrt(-I*b*e^(-2/3)))*e^(I*a)/sqrt(-I*b*e^(-2/3)) + I*sqrt(pi)*e
rf(-(d*x*e + c*e)^(1/3)*sqrt(I*b*e^(-2/3)))*e^(-I*a)/sqrt(I*b*e^(-2/3)))*c^2*e + (-8*I*sqrt(pi)*c^2*erf(-(d*x*
e + c*e)^(1/3)*sqrt(-I*b*e^(-2/3)))*e^(I*a)/(sqrt(-I*b*e^(-2/3))*d^2) + 8*I*sqrt(pi)*c^2*erf(-(d*x*e + c*e)^(1
/3)*sqrt(I*b*e^(-2/3)))*e^(-I*a)/(sqrt(I*b*e^(-2/3))*d^2) + (-2*I*(4*I*(d*x*e + c*e)^(5/3)*b^2*e^(-4/3) - 10*(
d*x*e + c*e)*b*e^(-2/3) - 15*I*(d*x*e + c*e)^(1/3))*e^(I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) + I*a)/b^3 - 15*sqrt(p
i)*erf(-(d*x*e + c*e)^(1/3)*sqrt(-I*b*e^(-2/3)))*e^(I*a)/(sqrt(-I*b*e^(-2/3))*b^3))/d^2 + (-2*I*(4*I*(d*x*e +
c*e)^(5/3)*b^2*e^(-4/3) + 10*(d*x*e + c*e)*b*e^(-2/3) - 15*I*(d*x*e + c*e)^(1/3))*e^(-I*(d*x*e + c*e)^(2/3)*b*
e^(-2/3) - I*a)/b^3 - 15*sqrt(pi)*erf(-(d*x*e + c*e)^(1/3)*sqrt(I*b*e^(-2/3)))*e^(-I*a)/(sqrt(I*b*e^(-2/3))*b^
3))/d^2 - 16*I*(-I*(d*x*e + c*e)^(2/3)*b*c*e^(-2/3) + c)*e^(I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) + I*a + 1/3)/(b^2
*d^2) - 16*I*(-I*(d*x*e + c*e)^(2/3)*b*c*e^(-2/3) - c)*e^(-I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) - I*a + 1/3)/(b^2*
d^2))*d^2*e + 16*(I*sqrt(pi)*c*erf(-(d*x*e + c*e)^(1/3)*sqrt(-I*b*e^(-2/3)))*e^(I*a + 1)/sqrt(-I*b*e^(-2/3)) -
I*sqrt(pi)*c*erf(-(d*x*e + c*e)^(1/3)*sqrt(I*b*e^(-2/3)))*e^(-I*a + 1)/sqrt(I*b*e^(-2/3)) - I*(I*(d*x*e + c*e
)^(2/3)*b*e^(-2/3) - 1)*e^(I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) + I*a + 4/3)/b^2 - I*(I*(d*x*e + c*e)^(2/3)*b*e^(-
2/3) + 1)*e^(-I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) - I*a + 4/3)/b^2)*c)/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+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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