3.68 \(\int (c+d x)^{2/3} \cos (a+b x) \, dx\)

Optimal. Leaf size=152 \[ \frac {d e^{i \left (a-\frac {b c}{d}\right )} \sqrt [3]{-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},-\frac {i b (c+d x)}{d}\right )}{3 b^2 \sqrt [3]{c+d x}}+\frac {d e^{-i \left (a-\frac {b c}{d}\right )} \sqrt [3]{\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},\frac {i b (c+d x)}{d}\right )}{3 b^2 \sqrt [3]{c+d x}}+\frac {(c+d x)^{2/3} \sin (a+b x)}{b} \]

[Out]

1/3*d*exp(I*(a-b*c/d))*(-I*b*(d*x+c)/d)^(1/3)*GAMMA(2/3,-I*b*(d*x+c)/d)/b^2/(d*x+c)^(1/3)+1/3*d*(I*b*(d*x+c)/d
)^(1/3)*GAMMA(2/3,I*b*(d*x+c)/d)/b^2/exp(I*(a-b*c/d))/(d*x+c)^(1/3)+(d*x+c)^(2/3)*sin(b*x+a)/b

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Rubi [A]  time = 0.15, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3296, 3308, 2181} \[ \frac {d e^{i \left (a-\frac {b c}{d}\right )} \sqrt [3]{-\frac {i b (c+d x)}{d}} \text {Gamma}\left (\frac {2}{3},-\frac {i b (c+d x)}{d}\right )}{3 b^2 \sqrt [3]{c+d x}}+\frac {d e^{-i \left (a-\frac {b c}{d}\right )} \sqrt [3]{\frac {i b (c+d x)}{d}} \text {Gamma}\left (\frac {2}{3},\frac {i b (c+d x)}{d}\right )}{3 b^2 \sqrt [3]{c+d x}}+\frac {(c+d x)^{2/3} \sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(2/3)*Cos[a + b*x],x]

[Out]

(d*E^(I*(a - (b*c)/d))*(((-I)*b*(c + d*x))/d)^(1/3)*Gamma[2/3, ((-I)*b*(c + d*x))/d])/(3*b^2*(c + d*x)^(1/3))
+ (d*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + d*x))/d])/(3*b^2*E^(I*(a - (b*c)/d))*(c + d*x)^(1/3)) + ((
c + d*x)^(2/3)*Sin[a + b*x])/b

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

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 3308

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

Rubi steps

\begin {align*} \int (c+d x)^{2/3} \cos (a+b x) \, dx &=\frac {(c+d x)^{2/3} \sin (a+b x)}{b}-\frac {(2 d) \int \frac {\sin (a+b x)}{\sqrt [3]{c+d x}} \, dx}{3 b}\\ &=\frac {(c+d x)^{2/3} \sin (a+b x)}{b}-\frac {(i d) \int \frac {e^{-i (a+b x)}}{\sqrt [3]{c+d x}} \, dx}{3 b}+\frac {(i d) \int \frac {e^{i (a+b x)}}{\sqrt [3]{c+d x}} \, dx}{3 b}\\ &=\frac {d e^{i \left (a-\frac {b c}{d}\right )} \sqrt [3]{-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},-\frac {i b (c+d x)}{d}\right )}{3 b^2 \sqrt [3]{c+d x}}+\frac {d e^{-i \left (a-\frac {b c}{d}\right )} \sqrt [3]{\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},\frac {i b (c+d x)}{d}\right )}{3 b^2 \sqrt [3]{c+d x}}+\frac {(c+d x)^{2/3} \sin (a+b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 124, normalized size = 0.82 \[ -\frac {i (c+d x)^{2/3} e^{-\frac {i (a d+b c)}{d}} \left (\frac {e^{2 i a} \Gamma \left (\frac {5}{3},-\frac {i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{2/3}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {5}{3},\frac {i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{2/3}}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/2*I)*(c + d*x)^(2/3)*((E^((2*I)*a)*Gamma[5/3, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(2/3) - (E^((
(2*I)*b*c)/d)*Gamma[5/3, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(2/3)))/(b*E^((I*(b*c + a*d))/d))

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fricas [A]  time = 0.86, size = 102, normalized size = 0.67 \[ \frac {d \left (\frac {i \, b}{d}\right )^{\frac {1}{3}} e^{\left (\frac {i \, b c - i \, a d}{d}\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b d x + i \, b c}{d}\right ) + d \left (-\frac {i \, b}{d}\right )^{\frac {1}{3}} e^{\left (\frac {-i \, b c + i \, a d}{d}\right )} \Gamma \left (\frac {2}{3}, \frac {-i \, b d x - i \, b c}{d}\right ) + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b \sin \left (b x + a\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(2/3)*cos(b*x+a),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{\frac {2}{3}} \cos \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^(2/3)*cos(b*x + a), x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{\frac {2}{3}} \cos \left (b x +a \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(2/3)*cos(b*x+a),x)

[Out]

int((d*x+c)^(2/3)*cos(b*x+a),x)

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maxima [A]  time = 1.87, size = 186, normalized size = 1.22 \[ \frac {6 \, {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {2}{3}} d \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} d \cos \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} d \sin \left (-\frac {b c - a d}{d}\right )\right )} {\left (d x + c\right )}^{\frac {2}{3}}}{6 \, b \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(d*x + c)^(2/3)*((d*x + c)*b/d)^(2/3)*d*sin(((d*x + c)*b - b*c + a*d)/d) + (((sqrt(3) + I)*gamma(2/3, I
*(d*x + c)*b/d) + (sqrt(3) - I)*gamma(2/3, -I*(d*x + c)*b/d))*d*cos(-(b*c - a*d)/d) + ((-I*sqrt(3) + 1)*gamma(
2/3, I*(d*x + c)*b/d) + (I*sqrt(3) + 1)*gamma(2/3, -I*(d*x + c)*b/d))*d*sin(-(b*c - a*d)/d))*(d*x + c)^(2/3))/
(b*((d*x + c)*b/d)^(2/3)*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{2/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)*(c + d*x)^(2/3),x)

[Out]

int(cos(a + b*x)*(c + d*x)^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(2/3)*cos(b*x+a),x)

[Out]

Integral((c + d*x)**(2/3)*cos(a + b*x), x)

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