### 3.34 $$\int (c+d x)^m (a+i a \tan (e+f x))^2 \, dx$$

Optimal. Leaf size=25 $\text{Unintegrable}\left ((c+d x)^m (a+i a \tan (e+f x))^2,x\right )$

[Out]

Unintegrable[(c + d*x)^m*(a + I*a*Tan[e + f*x])^2, x]

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Rubi [A]  time = 0.0498459, 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 = {} $\int (c+d x)^m (a+i a \tan (e+f x))^2 \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(c + d*x)^m*(a + I*a*Tan[e + f*x])^2,x]

[Out]

Defer[Int][(c + d*x)^m*(a + I*a*Tan[e + f*x])^2, x]

Rubi steps

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

Mathematica [A]  time = 22.7107, size = 0, normalized size = 0. $\int (c+d x)^m (a+i a \tan (e+f x))^2 \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(c + d*x)^m*(a + I*a*Tan[e + f*x])^2,x]

[Out]

Integrate[(c + d*x)^m*(a + I*a*Tan[e + f*x])^2, x]

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Maple [A]  time = 0.147, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{2}\, 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))^2,x)

[Out]

int((d*x+c)^m*(a+I*a*tan(f*x+e))^2,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))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**2,x)

[Out]

Timed out

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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 )}^{2}{\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))^2,x, algorithm="giac")

[Out]

integrate((I*a*tan(f*x + e) + a)^2*(d*x + c)^m, x)