3.22 \(\int (c+d x)^m (a+a \sec (e+f x)) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int (c+d x)^m (a+a \sec (e+f x)) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.67, size = 0, normalized size = 0.00 \[ \int (c+d x)^m (a+a \sec (e+f x)) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.76, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +a \sec \left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, a \int \frac {{\left (d x + c\right )}^{m} \cos \left (2 \, f x + 2 \, e\right ) \cos \left (f x + e\right ) + {\left (d x + c\right )}^{m} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + {\left (d x + c\right )}^{m} \cos \left (f x + e\right )}{\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1}\,{d x} + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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