3.3.81 \(\int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [281]
Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]
[Out]
Unintegrable(sech(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
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Rubi [A]
time = 0.05, 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 = {}
\begin {gather*} \int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Sech[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
[Out]
Defer[Int][Sech[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=\int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\\ \end {align*}
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Mathematica [A]
time = 102.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[Sech[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
[Out]
Integrate[Sech[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {sech}\left (d x +c \right )^{2}}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
[Out]
int(sech(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-4*I*f*integrate(1/(8*I*a*d*f^2*x^2 + 16*I*a*d*f*x*e + 8*I*a*d*e^2 + 8*(a*d*f^2*x^2*e^c + 2*a*d*f*x*e^(c + 1)
+ a*d*e^(c + 2))*e^(d*x)), x) - 1/3*(4*d^2*f^2*x^2 + 8*d^2*f*x*e + 4*d^2*e^2 - 2*f^2*e^(2*d*x + 2*c) - 2*f^2 +
(I*d*f^2*x*e^(3*c) - 2*I*f^2*e^(3*c) + I*d*f*e^(3*c + 1))*e^(3*d*x) + (8*I*d^2*f^2*x^2*e^c + 8*I*d^2*e^(c + 2
) + I*d*f*e^(c + 1) - 2*I*f^2*e^c + (16*I*d^2*f*e^(c + 1) + I*d*f^2*e^c)*x)*e^(d*x))/(a*d^3*f^3*x^3 + 3*a*d^3*
f^2*x^2*e + 3*a*d^3*f*x*e^2 + a*d^3*e^3 - (a*d^3*f^3*x^3*e^(4*c) + 3*a*d^3*f^2*x^2*e^(4*c + 1) + 3*a*d^3*f*x*e
^(4*c + 2) + a*d^3*e^(4*c + 3))*e^(4*d*x) + 2*(I*a*d^3*f^3*x^3*e^(3*c) + 3*I*a*d^3*f^2*x^2*e^(3*c + 1) + 3*I*a
*d^3*f*x*e^(3*c + 2) + I*a*d^3*e^(3*c + 3))*e^(3*d*x) + 2*(I*a*d^3*f^3*x^3*e^c + 3*I*a*d^3*f^2*x^2*e^(c + 1) +
3*I*a*d^3*f*x*e^(c + 2) + I*a*d^3*e^(c + 3))*e^(d*x)) - 4*integrate(1/24*(5*d^2*f^3*x^2 + 10*d^2*f^2*x*e + 5*
d^2*f*e^2 - 12*f^3)/(a*d^3*f^4*x^4 + 4*a*d^3*f^3*x^3*e + 6*a*d^3*f^2*x^2*e^2 + 4*a*d^3*f*x*e^3 + a*d^3*e^4 - (
-I*a*d^3*f^4*x^4*e^c - 4*I*a*d^3*f^3*x^3*e^(c + 1) - 6*I*a*d^3*f^2*x^2*e^(c + 2) - 4*I*a*d^3*f*x*e^(c + 3) - I
*a*d^3*e^(c + 4))*e^(d*x)), x)
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
[Out]
-1/3*(4*d^2*f^2*x^2 + 8*d^2*f*x*e + 4*d^2*e^2 - 2*f^2*e^(2*d*x + 2*c) - 2*f^2 + (I*d*f^2*x + I*d*f*e - 2*I*f^2
)*e^(3*d*x + 3*c) + (8*I*d^2*f^2*x^2 + I*d*f^2*x + 8*I*d^2*e^2 - 2*I*f^2 + (16*I*d^2*f*x + I*d*f)*e)*e^(d*x +
c) - 3*(a*d^3*f^3*x^3 + 3*a*d^3*f^2*x^2*e + 3*a*d^3*f*x*e^2 + a*d^3*e^3 - (a*d^3*f^3*x^3 + 3*a*d^3*f^2*x^2*e +
3*a*d^3*f*x*e^2 + a*d^3*e^3)*e^(4*d*x + 4*c) + 2*(I*a*d^3*f^3*x^3 + 3*I*a*d^3*f^2*x^2*e + 3*I*a*d^3*f*x*e^2 +
I*a*d^3*e^3)*e^(3*d*x + 3*c) + 2*(I*a*d^3*f^3*x^3 + 3*I*a*d^3*f^2*x^2*e + 3*I*a*d^3*f*x*e^2 + I*a*d^3*e^3)*e^
(d*x + c))*integral(-1/3*(4*d^2*f^3*x^2 + 8*d^2*f^2*x*e + 4*d^2*f*e^2 - 6*f^3 - (I*d^2*f^3*x^2 + 2*I*d^2*f^2*x
*e + I*d^2*f*e^2 - 6*I*f^3)*e^(d*x + c))/(a*d^3*f^4*x^4 + 4*a*d^3*f^3*x^3*e + 6*a*d^3*f^2*x^2*e^2 + 4*a*d^3*f*
x*e^3 + a*d^3*e^4 + (a*d^3*f^4*x^4 + 4*a*d^3*f^3*x^3*e + 6*a*d^3*f^2*x^2*e^2 + 4*a*d^3*f*x*e^3 + a*d^3*e^4)*e^
(2*d*x + 2*c)), x))/(a*d^3*f^3*x^3 + 3*a*d^3*f^2*x^2*e + 3*a*d^3*f*x*e^2 + a*d^3*e^3 - (a*d^3*f^3*x^3 + 3*a*d^
3*f^2*x^2*e + 3*a*d^3*f*x*e^2 + a*d^3*e^3)*e^(4*d*x + 4*c) + 2*(I*a*d^3*f^3*x^3 + 3*I*a*d^3*f^2*x^2*e + 3*I*a*
d^3*f*x*e^2 + I*a*d^3*e^3)*e^(3*d*x + 3*c) + 2*(I*a*d^3*f^3*x^3 + 3*I*a*d^3*f^2*x^2*e + 3*I*a*d^3*f*x*e^2 + I*
a*d^3*e^3)*e^(d*x + c))
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \int \frac {\operatorname {sech}^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)**2/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
[Out]
-I*Integral(sech(c + d*x)**2/(e*sinh(c + d*x) - I*e + f*x*sinh(c + d*x) - I*f*x), x)/a
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
[Out]
integrate(sech(d*x + c)^2/((f*x + e)*(I*a*sinh(d*x + c) + a)), x)
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cosh(c + d*x)^2*(e + f*x)*(a + a*sinh(c + d*x)*1i)),x)
[Out]
int(1/(cosh(c + d*x)^2*(e + f*x)*(a + a*sinh(c + d*x)*1i)), x)
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