∫ 1______ (a+b sin(c+dx3))2 dx [95]

3.1.95 \(\int \frac {1}{(a+b \sin (c+d x^3))^2} \, dx\) [95]

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{\left (a+b \sin \left (c+d x^3\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*sin(d*x^3+c))^2,x)

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Rubi [A]
time = 0.00, 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 {1}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 5.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \sin \left (c+d x^3\right )\right )^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(d*x^3+c))^2,x)

[Out]

int(1/(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(1/(a+b*sin(d*x^3+c))^2,x, algorithm="maxima")

[Out]

1/3*(4*a*b*cos(d*x^3)*cos(c) + 2*b^2*cos(2*c)*sin(2*d*x^3) + 2*b^2*cos(2*d*x^3)*sin(2*c) - 4*a*b*sin(d*x^3)*si

n(c) + 2*(a*b*cos(2*d*x^3)*cos(2*c) - 2*a^2*cos(c)*sin(d*x^3) - a*b*sin(2*d*x^3)*sin(2*c) - 2*a^2*cos(d*x^3)*s

in(c) - a*b)*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*co

s(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*(4*a*b*cos(d*x^3)*cos(c) + 2*b^2*cos(2*c)*sin(2*d*x^3) + 2*

b^2*cos(2*d*x^3)*sin(2*c) - 4*a*b*sin(d*x^3)*sin(c) - (2*a*b - (3*a*b*d*x^3*sin(2*c) + 2*a*b*cos(2*c))*cos(2*d

*x^3) - 2*(3*a^2*d*x^3*cos(c) - 2*a^2*sin(c))*cos(d*x^3) - (3*a*b*d*x^3*cos(2*c) - 2*a*b*sin(2*c))*sin(2*d*x^3

) + 2*(3*a^2*d*x^3*sin(c) + 2*a^2*cos(c))*sin(d*x^3))*cos(d*x^3 + c) + (3*a*b*d*x^3 - (3*a*b*d*x^3*cos(2*c) -

2*a*b*sin(2*c))*cos(2*d*x^3) + 2*(3*a^2*d*x^3*sin(c) + 2*a^2*cos(c))*cos(d*x^3) + (3*a*b*d*x^3*sin(2*c) + 2*a*

b*cos(2*c))*sin(2*d*x^3) + 2*(3*a^2*d*x^3*cos(c) - 2*a^2*sin(c))*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*s

in(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) + a*b*cos(2*c)*sin(2*d*x^3) + a*b*cos(2*d*x^3)*sin(2*c) - 2*a^2*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))*s

in(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(1/(a+b*sin(d*x^3+c))^2,x, algorithm="fricas")

[Out]

integral(-1/(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 {1}{\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(1/(a+b*sin(d*x**3+c))**2,x)

[Out]

Integral((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(1/(a+b*sin(d*x^3+c))^2,x, algorithm="giac")

[Out]

integrate((b*sin(d*x^3 + c) + a)^(-2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{{\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(1/(a + b*sin(c + d*x^3))^2,x)

[Out]

int(1/(a + b*sin(c + d*x^3))^2, x)

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