### 3.47 $$\int \frac{(a+b \tan (e+f x))^2}{c+d x} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0529595, 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 = {} $\int \frac{(a+b \tan (e+f x))^2}{c+d x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 18.9841, size = 0, normalized size = 0. $\int \frac{(a+b \tan (e+f x))^2}{c+d x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.105, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}{dx+c}}\, 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),x)

[Out]

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

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

[Out]

(((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*cos(2*f*x + 2*e)^2*log(d*x + c) + 2*b^2*d*sin(2*f*x + 2*e) + ((a^2 - b^
2)*d*f*x + (a^2 - b^2)*c*f)*log(d*x + c)*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)*log(d*x + c) + (d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(2*f*x + 2*e)^2 + (d^2*f*x + c*d*f)*sin(2*f*x
+ 2*e)^2 + 2*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e))*integrate(2*(2*a*b*d*f*x + 2*a*b*c*f + b^2*d)*sin(2*f*x + 2*e
)/(d^2*f*x^2 + 2*c*d*f*x + c^2*f + (d^2*f*x^2 + 2*c*d*f*x + c^2*f)*cos(2*f*x + 2*e)^2 + (d^2*f*x^2 + 2*c*d*f*x
+ c^2*f)*sin(2*f*x + 2*e)^2 + 2*(d^2*f*x^2 + 2*c*d*f*x + c^2*f)*cos(2*f*x + 2*e)), x) + ((a^2 - b^2)*d*f*x +
(a^2 - b^2)*c*f)*log(d*x + c))/(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(2*f*x + 2*e)^2 + (d^2*f*x + c*d*f)*sin
(2*f*x + 2*e)^2 + 2*(d^2*f*x + c*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 x + c}, 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),x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*tan(e + f*x))**2/(c + d*x), 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}}{d x + c}\,{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),x, algorithm="giac")

[Out]

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