3.3.87 \(\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [287]
Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]
[Out]
Unintegrable(sech(d*x+c)^3/(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}^3(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]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
[Out]
Defer[Int][Sech[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\\ \end {align*}
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Mathematica [A]
time = 94.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {sech}^3(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]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
[Out]
Integrate[Sech[c + d*x]^3/((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 \[\int \frac {\mathrm {sech}\left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
[Out]
int(sech(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
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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)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
[Out]
-1/12*(4*I*d^2*f^3*x^2 + 8*I*d^2*f^2*x*e + 4*I*d^2*f*e^2 - 6*I*f^3 + (9*d^3*f^3*x^3 - 9*d^2*f^3*x^2 - 2*d*f^3*
x + 9*d^3*e^3 + 6*f^3 + 9*(3*d^3*f*x - d^2*f)*e^2 + (27*d^3*f^2*x^2 - 18*d^2*f^2*x - 2*d*f^2)*e)*e^(5*d*x + 5*
c) - 6*(3*I*d^3*f^3*x^3 - 3*I*d^2*f^3*x^2 + 3*I*d^3*e^3 + I*f^3 + 3*(3*I*d^3*f*x - I*d^2*f)*e^2 + 3*(3*I*d^3*f
^2*x^2 - 2*I*d^2*f^2*x)*e)*e^(4*d*x + 4*c) + 2*(3*d^3*f^3*x^3 - 4*d^2*f^3*x^2 - 2*d*f^3*x + 3*d^3*e^3 + 6*f^3
+ (9*d^3*f*x - 4*d^2*f)*e^2 + (9*d^3*f^2*x^2 - 8*d^2*f^2*x - 2*d*f^2)*e)*e^(3*d*x + 3*c) - 2*(-9*I*d^3*f^3*x^3
- 11*I*d^2*f^3*x^2 - 9*I*d^3*e^3 + 6*I*f^3 + (-27*I*d^3*f*x - 11*I*d^2*f)*e^2 + (-27*I*d^3*f^2*x^2 - 22*I*d^2
*f^2*x)*e)*e^(2*d*x + 2*c) + (9*d^3*f^3*x^3 + d^2*f^3*x^2 - 2*d*f^3*x + 9*d^3*e^3 + 6*f^3 + (27*d^3*f*x + d^2*
f)*e^2 + (27*d^3*f^2*x^2 + 2*d^2*f^2*x - 2*d*f^2)*e)*e^(d*x + c) - 12*(a*d^4*f^4*x^4 + 4*a*d^4*f^3*x^3*e + 6*a
*d^4*f^2*x^2*e^2 + 4*a*d^4*f*x*e^3 + a*d^4*e^4 - (a*d^4*f^4*x^4 + 4*a*d^4*f^3*x^3*e + 6*a*d^4*f^2*x^2*e^2 + 4*
a*d^4*f*x*e^3 + a*d^4*e^4)*e^(6*d*x + 6*c) + 2*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*f^3*x^3*e + 6*I*a*d^4*f^2*x^2*e^2
+ 4*I*a*d^4*f*x*e^3 + I*a*d^4*e^4)*e^(5*d*x + 5*c) - (a*d^4*f^4*x^4 + 4*a*d^4*f^3*x^3*e + 6*a*d^4*f^2*x^2*e^2
+ 4*a*d^4*f*x*e^3 + a*d^4*e^4)*e^(4*d*x + 4*c) + 4*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*f^3*x^3*e + 6*I*a*d^4*f^2*x^2*
e^2 + 4*I*a*d^4*f*x*e^3 + I*a*d^4*e^4)*e^(3*d*x + 3*c) + (a*d^4*f^4*x^4 + 4*a*d^4*f^3*x^3*e + 6*a*d^4*f^2*x^2*
e^2 + 4*a*d^4*f*x*e^3 + a*d^4*e^4)*e^(2*d*x + 2*c) + 2*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*f^3*x^3*e + 6*I*a*d^4*f^2*
x^2*e^2 + 4*I*a*d^4*f*x*e^3 + I*a*d^4*e^4)*e^(d*x + c))*integral(1/12*(-8*I*d^2*f^4*x^2 - 16*I*d^2*f^3*x*e - 8
*I*d^2*f^2*e^2 + 24*I*f^4 + (9*d^4*f^4*x^4 - 20*d^2*f^4*x^2 + 36*d^4*f*x*e^3 + 9*d^4*e^4 + 24*f^4 + 2*(27*d^4*
f^2*x^2 - 10*d^2*f^2)*e^2 + 4*(9*d^4*f^3*x^3 - 10*d^2*f^3*x)*e)*e^(d*x + c))/(a*d^4*f^5*x^5 + 5*a*d^4*f^4*x^4*
e + 10*a*d^4*f^3*x^3*e^2 + 10*a*d^4*f^2*x^2*e^3 + 5*a*d^4*f*x*e^4 + a*d^4*e^5 + (a*d^4*f^5*x^5 + 5*a*d^4*f^4*x
^4*e + 10*a*d^4*f^3*x^3*e^2 + 10*a*d^4*f^2*x^2*e^3 + 5*a*d^4*f*x*e^4 + a*d^4*e^5)*e^(2*d*x + 2*c)), x))/(a*d^4
*f^4*x^4 + 4*a*d^4*f^3*x^3*e + 6*a*d^4*f^2*x^2*e^2 + 4*a*d^4*f*x*e^3 + a*d^4*e^4 - (a*d^4*f^4*x^4 + 4*a*d^4*f^
3*x^3*e + 6*a*d^4*f^2*x^2*e^2 + 4*a*d^4*f*x*e^3 + a*d^4*e^4)*e^(6*d*x + 6*c) + 2*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*
f^3*x^3*e + 6*I*a*d^4*f^2*x^2*e^2 + 4*I*a*d^4*f*x*e^3 + I*a*d^4*e^4)*e^(5*d*x + 5*c) - (a*d^4*f^4*x^4 + 4*a*d^
4*f^3*x^3*e + 6*a*d^4*f^2*x^2*e^2 + 4*a*d^4*f*x*e^3 + a*d^4*e^4)*e^(4*d*x + 4*c) + 4*(I*a*d^4*f^4*x^4 + 4*I*a*
d^4*f^3*x^3*e + 6*I*a*d^4*f^2*x^2*e^2 + 4*I*a*d^4*f*x*e^3 + I*a*d^4*e^4)*e^(3*d*x + 3*c) + (a*d^4*f^4*x^4 + 4*
a*d^4*f^3*x^3*e + 6*a*d^4*f^2*x^2*e^2 + 4*a*d^4*f*x*e^3 + a*d^4*e^4)*e^(2*d*x + 2*c) + 2*(I*a*d^4*f^4*x^4 + 4*
I*a*d^4*f^3*x^3*e + 6*I*a*d^4*f^2*x^2*e^2 + 4*I*a*d^4*f*x*e^3 + I*a*d^4*e^4)*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}^{3}{\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)**3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
[Out]
-I*Integral(sech(c + d*x)**3/(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)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
[Out]
integrate(sech(d*x + c)^3/((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 )}^3\,\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)^3*(e + f*x)*(a + a*sinh(c + d*x)*1i)),x)
[Out]
int(1/(cosh(c + d*x)^3*(e + f*x)*(a + a*sinh(c + d*x)*1i)), x)
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