3.282 \(\int \frac {\text {sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\)

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)

________________________________________________________________________________________

Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {\text {sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]

[Out]

Defer[Int][Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]

[Out]

$Aborted

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 0, normalized size = 0.00 \[ -\frac {4 \, d^{2} f^{2} x^{2} + 8 \, d^{2} e f x + 4 \, d^{2} e^{2} - 6 \, f^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, f^{2} + {\left (2 i \, d f^{2} x + 2 i \, d e f - 6 i \, f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (8 i \, d^{2} f^{2} x^{2} + 8 i \, d^{2} e^{2} + 2 i \, d e f - 6 i \, f^{2} + {\left (16 i \, d^{2} e f + 2 i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )} - {\left (3 \, a d^{3} f^{4} x^{4} + 12 \, a d^{3} e f^{3} x^{3} + 18 \, a d^{3} e^{2} f^{2} x^{2} + 12 \, a d^{3} e^{3} f x + 3 \, a d^{3} e^{4} - 3 \, {\left (a d^{3} f^{4} x^{4} + 4 \, a d^{3} e f^{3} x^{3} + 6 \, a d^{3} e^{2} f^{2} x^{2} + 4 \, a d^{3} e^{3} f x + a d^{3} e^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (-6 i \, a d^{3} f^{4} x^{4} - 24 i \, a d^{3} e f^{3} x^{3} - 36 i \, a d^{3} e^{2} f^{2} x^{2} - 24 i \, a d^{3} e^{3} f x - 6 i \, a d^{3} e^{4}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (-6 i \, a d^{3} f^{4} x^{4} - 24 i \, a d^{3} e f^{3} x^{3} - 36 i \, a d^{3} e^{2} f^{2} x^{2} - 24 i \, a d^{3} e^{3} f x - 6 i \, a d^{3} e^{4}\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (-\frac {8 \, d^{2} f^{3} x^{2} + 16 \, d^{2} e f^{2} x + 8 \, d^{2} e^{2} f - 24 \, f^{3} - {\left (2 i \, d^{2} f^{3} x^{2} + 4 i \, d^{2} e f^{2} x + 2 i \, d^{2} e^{2} f - 24 i \, f^{3}\right )} e^{\left (d x + c\right )}}{3 \, {\left (a d^{3} f^{5} x^{5} + 5 \, a d^{3} e f^{4} x^{4} + 10 \, a d^{3} e^{2} f^{3} x^{3} + 10 \, a d^{3} e^{3} f^{2} x^{2} + 5 \, a d^{3} e^{4} f x + a d^{3} e^{5} + {\left (a d^{3} f^{5} x^{5} + 5 \, a d^{3} e f^{4} x^{4} + 10 \, a d^{3} e^{2} f^{3} x^{3} + 10 \, a d^{3} e^{3} f^{2} x^{2} + 5 \, a d^{3} e^{4} f x + a d^{3} e^{5}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}, x\right )}{3 \, a d^{3} f^{4} x^{4} + 12 \, a d^{3} e f^{3} x^{3} + 18 \, a d^{3} e^{2} f^{2} x^{2} + 12 \, a d^{3} e^{3} f x + 3 \, a d^{3} e^{4} - 3 \, {\left (a d^{3} f^{4} x^{4} + 4 \, a d^{3} e f^{3} x^{3} + 6 \, a d^{3} e^{2} f^{2} x^{2} + 4 \, a d^{3} e^{3} f x + a d^{3} e^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (-6 i \, a d^{3} f^{4} x^{4} - 24 i \, a d^{3} e f^{3} x^{3} - 36 i \, a d^{3} e^{2} f^{2} x^{2} - 24 i \, a d^{3} e^{3} f x - 6 i \, a d^{3} e^{4}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (-6 i \, a d^{3} f^{4} x^{4} - 24 i \, a d^{3} e f^{3} x^{3} - 36 i \, a d^{3} e^{2} f^{2} x^{2} - 24 i \, a d^{3} e^{3} f x - 6 i \, a d^{3} e^{4}\right )} e^{\left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(4*d^2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 - 6*f^2*e^(2*d*x + 2*c) - 6*f^2 + (2*I*d*f^2*x + 2*I*d*e*f - 6*I*f^2
)*e^(3*d*x + 3*c) + (8*I*d^2*f^2*x^2 + 8*I*d^2*e^2 + 2*I*d*e*f - 6*I*f^2 + (16*I*d^2*e*f + 2*I*d*f^2)*x)*e^(d*
x + c) - (3*a*d^3*f^4*x^4 + 12*a*d^3*e*f^3*x^3 + 18*a*d^3*e^2*f^2*x^2 + 12*a*d^3*e^3*f*x + 3*a*d^3*e^4 - 3*(a*
d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*e^(4*d*x + 4*c) - (-6*I*a
*d^3*f^4*x^4 - 24*I*a*d^3*e*f^3*x^3 - 36*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^4)*e^(3*d*x +
3*c) - (-6*I*a*d^3*f^4*x^4 - 24*I*a*d^3*e*f^3*x^3 - 36*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^
4)*e^(d*x + c))*integral(-1/3*(8*d^2*f^3*x^2 + 16*d^2*e*f^2*x + 8*d^2*e^2*f - 24*f^3 - (2*I*d^2*f^3*x^2 + 4*I*
d^2*e*f^2*x + 2*I*d^2*e^2*f - 24*I*f^3)*e^(d*x + c))/(a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*x^3
+ 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*e^4*f*x + a*d^3*e^5 + (a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*
x^3 + 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*e^4*f*x + a*d^3*e^5)*e^(2*d*x + 2*c)), x))/(3*a*d^3*f^4*x^4 + 12*a*d^3*e*
f^3*x^3 + 18*a*d^3*e^2*f^2*x^2 + 12*a*d^3*e^3*f*x + 3*a*d^3*e^4 - 3*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d
^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*e^(4*d*x + 4*c) - (-6*I*a*d^3*f^4*x^4 - 24*I*a*d^3*e*f^3*x^3 - 3
6*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^4)*e^(3*d*x + 3*c) - (-6*I*a*d^3*f^4*x^4 - 24*I*a*d^3
*e*f^3*x^3 - 36*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^4)*e^(d*x + c))

________________________________________________________________________________________

giac [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(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 1.98, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (d x +c \right )^{2}}{\left (f x +e \right )^{2} \left (a +i a \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)

[Out]

int(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -4 i \, f \int \frac {1}{4 i \, a d f^{3} x^{3} + 12 i \, a d e f^{2} x^{2} + 12 i \, a d e^{2} f x + 4 i \, a d e^{3} + 4 \, {\left (a d f^{3} x^{3} e^{c} + 3 \, a d e f^{2} x^{2} e^{c} + 3 \, a d e^{2} f x e^{c} + a d e^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac {4 \, {\left (2 \, d^{2} f^{2} x^{2} + 4 \, d^{2} e f x + 2 \, d^{2} e^{2} - 3 \, f^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, f^{2} + {\left (i \, d f^{2} x e^{\left (3 \, c\right )} + {\left (i \, d e f - 3 i \, f^{2}\right )} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (4 i \, d^{2} f^{2} x^{2} e^{c} + {\left (8 i \, d^{2} e f + i \, d f^{2}\right )} x e^{c} + {\left (4 i \, d^{2} e^{2} + i \, d e f - 3 i \, f^{2}\right )} e^{c}\right )} e^{\left (d x\right )}\right )}}{6 \, a d^{3} f^{4} x^{4} + 24 \, a d^{3} e f^{3} x^{3} + 36 \, a d^{3} e^{2} f^{2} x^{2} + 24 \, a d^{3} e^{3} f x + 6 \, a d^{3} e^{4} - 6 \, {\left (a d^{3} f^{4} x^{4} e^{\left (4 \, c\right )} + 4 \, a d^{3} e f^{3} x^{3} e^{\left (4 \, c\right )} + 6 \, a d^{3} e^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 4 \, a d^{3} e^{3} f x e^{\left (4 \, c\right )} + a d^{3} e^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - {\left (-12 i \, a d^{3} f^{4} x^{4} e^{\left (3 \, c\right )} - 48 i \, a d^{3} e f^{3} x^{3} e^{\left (3 \, c\right )} - 72 i \, a d^{3} e^{2} f^{2} x^{2} e^{\left (3 \, c\right )} - 48 i \, a d^{3} e^{3} f x e^{\left (3 \, c\right )} - 12 i \, a d^{3} e^{4} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (-12 i \, a d^{3} f^{4} x^{4} e^{c} - 48 i \, a d^{3} e f^{3} x^{3} e^{c} - 72 i \, a d^{3} e^{2} f^{2} x^{2} e^{c} - 48 i \, a d^{3} e^{3} f x e^{c} - 12 i \, a d^{3} e^{4} e^{c}\right )} e^{\left (d x\right )}} - 4 \, \int \frac {5 \, d^{2} f^{3} x^{2} + 10 \, d^{2} e f^{2} x + 5 \, d^{2} e^{2} f - 24 \, f^{3}}{12 \, a d^{3} f^{5} x^{5} + 60 \, a d^{3} e f^{4} x^{4} + 120 \, a d^{3} e^{2} f^{3} x^{3} + 120 \, a d^{3} e^{3} f^{2} x^{2} + 60 \, a d^{3} e^{4} f x + 12 \, a d^{3} e^{5} + {\left (12 i \, a d^{3} f^{5} x^{5} e^{c} + 60 i \, a d^{3} e f^{4} x^{4} e^{c} + 120 i \, a d^{3} e^{2} f^{3} x^{3} e^{c} + 120 i \, a d^{3} e^{3} f^{2} x^{2} e^{c} + 60 i \, a d^{3} e^{4} f x e^{c} + 12 i \, a d^{3} e^{5} e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-4*I*f*integrate(1/(4*I*a*d*f^3*x^3 + 12*I*a*d*e*f^2*x^2 + 12*I*a*d*e^2*f*x + 4*I*a*d*e^3 + 4*(a*d*f^3*x^3*e^c
+ 3*a*d*e*f^2*x^2*e^c + 3*a*d*e^2*f*x*e^c + a*d*e^3*e^c)*e^(d*x)), x) - 4*(2*d^2*f^2*x^2 + 4*d^2*e*f*x + 2*d^
2*e^2 - 3*f^2*e^(2*d*x + 2*c) - 3*f^2 + (I*d*f^2*x*e^(3*c) + (I*d*e*f - 3*I*f^2)*e^(3*c))*e^(3*d*x) + (4*I*d^2
*f^2*x^2*e^c + (8*I*d^2*e*f + I*d*f^2)*x*e^c + (4*I*d^2*e^2 + I*d*e*f - 3*I*f^2)*e^c)*e^(d*x))/(6*a*d^3*f^4*x^
4 + 24*a*d^3*e*f^3*x^3 + 36*a*d^3*e^2*f^2*x^2 + 24*a*d^3*e^3*f*x + 6*a*d^3*e^4 - 6*(a*d^3*f^4*x^4*e^(4*c) + 4*
a*d^3*e*f^3*x^3*e^(4*c) + 6*a*d^3*e^2*f^2*x^2*e^(4*c) + 4*a*d^3*e^3*f*x*e^(4*c) + a*d^3*e^4*e^(4*c))*e^(4*d*x)
- (-12*I*a*d^3*f^4*x^4*e^(3*c) - 48*I*a*d^3*e*f^3*x^3*e^(3*c) - 72*I*a*d^3*e^2*f^2*x^2*e^(3*c) - 48*I*a*d^3*e
^3*f*x*e^(3*c) - 12*I*a*d^3*e^4*e^(3*c))*e^(3*d*x) - (-12*I*a*d^3*f^4*x^4*e^c - 48*I*a*d^3*e*f^3*x^3*e^c - 72*
I*a*d^3*e^2*f^2*x^2*e^c - 48*I*a*d^3*e^3*f*x*e^c - 12*I*a*d^3*e^4*e^c)*e^(d*x)) - 4*integrate((5*d^2*f^3*x^2 +
10*d^2*e*f^2*x + 5*d^2*e^2*f - 24*f^3)/(12*a*d^3*f^5*x^5 + 60*a*d^3*e*f^4*x^4 + 120*a*d^3*e^2*f^3*x^3 + 120*a
*d^3*e^3*f^2*x^2 + 60*a*d^3*e^4*f*x + 12*a*d^3*e^5 + (12*I*a*d^3*f^5*x^5*e^c + 60*I*a*d^3*e*f^4*x^4*e^c + 120*
I*a*d^3*e^2*f^3*x^3*e^c + 120*I*a*d^3*e^3*f^2*x^2*e^c + 60*I*a*d^3*e^4*f*x*e^c + 12*I*a*d^3*e^5*e^c)*e^(d*x)),
x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^2*(e + f*x)^2*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(1/(cosh(c + d*x)^2*(e + f*x)^2*(a + a*sinh(c + d*x)*1i)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**2/(f*x+e)**2/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*Integral(sech(c + d*x)**2/(e**2*sinh(c + d*x) - I*e**2 + 2*e*f*x*sinh(c + d*x) - 2*I*e*f*x + f**2*x**2*sinh
(c + d*x) - I*f**2*x**2), x)/a

________________________________________________________________________________________