3.40 \(\int \frac{\text{sech}^3(a+b x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\text{sech}^3(a+b x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable[Sech[a + b*x]^3/(c + d*x)^2, x]

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Rubi [A]  time = 0.034078, 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 = {} \[ \int \frac{\text{sech}^3(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sech[a + b*x]^3/(c + d*x)^2,x]

[Out]

Defer[Int][Sech[a + b*x]^3/(c + d*x)^2, x]

Rubi steps

\begin{align*} \int \frac{\text{sech}^3(a+b x)}{(c+d x)^2} \, dx &=\int \frac{\text{sech}^3(a+b x)}{(c+d x)^2} \, dx\\ \end{align*}

Mathematica [F]  time = 180.015, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[Sech[a + b*x]^3/(c + d*x)^2,x]

[Out]

$Aborted

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Maple [A]  time = 0.422, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^3/(d*x+c)^2,x)

[Out]

int(sech(b*x+a)^3/(d*x+c)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b d x e^{\left (3 \, a\right )} +{\left (b c - 2 \, d\right )} e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} -{\left (b d x e^{a} +{\left (b c + 2 \, d\right )} e^{a}\right )} e^{\left (b x\right )}}{b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3} +{\left (b^{2} d^{3} x^{3} e^{\left (4 \, a\right )} + 3 \, b^{2} c d^{2} x^{2} e^{\left (4 \, a\right )} + 3 \, b^{2} c^{2} d x e^{\left (4 \, a\right )} + b^{2} c^{3} e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )} + 2 \,{\left (b^{2} d^{3} x^{3} e^{\left (2 \, a\right )} + 3 \, b^{2} c d^{2} x^{2} e^{\left (2 \, a\right )} + 3 \, b^{2} c^{2} d x e^{\left (2 \, a\right )} + b^{2} c^{3} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}} + 8 \, \int \frac{{\left (b^{2} d^{2} x^{2} e^{a} + 2 \, b^{2} c d x e^{a} +{\left (b^{2} c^{2} - 6 \, d^{2}\right )} e^{a}\right )} e^{\left (b x\right )}}{8 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} +{\left (b^{2} d^{4} x^{4} e^{\left (2 \, a\right )} + 4 \, b^{2} c d^{3} x^{3} e^{\left (2 \, a\right )} + 6 \, b^{2} c^{2} d^{2} x^{2} e^{\left (2 \, a\right )} + 4 \, b^{2} c^{3} d x e^{\left (2 \, a\right )} + b^{2} c^{4} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

((b*d*x*e^(3*a) + (b*c - 2*d)*e^(3*a))*e^(3*b*x) - (b*d*x*e^a + (b*c + 2*d)*e^a)*e^(b*x))/(b^2*d^3*x^3 + 3*b^2
*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3*e^(4*a) + 3*b^2*c*d^2*x^2*e^(4*a) + 3*b^2*c^2*d*x*e^(4*a)
+ b^2*c^3*e^(4*a))*e^(4*b*x) + 2*(b^2*d^3*x^3*e^(2*a) + 3*b^2*c*d^2*x^2*e^(2*a) + 3*b^2*c^2*d*x*e^(2*a) + b^2*
c^3*e^(2*a))*e^(2*b*x)) + 8*integrate(1/8*(b^2*d^2*x^2*e^a + 2*b^2*c*d*x*e^a + (b^2*c^2 - 6*d^2)*e^a)*e^(b*x)/
(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d^4*x^4*e^(2*a) + 4*b^2*c*
d^3*x^3*e^(2*a) + 6*b^2*c^2*d^2*x^2*e^(2*a) + 4*b^2*c^3*d*x*e^(2*a) + b^2*c^4*e^(2*a))*e^(2*b*x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{sech}\left (b x + a\right )^{3}}{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(sech(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(sech(b*x + a)^3/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**3/(d*x+c)**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(sech(b*x + a)^3/(d*x + c)^2, x)