### 3.42 $$\int \frac{a+b \tan (e+f x)}{c+d x} \, dx$$

Optimal. Leaf size=20 $\text{Unintegrable}\left (\frac{a+b \tan (e+f x)}{c+d x},x\right )$

[Out]

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

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Rubi [A]  time = 0.0279738, 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 = {} $\int \frac{a+b \tan (e+f x)}{c+d x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(a + b*Tan[e + f*x])/(c + d*x),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.9219, size = 0, normalized size = 0. $\int \frac{a+b \tan (e+f x)}{c+d x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\tan \left ( fx+e \right ) }{dx+c}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b d \int \frac{\sin \left (2 \, f x + 2 \, e\right )}{{\left (d x + c\right )}{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}\,{d x} + a \log \left (d x + c\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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