3.1.98 \(\int (e x)^m (a+b \sin (c+d x^3))^3 \, dx\) [98]

Optimal. Leaf size=442 \[ \frac {a \left (2 a^2+3 b^2\right ) (e x)^{1+m}}{2 e (1+m)}+\frac {i b \left (4 a^2+b^2\right ) e^{i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-i d x^3\right )}{8 e}-\frac {i b \left (4 a^2+b^2\right ) e^{-i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},i d x^3\right )}{8 e}+\frac {2^{-\frac {7}{3}-\frac {m}{3}} a b^2 e^{2 i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-2 i d x^3\right )}{e}+\frac {2^{-\frac {7}{3}-\frac {m}{3}} a b^2 e^{-2 i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},2 i d x^3\right )}{e}-\frac {i 3^{-\frac {4}{3}-\frac {m}{3}} b^3 e^{3 i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-3 i d x^3\right )}{8 e}+\frac {i 3^{-\frac {4}{3}-\frac {m}{3}} b^3 e^{-3 i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},3 i d x^3\right )}{8 e} \]

[Out]

1/2*a*(2*a^2+3*b^2)*(e*x)^(1+m)/e/(1+m)+1/8*I*b*(4*a^2+b^2)*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-1/8*I*b*(4*a^2+b^2)*(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)+2^(-7/3-1/3*m)*a*b^2*exp(2*I*c)*(e*x)^(1+m)*(-I*d*x^3)^(-1/3-1/3*m)*GAMMA(1/3+1/3*m,-2*I*d*x^3)/e+2^(-7/3-
1/3*m)*a*b^2*(e*x)^(1+m)*(I*d*x^3)^(-1/3-1/3*m)*GAMMA(1/3+1/3*m,2*I*d*x^3)/e/exp(2*I*c)-1/8*I*3^(-4/3-1/3*m)*b
^3*exp(3*I*c)*(e*x)^(1+m)*(-I*d*x^3)^(-1/3-1/3*m)*GAMMA(1/3+1/3*m,-3*I*d*x^3)/e+1/8*I*3^(-4/3-1/3*m)*b^3*(e*x)
^(1+m)*(I*d*x^3)^(-1/3-1/3*m)*GAMMA(1/3+1/3*m,3*I*d*x^3)/e/exp(3*I*c)

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Rubi [A]
time = 0.28, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3484, 6, 3471, 2250, 3470} \begin {gather*} \frac {i b e^{i c} \left (4 a^2+b^2\right ) \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 )}{8 e}-\frac {i b e^{-i c} \left (4 a^2+b^2\right ) \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 )}{8 e}+\frac {a b^2 e^{2 i c} 2^{-\frac {m}{3}-\frac {7}{3}} \left (-i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{3},-2 i d x^3\right )}{e}+\frac {a b^2 e^{-2 i c} 2^{-\frac {m}{3}-\frac {7}{3}} \left (i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{3},2 i d x^3\right )}{e}-\frac {i b^3 e^{3 i c} 3^{-\frac {m}{3}-\frac {4}{3}} \left (-i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{3},-3 i d x^3\right )}{8 e}+\frac {i b^3 e^{-3 i c} 3^{-\frac {m}{3}-\frac {4}{3}} \left (i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \text {Gamma}\left (\frac {m+1}{3},3 i d x^3\right )}{8 e}+\frac {a \left (2 a^2+3 b^2\right ) (e x)^{m+1}}{2 e (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*a^2 + 3*b^2)*(e*x)^(1 + m))/(2*e*(1 + m)) + ((I/8)*b*(4*a^2 + b^2)*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((
-1 - m)/3)*Gamma[(1 + m)/3, (-I)*d*x^3])/e - ((I/8)*b*(4*a^2 + b^2)*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma
[(1 + m)/3, I*d*x^3])/(e*E^(I*c)) + (2^(-7/3 - m/3)*a*b^2*E^((2*I)*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*
Gamma[(1 + m)/3, (-2*I)*d*x^3])/e + (2^(-7/3 - m/3)*a*b^2*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3
, (2*I)*d*x^3])/(e*E^((2*I)*c)) - ((I/8)*3^(-4/3 - m/3)*b^3*E^((3*I)*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3
)*Gamma[(1 + m)/3, (-3*I)*d*x^3])/e + ((I/8)*3^(-4/3 - m/3)*b^3*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1
+ m)/3, (3*I)*d*x^3])/(e*E^((3*I)*c))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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]

Rule 3471

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^((-c)*I - d*I*x^n),
x], x] + Dist[1/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]

Rule 3484

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (e x)^m \left (a+b \sin \left (c+d x^3\right )\right )^3 \, dx &=\int \left (a^3 (e x)^m+\frac {3}{2} a b^2 (e x)^m-\frac {3}{2} a b^2 (e x)^m \cos \left (2 c+2 d x^3\right )+3 a^2 b (e x)^m \sin \left (c+d x^3\right )+\frac {3}{4} b^3 (e x)^m \sin \left (c+d x^3\right )-\frac {1}{4} b^3 (e x)^m \sin \left (3 c+3 d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^3+\frac {3 a b^2}{2}\right ) (e x)^m-\frac {3}{2} a b^2 (e x)^m \cos \left (2 c+2 d x^3\right )+3 a^2 b (e x)^m \sin \left (c+d x^3\right )+\frac {3}{4} b^3 (e x)^m \sin \left (c+d x^3\right )-\frac {1}{4} b^3 (e x)^m \sin \left (3 c+3 d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^3+\frac {3 a b^2}{2}\right ) (e x)^m-\frac {3}{2} a b^2 (e x)^m \cos \left (2 c+2 d x^3\right )+\left (3 a^2 b+\frac {3 b^3}{4}\right ) (e x)^m \sin \left (c+d x^3\right )-\frac {1}{4} b^3 (e x)^m \sin \left (3 c+3 d x^3\right )\right ) \, dx\\ &=\frac {a \left (2 a^2+3 b^2\right ) (e x)^{1+m}}{2 e (1+m)}-\frac {1}{2} \left (3 a b^2\right ) \int (e x)^m \cos \left (2 c+2 d x^3\right ) \, dx-\frac {1}{4} b^3 \int (e x)^m \sin \left (3 c+3 d x^3\right ) \, dx+\frac {1}{4} \left (3 b \left (4 a^2+b^2\right )\right ) \int (e x)^m \sin \left (c+d x^3\right ) \, dx\\ &=\frac {a \left (2 a^2+3 b^2\right ) (e x)^{1+m}}{2 e (1+m)}-\frac {1}{4} \left (3 a b^2\right ) \int e^{-2 i c-2 i d x^3} (e x)^m \, dx-\frac {1}{4} \left (3 a b^2\right ) \int e^{2 i c+2 i d x^3} (e x)^m \, dx-\frac {1}{8} \left (i b^3\right ) \int e^{-3 i c-3 i d x^3} (e x)^m \, dx+\frac {1}{8} \left (i b^3\right ) \int e^{3 i c+3 i d x^3} (e x)^m \, dx+\frac {1}{8} \left (3 i b \left (4 a^2+b^2\right )\right ) \int e^{-i c-i d x^3} (e x)^m \, dx-\frac {1}{8} \left (3 i b \left (4 a^2+b^2\right )\right ) \int e^{i c+i d x^3} (e x)^m \, dx\\ &=\frac {a \left (2 a^2+3 b^2\right ) (e x)^{1+m}}{2 e (1+m)}+\frac {i b \left (4 a^2+b^2\right ) 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 )}{8 e}-\frac {i b \left (4 a^2+b^2\right ) 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 )}{8 e}+\frac {2^{-\frac {7}{3}-\frac {m}{3}} a b^2 e^{2 i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-2 i d x^3\right )}{e}+\frac {2^{-\frac {7}{3}-\frac {m}{3}} a b^2 e^{-2 i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},2 i d x^3\right )}{e}-\frac {i 3^{-\frac {4}{3}-\frac {m}{3}} b^3 e^{3 i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-3 i d x^3\right )}{8 e}+\frac {i 3^{-\frac {4}{3}-\frac {m}{3}} b^3 e^{-3 i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},3 i d x^3\right )}{8 e}\\ \end {align*}

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Mathematica [A]
time = 10.86, size = 451, normalized size = 1.02 \begin {gather*} \frac {1}{24} i x (e x)^m \left (3 b \left (4 a^2+b^2\right ) e^{i c} \left (-i d x^3\right )^{-\frac {1}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},-i d x^3\right )+\frac {-24 i a^3 d x^3-36 i a b^2 d x^3+12 i a^2 b e^{-i c} (1+m) \left (i d x^3\right )^{\frac {2}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},i d x^3\right )+3 i b^3 e^{-i c} (1+m) \left (i d x^3\right )^{\frac {2}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},i d x^3\right )+3\ 2^{\frac {2}{3}-\frac {m}{3}} a b^2 e^{2 i c} (1+m) \left (-i d x^3\right )^{\frac {2}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},-2 i d x^3\right )-3\ 2^{\frac {2}{3}-\frac {m}{3}} a b^2 e^{-2 i c} (1+m) \left (i d x^3\right )^{\frac {2}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},2 i d x^3\right )-i 3^{-\frac {1}{3}-\frac {m}{3}} b^3 e^{3 i c} (1+m) \left (-i d x^3\right )^{\frac {2}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},-3 i d x^3\right )-i 3^{-\frac {1}{3}-\frac {m}{3}} b^3 e^{-3 i c} (1+m) \left (i d x^3\right )^{\frac {2}{3}-\frac {m}{3}} \Gamma \left (\frac {1+m}{3},3 i d x^3\right )}{d (1+m) x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I/24)*x*(e*x)^m*(3*b*(4*a^2 + b^2)*E^(I*c)*((-I)*d*x^3)^(-1/3 - m/3)*Gamma[(1 + m)/3, (-I)*d*x^3] + ((-24*I)*
a^3*d*x^3 - (36*I)*a*b^2*d*x^3 + ((12*I)*a^2*b*(1 + m)*(I*d*x^3)^(2/3 - m/3)*Gamma[(1 + m)/3, I*d*x^3])/E^(I*c
) + ((3*I)*b^3*(1 + m)*(I*d*x^3)^(2/3 - m/3)*Gamma[(1 + m)/3, I*d*x^3])/E^(I*c) + 3*2^(2/3 - m/3)*a*b^2*E^((2*
I)*c)*(1 + m)*((-I)*d*x^3)^(2/3 - m/3)*Gamma[(1 + m)/3, (-2*I)*d*x^3] - (3*2^(2/3 - m/3)*a*b^2*(1 + m)*(I*d*x^
3)^(2/3 - m/3)*Gamma[(1 + m)/3, (2*I)*d*x^3])/E^((2*I)*c) - I*3^(-1/3 - m/3)*b^3*E^((3*I)*c)*(1 + m)*((-I)*d*x
^3)^(2/3 - m/3)*Gamma[(1 + m)/3, (-3*I)*d*x^3] - (I*3^(-1/3 - m/3)*b^3*(1 + m)*(I*d*x^3)^(2/3 - m/3)*Gamma[(1
+ m)/3, (3*I)*d*x^3])/E^((3*I)*c))/(d*(1 + m)*x^3))

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))^3,x, algorithm="maxima")

[Out]

(x*e)^(m + 1)*a^3*e^(-1)/(m + 1) + 1/8*(12*a*b^2*x*e^(m*log(x) + m) + 3*((4*a^2*b + b^3)*m*sin(c) + (4*a^2*b +
b^3)*sin(c))*e^m*integrate(x^m*cos(d*x^3), x) + 3*(4*a^2*b + b^3 + (4*a^2*b + b^3)*m)*e^m*integrate(x^m*sin(d
*x^3 + c), x) + 3*((4*a^2*b + b^3)*m*cos(c) + (4*a^2*b + b^3)*cos(c))*e^m*integrate(x^m*sin(d*x^3), x) - 12*(a
*b^2*m + a*b^2)*integrate(cos(2*d*x^3 + 2*c)*e^(m*log(x) + m), x) - 2*(b^3*m + b^3)*integrate(e^(m*log(x) + m)
*sin(3*d*x^3 + 3*c), x))/(m + 1)

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Fricas [A]
time = 0.13, size = 302, normalized size = 0.68 \begin {gather*} \frac {36 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \left (x e\right )^{m} d x + {\left (b^{3} m + b^{3}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (3 i \, d e^{\left (-3\right )}\right ) - 3 i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, 3 i \, d x^{3}\right ) - 9 i \, {\left (a b^{2} m + a b^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (2 i \, d e^{\left (-3\right )}\right ) - 2 i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, 2 i \, d x^{3}\right ) - 9 \, {\left (4 \, a^{2} b + b^{3} + {\left (4 \, a^{2} b + b^{3}\right )} m\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 ) - 9 \, {\left (4 \, a^{2} b + b^{3} + {\left (4 \, a^{2} b + b^{3}\right )} m\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 ) + 9 i \, {\left (a b^{2} m + a b^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-2 i \, d e^{\left (-3\right )}\right ) + 2 i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -2 i \, d x^{3}\right ) + {\left (b^{3} m + b^{3}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-3 i \, d e^{\left (-3\right )}\right ) + 3 i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -3 i \, d x^{3}\right )}{72 \, {\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))^3,x, algorithm="fricas")

[Out]

1/72*(36*(2*a^3 + 3*a*b^2)*(x*e)^m*d*x + (b^3*m + b^3)*e^(-1/3*(m - 2)*log(3*I*d*e^(-3)) - 3*I*c + 2)*gamma(1/
3*m + 1/3, 3*I*d*x^3) - 9*I*(a*b^2*m + a*b^2)*e^(-1/3*(m - 2)*log(2*I*d*e^(-3)) - 2*I*c + 2)*gamma(1/3*m + 1/3
, 2*I*d*x^3) - 9*(4*a^2*b + b^3 + (4*a^2*b + b^3)*m)*e^(-1/3*(m - 2)*log(I*d*e^(-3)) - I*c + 2)*gamma(1/3*m +
1/3, I*d*x^3) - 9*(4*a^2*b + b^3 + (4*a^2*b + b^3)*m)*e^(-1/3*(m - 2)*log(-I*d*e^(-3)) + I*c + 2)*gamma(1/3*m
+ 1/3, -I*d*x^3) + 9*I*(a*b^2*m + a*b^2)*e^(-1/3*(m - 2)*log(-2*I*d*e^(-3)) + 2*I*c + 2)*gamma(1/3*m + 1/3, -2
*I*d*x^3) + (b^3*m + b^3)*e^(-1/3*(m - 2)*log(-3*I*d*e^(-3)) + 3*I*c + 2)*gamma(1/3*m + 1/3, -3*I*d*x^3))/(d*m
+ d)

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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 )^{3}\, 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))**3,x)

[Out]

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

[Out]

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

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

[Out]

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

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