3.43 $$\int \frac{a+b \tan (e+f x)}{(c+d x)^2} \, dx$$

Optimal. Leaf size=20 $\text{Unintegrable}\left (\frac{a+b \tan (e+f x)}{(c+d x)^2},x\right )$

[Out]

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

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Rubi [A]  time = 0.0271825, 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 = {} $\int \frac{a+b \tan (e+f x)}{(c+d x)^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.7175, size = 0, normalized size = 0. $\int \frac{a+b \tan (e+f x)}{(c+d x)^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\tan \left ( fx+e \right ) }{ \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))/(d*x+c)^2,x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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