3.252 \(\int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\tan (a+b x) \sec (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

CannotIntegrate(sec(b*x+a)*tan(b*x+a)/(d*x+c)^2,x)

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 19.12, size = 0, normalized size = 0.00 \[ \int \frac {\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sec(b*x + a)*tan(b*x + 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 \[ \int \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x +a \right ) \tan \left (b x +a \right )}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {tan}\left (a+b\,x\right )}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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