3.1.56 \(\int \frac {(e x)^m}{(a+b \sin (c+d x^2))^2} \, dx\) [56]

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

[Out]

Unintegrable((e*x)^m/(a+b*sin(d*x^2+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^2\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e x)^m}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m}}{\left (a +b \sin \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x)^m/(a+b*sin(d*x^2+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^2+c))^2,x, algorithm="maxima")

[Out]

(a^3*b*cos(2*d*x^2 + 2*c)*cos(d*x^2 + c)*e^(m*log(x) + m) - b^4*cos(2*c)*e^(m*log(x) + m)*sin(2*d*x^2) - b^4*c
os(2*d*x^2)*e^(m*log(x) + m)*sin(2*c) + 2*(a^3*b - a*b^3)*cos(d*x^2)*cos(c)*e^(m*log(x) + m) - 2*(a^3*b - a*b^
3)*e^(m*log(x) + m)*sin(d*x^2)*sin(c) - (a*b^3*cos(2*d*x^2)*cos(2*c)*e^(m*log(x) + m) - a*b^3*e^(m*log(x) + m)
*sin(2*d*x^2)*sin(2*c) + 2*(a^4 - a^2*b^2)*cos(c)*e^(m*log(x) + m)*sin(d*x^2) + 2*(a^4 - a^2*b^2)*cos(d*x^2)*e
^(m*log(x) + m)*sin(c) + (a^3*b - a*b^3)*e^(m*log(x) + m))*cos(d*x^2 + c) - (a^4*b^2*d*x*cos(2*d*x^2 + 2*c)^2
+ a^4*b^2*d*x*sin(2*d*x^2 + 2*c)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x*cos(2*d*x^2)^2 + 4*((a^6 - 2*a^4*b^
2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2)*d*x*cos(d*x^2)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c
)^2)*d*x*sin(2*d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x*cos(c)*sin(d*x^2) + 4*((a^6 - 2*a^4*b^2 + a^2*b^4)
*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2)*d*x*sin(d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x*cos(d*x
^2)*sin(c) + (a^4*b^2 - 2*a^2*b^4 + b^6)*d*x - 2*(2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos
(2*c)*sin(c))*d*x*cos(d*x^2) - (a^2*b^4 - b^6)*d*x*cos(2*c) - 2*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3
- a*b^5)*sin(2*c)*sin(c))*d*x*sin(d*x^2))*cos(2*d*x^2) - 2*(a^2*b^4*d*x*cos(2*d*x^2)*cos(2*c) - a^2*b^4*d*x*si
n(2*d*x^2)*sin(2*c) + 2*(a^5*b - a^3*b^3)*d*x*cos(c)*sin(d*x^2) + 2*(a^5*b - a^3*b^3)*d*x*cos(d*x^2)*sin(c) +
(a^4*b^2 - a^2*b^4)*d*x)*cos(2*d*x^2 + 2*c) - 2*(2*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(
2*c)*sin(c))*d*x*cos(d*x^2) + 2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*x*si
n(d*x^2) + (a^2*b^4 - b^6)*d*x*sin(2*c))*sin(2*d*x^2) - 2*(a^2*b^4*d*x*cos(2*c)*sin(2*d*x^2) + a^2*b^4*d*x*cos
(2*d*x^2)*sin(2*c) - 2*(a^5*b - a^3*b^3)*d*x*cos(d*x^2)*cos(c) + 2*(a^5*b - a^3*b^3)*d*x*sin(d*x^2)*sin(c))*si
n(2*d*x^2 + 2*c))*integrate(-((b^4*m*sin(2*c) - b^4*sin(2*c))*cos(2*d*x^2)*e^(m*log(x) + m) - 2*((a^3*b - a*b^
3)*m*cos(c) - (a^3*b - a*b^3)*cos(c))*cos(d*x^2)*e^(m*log(x) + m) + (b^4*m*cos(2*c) - b^4*cos(2*c))*e^(m*log(x
) + m)*sin(2*d*x^2) + 2*((a^3*b - a*b^3)*m*sin(c) - (a^3*b - a*b^3)*sin(c))*e^(m*log(x) + m)*sin(d*x^2) - (2*a
^3*b*d*x^2*e^(m*log(x) + m)*sin(d*x^2 + c) + (a^3*b*m - a^3*b)*cos(d*x^2 + c)*e^(m*log(x) + m))*cos(2*d*x^2 +
2*c) - ((2*a*b^3*d*x^2*e^m*sin(2*c) - (a*b^3*m*cos(2*c) - a*b^3*cos(2*c))*e^m)*x^m*cos(2*d*x^2) - 2*(2*(a^4 -
a^2*b^2)*d*x^2*cos(c)*e^m + ((a^4 - a^2*b^2)*m*sin(c) - (a^4 - a^2*b^2)*sin(c))*e^m)*x^m*cos(d*x^2) + (2*a*b^3
*d*x^2*cos(2*c)*e^m + (a*b^3*m*sin(2*c) - a*b^3*sin(2*c))*e^m)*x^m*sin(2*d*x^2) + 2*(2*(a^4 - a^2*b^2)*d*x^2*e
^m*sin(c) - ((a^4 - a^2*b^2)*m*cos(c) - (a^4 - a^2*b^2)*cos(c))*e^m)*x^m*sin(d*x^2) + (a^3*b - a*b^3 - (a^3*b
- a*b^3)*m)*e^(m*log(x) + m))*cos(d*x^2 + c) + (2*a^3*b*d*x^2*cos(d*x^2 + c)*e^(m*log(x) + m) - (a^3*b*m - a^3
*b)*e^(m*log(x) + m)*sin(d*x^2 + c) - (a^2*b^2*m - a^2*b^2)*e^(m*log(x) + m))*sin(2*d*x^2 + 2*c) + (2*(a^3*b -
a*b^3)*d*x^2*e^(m*log(x) + m) + (2*a*b^3*d*x^2*cos(2*c)*e^m + (a*b^3*m*sin(2*c) - a*b^3*sin(2*c))*e^m)*x^m*co
s(2*d*x^2) + 2*(2*(a^4 - a^2*b^2)*d*x^2*e^m*sin(c) - ((a^4 - a^2*b^2)*m*cos(c) - (a^4 - a^2*b^2)*cos(c))*e^m)*
x^m*cos(d*x^2) - (2*a*b^3*d*x^2*e^m*sin(2*c) - (a*b^3*m*cos(2*c) - a*b^3*cos(2*c))*e^m)*x^m*sin(2*d*x^2) + 2*(
2*(a^4 - a^2*b^2)*d*x^2*cos(c)*e^m + ((a^4 - a^2*b^2)*m*sin(c) - (a^4 - a^2*b^2)*sin(c))*e^m)*x^m*sin(d*x^2))*
sin(d*x^2 + c))/(a^4*b^2*d*x^2*cos(2*d*x^2 + 2*c)^2 + a^4*b^2*d*x^2*sin(2*d*x^2 + 2*c)^2 + (b^6*cos(2*c)^2 + b
^6*sin(2*c)^2)*d*x^2*cos(2*d*x^2)^2 + 4*((a^6 - 2*a^4*b^2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*si
n(c)^2)*d*x^2*cos(d*x^2)^2 + (b^6*cos(2*c)^2 + b^6*sin(2*c)^2)*d*x^2*sin(2*d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a
*b^5)*d*x^2*cos(c)*sin(d*x^2) + 4*((a^6 - 2*a^4*b^2 + a^2*b^4)*cos(c)^2 + (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(c)^2
)*d*x^2*sin(d*x^2)^2 + 4*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(d*x^2)*sin(c) + (a^4*b^2 - 2*a^2*b^4 + b^6)*d*x
^2 - 2*(2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*x^2*cos(d*x^2) - (a^2*b^4
- b^6)*d*x^2*cos(2*c) - 2*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(2*c)*sin(c))*d*x^2*sin(d*
x^2))*cos(2*d*x^2) - 2*(a^2*b^4*d*x^2*cos(2*d*x^2)*cos(2*c) - a^2*b^4*d*x^2*sin(2*d*x^2)*sin(2*c) + 2*(a^5*b -
a^3*b^3)*d*x^2*cos(c)*sin(d*x^2) + 2*(a^5*b - a^3*b^3)*d*x^2*cos(d*x^2)*sin(c) + (a^4*b^2 - a^2*b^4)*d*x^2)*c
os(2*d*x^2 + 2*c) - 2*(2*((a^3*b^3 - a*b^5)*cos(2*c)*cos(c) + (a^3*b^3 - a*b^5)*sin(2*c)*sin(c))*d*x^2*cos(d*x
^2) + 2*((a^3*b^3 - a*b^5)*cos(c)*sin(2*c) - (a^3*b^3 - a*b^5)*cos(2*c)*sin(c))*d*x^2*sin(d*x^2) + (a^2*b^4 -
b^6)*d*x^2*sin(2*c))*sin(2*d*x^2) - 2*(a^2*b^4*d*x^2*cos(2*c)*sin(2*d*x^2) + a^2*b^4*d*x^2*cos(2*d*x^2)*sin(2*
c) - 2*(a^5*b - a^3*b^3)*d*x^2*cos(d*x^2)*cos(c) + 2*(a^5*b - a^3*b^3)*d*x^2*sin(d*x^2)*sin(c))*sin(2*d*x^2 +
2*c)), x) + (a^3*b*e^(m*log(x) + m)*sin(d*x^2 + c) + a^2*b^2*e^(m*log(x) + m))*sin(2*d*x^2 + 2*c) - (a*b^3*cos
(2*c)*e^(m*log(x) + m)*sin(2*d*x^2) + a*b^3*cos...

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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^2+c))^2,x, algorithm="fricas")

[Out]

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

[Out]

Integral((e*x)**m/(a + b*sin(c + d*x**2))**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^2+c))^2,x, algorithm="giac")

[Out]

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

[Out]

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

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