### 3.48 $$\int \frac{(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx$$

Optimal. Leaf size=22 $\text{Unintegrable}\left (\frac{(a+b \tan (e+f x))^2}{(c+d x)^2},x\right )$

[Out]

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

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Rubi [A]  time = 0.051245, 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 = {} $\int \frac{(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 14.3644, size = 0, normalized size = 0. $\int \frac{(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.779, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^2/(d*x+c)^2,x)

[Out]

int((a+b*tan(f*x+e))^2/(d*x+c)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (a^{2} - b^{2}\right )} d f x - 2 \, b^{2} d \sin \left (2 \, f x + 2 \, e\right ) +{\left (a^{2} - b^{2}\right )} c f +{\left ({\left (a^{2} - b^{2}\right )} d f x +{\left (a^{2} - b^{2}\right )} c f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left ({\left (a^{2} - b^{2}\right )} d f x +{\left (a^{2} - b^{2}\right )} c f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left ({\left (a^{2} - b^{2}\right )} d f x +{\left (a^{2} - b^{2}\right )} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \,{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )\right )} \int \frac{{\left (a b d f x + a b c f + b^{2} d\right )} \sin \left (2 \, f x + 2 \, e\right )}{d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f +{\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \cos \left (2 \, f x + 2 \, e\right )}\,{d x}}{d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \,{\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

-((a^2 - b^2)*d*f*x - 2*b^2*d*sin(2*f*x + 2*e) + (a^2 - b^2)*c*f + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*cos(2
*f*x + 2*e)^2 + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*sin(2*f*x + 2*e)^2 + 2*((a^2 - b^2)*d*f*x + (a^2 - b^2)*
c*f)*cos(2*f*x + 2*e) - (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f + (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*cos(2*f*x + 2
*e)^2 + (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*sin(2*f*x + 2*e)^2 + 2*(d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*cos(2*f
*x + 2*e))*integrate(4*(a*b*d*f*x + a*b*c*f + b^2*d)*sin(2*f*x + 2*e)/(d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x
+ c^3*f + (d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x + c^3*f)*cos(2*f*x + 2*e)^2 + (d^3*f*x^3 + 3*c*d^2*f*x^2 +
3*c^2*d*f*x + c^3*f)*sin(2*f*x + 2*e)^2 + 2*(d^3*f*x^3 + 3*c*d^2*f*x^2 + 3*c^2*d*f*x + c^3*f)*cos(2*f*x + 2*e
)), x))/(d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f + (d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*cos(2*f*x + 2*e)^2 + (d^3*f*x
^2 + 2*c*d^2*f*x + c^2*d*f)*sin(2*f*x + 2*e)^2 + 2*(d^3*f*x^2 + 2*c*d^2*f*x + c^2*d*f)*cos(2*f*x + 2*e))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2)/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \tan{\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**2/(d*x+c)**2,x)

[Out]

Integral((a + b*tan(e + f*x))**2/(c + d*x)**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^2/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e) + a)^2/(d*x + c)^2, x)