3.4.18 \(\int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [318]

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

[Out]

Unintegrable((f*x+e)^m*sec(d*x+c)^2/(a+b*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 = {} \begin {gather*} \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 8.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{m} \left (\sec ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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((f*x+e)^m*sec(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**m*sec(d*x+c)**2/(a+b*sin(d*x+c)),x)

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^m}{{\cos \left (c+d\,x\right )}^2\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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