3.1.45 \(\int (c+d x)^m (a+b \sec (e+f x)) \, dx\) [45]

Optimal. Leaf size=21 \[ \text {Int}\left ((c+d x)^m (a+b \sec (e+f x)),x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 (c+d x)^m (a+b \sec (e+f x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int (c+d x)^m (a+b \sec (e+f x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +b \sec \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

2*b*integrate(((d*x + c)^m*cos(2*f*x + 2*e)*cos(f*x + e) + (d*x + c)^m*sin(2*f*x + 2*e)*sin(f*x + e) + (d*x +
c)^m*cos(f*x + e))/(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1), x) + (d*x + c)^(m + 1)*
a/(d*(m + 1))

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )\,{\left (c+d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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