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3.287 \(\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\)

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 = 0.08, 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 = {} \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

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 = 118.30, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

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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fricas [A]  time = 0.73, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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]

-(4*I*d^2*f^3*x^2 + 8*I*d^2*e*f^2*x + 4*I*d^2*e^2*f - 6*I*f^3 + (9*d^3*f^3*x^3 + 9*d^3*e^3 - 9*d^2*e^2*f - 2*d
*e*f^2 + 6*f^3 + 9*(3*d^3*e*f^2 - d^2*f^3)*x^2 + (27*d^3*e^2*f - 18*d^2*e*f^2 - 2*d*f^3)*x)*e^(5*d*x + 5*c) +
(-18*I*d^3*f^3*x^3 - 18*I*d^3*e^3 + 18*I*d^2*e^2*f - 6*I*f^3 + (-54*I*d^3*e*f^2 + 18*I*d^2*f^3)*x^2 + (-54*I*d
^3*e^2*f + 36*I*d^2*e*f^2)*x)*e^(4*d*x + 4*c) + 2*(3*d^3*f^3*x^3 + 3*d^3*e^3 - 4*d^2*e^2*f - 2*d*e*f^2 + 6*f^3
 + (9*d^3*e*f^2 - 4*d^2*f^3)*x^2 + (9*d^3*e^2*f - 8*d^2*e*f^2 - 2*d*f^3)*x)*e^(3*d*x + 3*c) + (18*I*d^3*f^3*x^
3 + 18*I*d^3*e^3 + 22*I*d^2*e^2*f - 12*I*f^3 + (54*I*d^3*e*f^2 + 22*I*d^2*f^3)*x^2 + (54*I*d^3*e^2*f + 44*I*d^
2*e*f^2)*x)*e^(2*d*x + 2*c) + (9*d^3*f^3*x^3 + 9*d^3*e^3 + d^2*e^2*f - 2*d*e*f^2 + 6*f^3 + (27*d^3*e*f^2 + d^2
*f^3)*x^2 + (27*d^3*e^2*f + 2*d^2*e*f^2 - 2*d*f^3)*x)*e^(d*x + c) - (12*a*d^4*f^4*x^4 + 48*a*d^4*e*f^3*x^3 + 7
2*a*d^4*e^2*f^2*x^2 + 48*a*d^4*e^3*f*x + 12*a*d^4*e^4 - 12*(a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^
2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(6*d*x + 6*c) - (-24*I*a*d^4*f^4*x^4 - 96*I*a*d^4*e*f^3*x^3 - 144*I*a*d
^4*e^2*f^2*x^2 - 96*I*a*d^4*e^3*f*x - 24*I*a*d^4*e^4)*e^(5*d*x + 5*c) - 12*(a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3
+ 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(4*d*x + 4*c) - (-48*I*a*d^4*f^4*x^4 - 192*I*a*d^4*e*f^
3*x^3 - 288*I*a*d^4*e^2*f^2*x^2 - 192*I*a*d^4*e^3*f*x - 48*I*a*d^4*e^4)*e^(3*d*x + 3*c) + 12*(a*d^4*f^4*x^4 +
4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(2*d*x + 2*c) - (-24*I*a*d^4*f^4*x^4
- 96*I*a*d^4*e*f^3*x^3 - 144*I*a*d^4*e^2*f^2*x^2 - 96*I*a*d^4*e^3*f*x - 24*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*e*f^3*x - 8*I*d^2*e^2*f^2 + 24*I*f^4 + (9*d^4*f^4*x^4 + 36*d^4*e*f^3*x^3 + 9
*d^4*e^4 - 20*d^2*e^2*f^2 + 24*f^4 + 2*(27*d^4*e^2*f^2 - 10*d^2*f^4)*x^2 + 4*(9*d^4*e^3*f - 10*d^2*e*f^3)*x)*e
^(d*x + c))/(a*d^4*f^5*x^5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*f*x
 + a*d^4*e^5 + (a*d^4*f^5*x^5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*
f*x + a*d^4*e^5)*e^(2*d*x + 2*c)), x))/(12*a*d^4*f^4*x^4 + 48*a*d^4*e*f^3*x^3 + 72*a*d^4*e^2*f^2*x^2 + 48*a*d^
4*e^3*f*x + 12*a*d^4*e^4 - 12*(a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d
^4*e^4)*e^(6*d*x + 6*c) - (-24*I*a*d^4*f^4*x^4 - 96*I*a*d^4*e*f^3*x^3 - 144*I*a*d^4*e^2*f^2*x^2 - 96*I*a*d^4*e
^3*f*x - 24*I*a*d^4*e^4)*e^(5*d*x + 5*c) - 12*(a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d
^4*e^3*f*x + a*d^4*e^4)*e^(4*d*x + 4*c) - (-48*I*a*d^4*f^4*x^4 - 192*I*a*d^4*e*f^3*x^3 - 288*I*a*d^4*e^2*f^2*x
^2 - 192*I*a*d^4*e^3*f*x - 48*I*a*d^4*e^4)*e^(3*d*x + 3*c) + 12*(a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e
^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(2*d*x + 2*c) - (-24*I*a*d^4*f^4*x^4 - 96*I*a*d^4*e*f^3*x^3 - 144*
I*a*d^4*e^2*f^2*x^2 - 96*I*a*d^4*e^3*f*x - 24*I*a*d^4*e^4)*e^(d*x + c))

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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)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 2.02, 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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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]

-8*(4*I*d^2*f^3*x^2 + 8*I*d^2*e*f^2*x + 4*I*d^2*e^2*f - 6*I*f^3 + (9*d^3*f^3*x^3*e^(5*c) + 9*(3*d^3*e*f^2 - d^
2*f^3)*x^2*e^(5*c) + (27*d^3*e^2*f - 18*d^2*e*f^2 - 2*d*f^3)*x*e^(5*c) + (9*d^3*e^3 - 9*d^2*e^2*f - 2*d*e*f^2
+ 6*f^3)*e^(5*c))*e^(5*d*x) + (-18*I*d^3*f^3*x^3*e^(4*c) + (-54*I*d^3*e*f^2 + 18*I*d^2*f^3)*x^2*e^(4*c) + (-54
*I*d^3*e^2*f + 36*I*d^2*e*f^2)*x*e^(4*c) + (-18*I*d^3*e^3 + 18*I*d^2*e^2*f - 6*I*f^3)*e^(4*c))*e^(4*d*x) + 2*(
3*d^3*f^3*x^3*e^(3*c) + (9*d^3*e*f^2 - 4*d^2*f^3)*x^2*e^(3*c) + (9*d^3*e^2*f - 8*d^2*e*f^2 - 2*d*f^3)*x*e^(3*c
) + (3*d^3*e^3 - 4*d^2*e^2*f - 2*d*e*f^2 + 6*f^3)*e^(3*c))*e^(3*d*x) + (18*I*d^3*f^3*x^3*e^(2*c) + (54*I*d^3*e
*f^2 + 22*I*d^2*f^3)*x^2*e^(2*c) + (54*I*d^3*e^2*f + 44*I*d^2*e*f^2)*x*e^(2*c) + (18*I*d^3*e^3 + 22*I*d^2*e^2*
f - 12*I*f^3)*e^(2*c))*e^(2*d*x) + (9*d^3*f^3*x^3*e^c + (27*d^3*e*f^2 + d^2*f^3)*x^2*e^c + (27*d^3*e^2*f + 2*d
^2*e*f^2 - 2*d*f^3)*x*e^c + (9*d^3*e^3 + d^2*e^2*f - 2*d*e*f^2 + 6*f^3)*e^c)*e^(d*x))/(96*a*d^4*f^4*x^4 + 384*
a*d^4*e*f^3*x^3 + 576*a*d^4*e^2*f^2*x^2 + 384*a*d^4*e^3*f*x + 96*a*d^4*e^4 - 96*(a*d^4*f^4*x^4*e^(6*c) + 4*a*d
^4*e*f^3*x^3*e^(6*c) + 6*a*d^4*e^2*f^2*x^2*e^(6*c) + 4*a*d^4*e^3*f*x*e^(6*c) + a*d^4*e^4*e^(6*c))*e^(6*d*x) -
(-192*I*a*d^4*f^4*x^4*e^(5*c) - 768*I*a*d^4*e*f^3*x^3*e^(5*c) - 1152*I*a*d^4*e^2*f^2*x^2*e^(5*c) - 768*I*a*d^4
*e^3*f*x*e^(5*c) - 192*I*a*d^4*e^4*e^(5*c))*e^(5*d*x) - 96*(a*d^4*f^4*x^4*e^(4*c) + 4*a*d^4*e*f^3*x^3*e^(4*c)
+ 6*a*d^4*e^2*f^2*x^2*e^(4*c) + 4*a*d^4*e^3*f*x*e^(4*c) + a*d^4*e^4*e^(4*c))*e^(4*d*x) - (-384*I*a*d^4*f^4*x^4
*e^(3*c) - 1536*I*a*d^4*e*f^3*x^3*e^(3*c) - 2304*I*a*d^4*e^2*f^2*x^2*e^(3*c) - 1536*I*a*d^4*e^3*f*x*e^(3*c) -
384*I*a*d^4*e^4*e^(3*c))*e^(3*d*x) + 96*(a*d^4*f^4*x^4*e^(2*c) + 4*a*d^4*e*f^3*x^3*e^(2*c) + 6*a*d^4*e^2*f^2*x
^2*e^(2*c) + 4*a*d^4*e^3*f*x*e^(2*c) + a*d^4*e^4*e^(2*c))*e^(2*d*x) - (-192*I*a*d^4*f^4*x^4*e^c - 768*I*a*d^4*
e*f^3*x^3*e^c - 1152*I*a*d^4*e^2*f^2*x^2*e^c - 768*I*a*d^4*e^3*f*x*e^c - 192*I*a*d^4*e^4*e^c)*e^(d*x)) + 8*int
egrate((9*d^4*f^4*x^4 + 36*d^4*e*f^3*x^3 + 9*d^4*e^4 - 28*d^2*e^2*f^2 + 48*f^4 + 2*(27*d^4*e^2*f^2 - 14*d^2*f^
4)*x^2 + 4*(9*d^4*e^3*f - 14*d^2*e*f^3)*x)/(-192*I*a*d^4*f^5*x^5 - 960*I*a*d^4*e*f^4*x^4 - 1920*I*a*d^4*e^2*f^
3*x^3 - 1920*I*a*d^4*e^3*f^2*x^2 - 960*I*a*d^4*e^4*f*x - 192*I*a*d^4*e^5 + 192*(a*d^4*f^5*x^5*e^c + 5*a*d^4*e*
f^4*x^4*e^c + 10*a*d^4*e^2*f^3*x^3*e^c + 10*a*d^4*e^3*f^2*x^2*e^c + 5*a*d^4*e^4*f*x*e^c + a*d^4*e^5*e^c)*e^(d*
x)), x) + 8*integrate((3*d^2*f^2*x^2 + 6*d^2*e*f*x + 3*d^2*e^2 - 4*f^2)/(64*I*a*d^2*f^3*x^3 + 192*I*a*d^2*e*f^
2*x^2 + 192*I*a*d^2*e^2*f*x + 64*I*a*d^2*e^3 + 64*(a*d^2*f^3*x^3*e^c + 3*a*d^2*e*f^2*x^2*e^c + 3*a*d^2*e^2*f*x
*e^c + a*d^2*e^3*e^c)*e^(d*x)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

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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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]

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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