### 3.35 $$\int (c+d x)^m (a+i a \tan (e+f x)) \, dx$$

Optimal. Leaf size=23 $\text{Unintegrable}\left ((c+d x)^m (a+i a \tan (e+f x)),x\right )$

[Out]

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

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Rubi [A]  time = 0.0280646, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int (c+d x)^m (a+i a \tan (e+f x)) \, dx$

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*}

Mathematica [A]  time = 13.8856, size = 0, normalized size = 0. $\int (c+d x)^m (a+i a \tan (e+f x)) \, dx$

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 = 0.185, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+ia\tan \left ( fx+e \right ) \right ) \, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{2 \,{\left (d x + c\right )}^{m} a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} a \left (\int i \left (c + d x\right )^{m} \tan{\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m}\, dx\right ) \end{align*}

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]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}{\left (d x + c\right )}^{m}\,{d x} \end{align*}

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)