3.196 \(\int \frac {\sin ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sin ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {\sin ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.19, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{a f^{2} x^{2} + 2 \, a e f x + a e^{2} + {\left (a f^{2} x^{2} + 2 \, a e f x + a e^{2}\right )} \sin \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c)^2 - 1)*sin(d*x + c)/(a*f^2*x^2 + 2*a*e*f*x + a*e^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*s
in(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 1.49, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}\left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\sin \left (c+d\,x\right )}^3}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**3/(f*x+e)**2/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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