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

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sec (c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 164.93, size = 0, normalized size = 0.00 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )}{a \sin \left (d x + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{m} \sec \left (d x +c \right )}{a +a \sin \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)^m*sec(d*x+c)/(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((f*x+e)^m*sec(d*x+c)/(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 \[ \int \frac {{\left (e+f\,x\right )}^m}{\cos \left (c+d\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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