3.35 \(\int \frac {\text {sech}^2(a+b x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\text {sech}^2(a+b x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable(sech(b*x+a)^2/(d*x+c)^2,x)

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {\text {sech}^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sech[a + b*x]^2/(c + d*x)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 17.96, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right )^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -4 \, d \int \frac {1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + {\left (b d^{3} x^{3} e^{\left (2 \, a\right )} + 3 \, b c d^{2} x^{2} e^{\left (2 \, a\right )} + 3 \, b c^{2} d x e^{\left (2 \, a\right )} + b c^{3} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}\,{d x} - \frac {2}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + {\left (b d^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b c d x e^{\left (2 \, a\right )} + b c^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*d*integrate(1/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + (b*d^3*x^3*e^(2*a) + 3*b*c*d^2*x^2*e^(2*a)
+ 3*b*c^2*d*x*e^(2*a) + b*c^3*e^(2*a))*e^(2*b*x)), x) - 2/(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + (b*d^2*x^2*e^(2*a)
+ 2*b*c*d*x*e^(2*a) + b*c^2*e^(2*a))*e^(2*b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^2*(c + d*x)^2),x)

[Out]

int(1/(cosh(a + b*x)^2*(c + d*x)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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