3.2.2 \(\int \frac {(e x)^m}{(a+b \sin (c+d x^3))^2} \, dx\) [102]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(e x)^m}{\left (a+b \sin \left (c+d x^3\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(e x)^m}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e x)^m}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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))^2,x, algorithm="maxima")

[Out]

1/3*(4*a*b*cos(d*x^3)*cos(c)*e^(m*log(x) + m) + 2*b^2*cos(2*c)*e^(m*log(x) + m)*sin(2*d*x^3) + 2*b^2*cos(2*d*x
^3)*e^(m*log(x) + m)*sin(2*c) - 4*a*b*e^(m*log(x) + m)*sin(d*x^3)*sin(c) + 2*(a*b*cos(2*d*x^3)*cos(2*c)*e^(m*l
og(x) + m) - 2*a^2*cos(c)*e^(m*log(x) + m)*sin(d*x^3) - a*b*e^(m*log(x) + m)*sin(2*d*x^3)*sin(2*c) - 2*a^2*cos
(d*x^3)*e^(m*log(x) + m)*sin(c) - a*b*e^(m*log(x) + m))*cos(d*x^3 + c) - 3*(((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2
*b^2 - b^4)*sin(2*c)^2)*d*x^2*cos(2*d*x^3)^2 + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^2*c
os(d*x^3)^2 + ((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^2*sin(2*d*x^3)^2 + 4*(a^3*b - a*b^
3)*d*x^2*cos(c)*sin(d*x^3) + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^2*sin(d*x^3)^2 + 4*(a
^3*b - a*b^3)*d*x^2*cos(d*x^3)*sin(c) + (a^2*b^2 - b^4)*d*x^2 + 2*(2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b
- a*b^3)*cos(2*c)*sin(c))*d*x^2*cos(d*x^3) - (a^2*b^2 - b^4)*d*x^2*cos(2*c) - 2*((a^3*b - a*b^3)*cos(2*c)*cos
(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^2*sin(d*x^3))*cos(2*d*x^3) + 2*(2*((a^3*b - a*b^3)*cos(2*c)*cos(c)
+ (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^2*cos(d*x^3) + 2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos
(2*c)*sin(c))*d*x^2*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^2*sin(2*c))*sin(2*d*x^3))*integrate(2/3*((b^2*m*sin(2*c)
- 2*b^2*sin(2*c))*cos(2*d*x^3)*e^(m*log(x) + m) + 2*(a*b*m*cos(c) - 2*a*b*cos(c))*cos(d*x^3)*e^(m*log(x) + m)
+ (b^2*m*cos(2*c) - 2*b^2*cos(2*c))*e^(m*log(x) + m)*sin(2*d*x^3) - 2*(a*b*m*sin(c) - 2*a*b*sin(c))*e^(m*log(x
) + m)*sin(d*x^3) - ((3*a*b*d*x^3*e^m*sin(2*c) - (a*b*m*cos(2*c) - 2*a*b*cos(2*c))*e^m)*x^m*cos(2*d*x^3) + 2*(
3*a^2*d*x^3*cos(c)*e^m + (a^2*m*sin(c) - 2*a^2*sin(c))*e^m)*x^m*cos(d*x^3) + (3*a*b*d*x^3*cos(2*c)*e^m + (a*b*
m*sin(2*c) - 2*a*b*sin(2*c))*e^m)*x^m*sin(2*d*x^3) - 2*(3*a^2*d*x^3*e^m*sin(c) - (a^2*m*cos(c) - 2*a^2*cos(c))
*e^m)*x^m*sin(d*x^3) + (a*b*m - 2*a*b)*e^(m*log(x) + m))*cos(d*x^3 + c) - (3*a*b*d*x^3*e^(m*log(x) + m) - (3*a
*b*d*x^3*cos(2*c)*e^m + (a*b*m*sin(2*c) - 2*a*b*sin(2*c))*e^m)*x^m*cos(2*d*x^3) + 2*(3*a^2*d*x^3*e^m*sin(c) -
(a^2*m*cos(c) - 2*a^2*cos(c))*e^m)*x^m*cos(d*x^3) + (3*a*b*d*x^3*e^m*sin(2*c) - (a*b*m*cos(2*c) - 2*a*b*cos(2*
c))*e^m)*x^m*sin(2*d*x^3) + 2*(3*a^2*d*x^3*cos(c)*e^m + (a^2*m*sin(c) - 2*a^2*sin(c))*e^m)*x^m*sin(d*x^3))*sin
(d*x^3 + c))/(((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^3*cos(2*d*x^3)^2 + 4*((a^4 - a^2*b
^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin(c)^2)*d*x^3*cos(d*x^3)^2 + ((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*si
n(2*c)^2)*d*x^3*sin(2*d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^3*cos(c)*sin(d*x^3) + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^
4 - a^2*b^2)*sin(c)^2)*d*x^3*sin(d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^3*cos(d*x^3)*sin(c) + (a^2*b^2 - b^4)*d*x^3
+ 2*(2*((a^3*b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^3*cos(d*x^3) - (a^2*b^2 - b^4)*
d*x^3*cos(2*c) - 2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^3*sin(d*x^3))*cos(2
*d*x^3) + 2*(2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^3*cos(d*x^3) + 2*((a^3*
b - a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^3*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^3*sin(2*c
))*sin(2*d*x^3)), x) + 2*(2*a^2*cos(d*x^3)*cos(c)*e^(m*log(x) + m) + a*b*cos(2*c)*e^(m*log(x) + m)*sin(2*d*x^3
) + a*b*cos(2*d*x^3)*e^(m*log(x) + m)*sin(2*c) - 2*a^2*e^(m*log(x) + m)*sin(d*x^3)*sin(c))*sin(d*x^3 + c))/(((
a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^2*cos(2*d*x^3)^2 + 4*((a^4 - a^2*b^2)*cos(c)^2 + (
a^4 - a^2*b^2)*sin(c)^2)*d*x^2*cos(d*x^3)^2 + ((a^2*b^2 - b^4)*cos(2*c)^2 + (a^2*b^2 - b^4)*sin(2*c)^2)*d*x^2*
sin(2*d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^2*cos(c)*sin(d*x^3) + 4*((a^4 - a^2*b^2)*cos(c)^2 + (a^4 - a^2*b^2)*sin
(c)^2)*d*x^2*sin(d*x^3)^2 + 4*(a^3*b - a*b^3)*d*x^2*cos(d*x^3)*sin(c) + (a^2*b^2 - b^4)*d*x^2 + 2*(2*((a^3*b -
a*b^3)*cos(c)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^2*cos(d*x^3) - (a^2*b^2 - b^4)*d*x^2*cos(2*c) -
2*((a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^2*sin(d*x^3))*cos(2*d*x^3) + 2*(2*(
(a^3*b - a*b^3)*cos(2*c)*cos(c) + (a^3*b - a*b^3)*sin(2*c)*sin(c))*d*x^2*cos(d*x^3) + 2*((a^3*b - a*b^3)*cos(c
)*sin(2*c) - (a^3*b - a*b^3)*cos(2*c)*sin(c))*d*x^2*sin(d*x^3) + (a^2*b^2 - b^4)*d*x^2*sin(2*c))*sin(2*d*x^3))

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Fricas [A]
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))^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]
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))^2,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (e\,x\right )}^m}{{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2} \,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))^2,x)

[Out]

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

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