3.149 \(\int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {(c+d x)^m}{a+a \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{m}}{a \sin \left (f x + e\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x + c)^m/(a*sin(f*x + e) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{a +a \sin \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (c+d\,x\right )}^m}{a+a\,\sin \left (e+f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (c + d x\right )^{m}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)**m/(sin(e + f*x) + 1), x)/a

________________________________________________________________________________________