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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {a+b \tan (e+f x)}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 6.14, size = 0, normalized size = 0.00 \[ \int \frac {a+b \tan (e+f x)}{(c+d x)^2} \, dx \]

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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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \tan \left (f x + e\right ) + a}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \tan \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {a +b \tan \left (f x +e \right )}{\left (d x +c \right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b d^{2} x + b c d\right )} \int \frac {\sin \left (2 \, f x + 2 \, e\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}\,{d x} - a}{d^{2} x + c d} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \tan {\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]

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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