3.1.100 \(\int (e x)^m (a+b \sin (c+d x^3)) \, dx\) [100]

Optimal. Leaf size=134 \[ \frac {a (e x)^{1+m}}{e (1+m)}+\frac {i b e^{i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-i d x^3\right )}{6 e}-\frac {i b e^{-i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},i d x^3\right )}{6 e} \]

[Out]

a*(e*x)^(1+m)/e/(1+m)+1/6*I*b*exp(I*c)*(e*x)^(1+m)*(-I*d*x^3)^(-1/3-1/3*m)*GAMMA(1/3+1/3*m,-I*d*x^3)/e-1/6*I*b
*(e*x)^(1+m)*(I*d*x^3)^(-1/3-1/3*m)*GAMMA(1/3+1/3*m,I*d*x^3)/e/exp(I*c)

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {14, 3470, 2250} \begin {gather*} \frac {i b e^{i c} \left (-i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{3},-i d x^3\right )}{6 e}-\frac {i b e^{-i c} \left (i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{3},i d x^3\right )}{6 e}+\frac {a (e x)^{m+1}}{e (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*Sin[c + d*x^3]),x]

[Out]

(a*(e*x)^(1 + m))/(e*(1 + m)) + ((I/6)*b*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-I)
*d*x^3])/e - ((I/6)*b*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, I*d*x^3])/(e*E^(I*c))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3470

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[I/2, Int[(e*x)^m*E^((-c)*I - d*I*x^n),
x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (e x)^m \left (a+b \sin \left (c+d x^3\right )\right ) \, dx &=\int \left (a (e x)^m+b (e x)^m \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac {a (e x)^{1+m}}{e (1+m)}+b \int (e x)^m \sin \left (c+d x^3\right ) \, dx\\ &=\frac {a (e x)^{1+m}}{e (1+m)}+\frac {1}{2} (i b) \int e^{-i c-i d x^3} (e x)^m \, dx-\frac {1}{2} (i b) \int e^{i c+i d x^3} (e x)^m \, dx\\ &=\frac {a (e x)^{1+m}}{e (1+m)}+\frac {i b e^{i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-i d x^3\right )}{6 e}-\frac {i b e^{-i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},i d x^3\right )}{6 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.01, size = 149, normalized size = 1.11 \begin {gather*} \frac {x (e x)^m \left (d^2 x^6\right )^{\frac {1}{3} (-1-m)} \left (6 a \left (d^2 x^6\right )^{\frac {1+m}{3}}-i b (1+m) \left (-i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},i d x^3\right ) (\cos (c)-i \sin (c))+i b (1+m) \left (i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},-i d x^3\right ) (\cos (c)+i \sin (c))\right )}{6 (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a+b*sin(d*x^3+c)),x)

[Out]

int((e*x)^m*(a+b*sin(d*x^3+c)),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.11, size = 93, normalized size = 0.69 \begin {gather*} \frac {6 \, \left (x e\right )^{m} a d x - {\left (b m + b\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (i \, d e^{\left (-3\right )}\right ) - i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, i \, d x^{3}\right ) - {\left (b m + b\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-i \, d e^{\left (-3\right )}\right ) + i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -i \, d x^{3}\right )}{6 \, {\left (d m + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(x*e)^m*a*d*x - (b*m + b)*e^(-1/3*(m - 2)*log(I*d*e^(-3)) - I*c + 2)*gamma(1/3*m + 1/3, I*d*x^3) - (b*m
+ b)*e^(-1/3*(m - 2)*log(-I*d*e^(-3)) + I*c + 2)*gamma(1/3*m + 1/3, -I*d*x^3))/(d*m + d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*sin(d*x**3+c)),x)

[Out]

Integral((e*x)**m*(a + b*sin(c + d*x**3)), x)

________________________________________________________________________________________

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((e*x)^m*(a+b*sin(d*x^3+c)),x, algorithm="giac")

[Out]

integrate((b*sin(d*x^3 + c) + a)*(x*e)^m, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (e\,x\right )}^m\,\left (a+b\,\sin \left (d\,x^3+c\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a + b*sin(c + d*x^3)),x)

[Out]

int((e*x)^m*(a + b*sin(c + d*x^3)), x)

________________________________________________________________________________________