3.102 \(\int \frac {(e x)^m}{(a+b \sin (c+d x^3))^2} \, dx\)
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 =
{} \[ \int \frac {(e x)^m}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]
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 = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^m}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \]
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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fricas [A] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (e x\right )^{m}}{b^{2} \cos \left (d x^{3} + c\right )^{2} - 2 \, a b \sin \left (d x^{3} + c\right ) - a^{2} - b^{2}}, x\right ) \]
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(-(e*x)^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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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{m}}{{\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2}}\,{d x} \]
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((e*x)^m/(b*sin(d*x^3 + c) + a)^2, x)
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maple [A] time = 0.54, 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 \[ \text {result too large to display} \]
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*e^m*x^m*cos(d*x^3)*cos(c) + 2*b^2*e^m*x^m*cos(2*c)*sin(2*d*x^3) + 2*b^2*e^m*x^m*cos(2*d*x^3)*sin(2*
c) - 4*a*b*e^m*x^m*sin(d*x^3)*sin(c) + 2*(a*b*e^m*x^m*cos(2*d*x^3)*cos(2*c) - 2*a^2*e^m*x^m*cos(c)*sin(d*x^3)
- a*b*e^m*x^m*sin(2*d*x^3)*sin(2*c) - 2*a^2*e^m*x^m*cos(d*x^3)*sin(c) - a*b*e^m*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*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)*s
in(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))*in
tegrate(2/3*((b^2*e^m*m*sin(2*c) - 2*b^2*e^m*sin(2*c))*x^m*cos(2*d*x^3) + 2*(a*b*e^m*m*cos(c) - 2*a*b*e^m*cos(
c))*x^m*cos(d*x^3) + (b^2*e^m*m*cos(2*c) - 2*b^2*e^m*cos(2*c))*x^m*sin(2*d*x^3) - 2*(a*b*e^m*m*sin(c) - 2*a*b*
e^m*sin(c))*x^m*sin(d*x^3) - ((3*a*b*d*e^m*x^3*sin(2*c) - a*b*e^m*m*cos(2*c) + 2*a*b*e^m*cos(2*c))*x^m*cos(2*d
*x^3) + 2*(3*a^2*d*e^m*x^3*cos(c) + a^2*e^m*m*sin(c) - 2*a^2*e^m*sin(c))*x^m*cos(d*x^3) + (3*a*b*d*e^m*x^3*cos
(2*c) + a*b*e^m*m*sin(2*c) - 2*a*b*e^m*sin(2*c))*x^m*sin(2*d*x^3) - 2*(3*a^2*d*e^m*x^3*sin(c) - a^2*e^m*m*cos(
c) + 2*a^2*e^m*cos(c))*x^m*sin(d*x^3) + (a*b*e^m*m - 2*a*b*e^m)*x^m)*cos(d*x^3 + c) - (3*a*b*d*e^m*x^3*x^m - (
3*a*b*d*e^m*x^3*cos(2*c) + a*b*e^m*m*sin(2*c) - 2*a*b*e^m*sin(2*c))*x^m*cos(2*d*x^3) + 2*(3*a^2*d*e^m*x^3*sin(
c) - a^2*e^m*m*cos(c) + 2*a^2*e^m*cos(c))*x^m*cos(d*x^3) + (3*a*b*d*e^m*x^3*sin(2*c) - a*b*e^m*m*cos(2*c) + 2*
a*b*e^m*cos(2*c))*x^m*sin(2*d*x^3) + 2*(3*a^2*d*e^m*x^3*cos(c) + a^2*e^m*m*sin(c) - 2*a^2*e^m*sin(c))*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)*sin(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*e^m*x^m*cos(d*x^3)*cos(c) + a*b*e^m*x^m*cos(2*c)*sin(2*d*x^3) + a*b
*e^m*x^m*cos(2*d*x^3)*sin(2*c) - 2*a^2*e^m*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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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (e\,x\right )}^m}{{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2} \,d x \]
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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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{m}}{\left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}}\, dx \]
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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