3.1.35 \(\int (c+d x)^m (a+i a \tan (e+f x)) \, dx\) [35]

Optimal. Leaf size=24 \[ \text {Int}\left ((c+d x)^m (a+i a \tan (e+f x)),x\right ) \]

[Out]

Unintegrable((d*x+c)^m*(a+I*a*tan(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+i a \tan (e+f x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 14.60, size = 0, normalized size = 0.00 \begin {gather*} \int (c+d x)^m (a+i a \tan (e+f x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.32, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +i a \tan \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

2*I*a*integrate((d*x + c)^m*sin(2*f*x + 2*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+I*a*tan(f*x+e)),x, algorithm="fricas")

[Out]

integral(2*(d*x + c)^m*a*e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*a*(Integral(-I*(c + d*x)**m, x) + Integral((c + d*x)**m*tan(e + f*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((d*x+c)^m*(a+I*a*tan(f*x+e)),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(e + f*x)*1i)*(c + d*x)^m,x)

[Out]

int((a + a*tan(e + f*x)*1i)*(c + d*x)^m, x)

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