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

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

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

Verification is Not applicable to the result.

[In]

Int[Sech[c + d*x]^2/((e + f*x)*(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 = 130.79, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sech[c + d*x]^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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fricas [A]  time = 0.51, 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} - 2 \, f^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f^{2} + {\left (i \, d f^{2} x + i \, d e f - 2 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} + i \, d e f - 2 i \, f^{2} + {\left (16 i \, d^{2} e f + i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )} - {\left (3 \, a d^{3} f^{3} x^{3} + 9 \, a d^{3} e f^{2} x^{2} + 9 \, a d^{3} e^{2} f x + 3 \, a d^{3} e^{3} - 3 \, {\left (a d^{3} f^{3} x^{3} + 3 \, a d^{3} e f^{2} x^{2} + 3 \, a d^{3} e^{2} f x + a d^{3} e^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (-6 i \, a d^{3} f^{3} x^{3} - 18 i \, a d^{3} e f^{2} x^{2} - 18 i \, a d^{3} e^{2} f x - 6 i \, a d^{3} e^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (-6 i \, a d^{3} f^{3} x^{3} - 18 i \, a d^{3} e f^{2} x^{2} - 18 i \, a d^{3} e^{2} f x - 6 i \, a d^{3} e^{3}\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (-\frac {4 \, d^{2} f^{3} x^{2} + 8 \, d^{2} e f^{2} x + 4 \, d^{2} e^{2} f - 6 \, f^{3} - {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} e f^{2} x + i \, d^{2} e^{2} f - 6 i \, f^{3}\right )} e^{\left (d x + c\right )}}{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} + {\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 (2 \, d x + 2 \, c\right )}\right )}}, x\right )}{3 \, a d^{3} f^{3} x^{3} + 9 \, a d^{3} e f^{2} x^{2} + 9 \, a d^{3} e^{2} f x + 3 \, a d^{3} e^{3} - 3 \, {\left (a d^{3} f^{3} x^{3} + 3 \, a d^{3} e f^{2} x^{2} + 3 \, a d^{3} e^{2} f x + a d^{3} e^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (-6 i \, a d^{3} f^{3} x^{3} - 18 i \, a d^{3} e f^{2} x^{2} - 18 i \, a d^{3} e^{2} f x - 6 i \, a d^{3} e^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (-6 i \, a d^{3} f^{3} x^{3} - 18 i \, a d^{3} e f^{2} x^{2} - 18 i \, a d^{3} e^{2} f x - 6 i \, a d^{3} e^{3}\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)/(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 - 2*f^2*e^(2*d*x + 2*c) - 2*f^2 + (I*d*f^2*x + I*d*e*f - 2*I*f^2)*e^
(3*d*x + 3*c) + (8*I*d^2*f^2*x^2 + 8*I*d^2*e^2 + I*d*e*f - 2*I*f^2 + (16*I*d^2*e*f + I*d*f^2)*x)*e^(d*x + c) -
(3*a*d^3*f^3*x^3 + 9*a*d^3*e*f^2*x^2 + 9*a*d^3*e^2*f*x + 3*a*d^3*e^3 - 3*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 +
3*a*d^3*e^2*f*x + a*d^3*e^3)*e^(4*d*x + 4*c) - (-6*I*a*d^3*f^3*x^3 - 18*I*a*d^3*e*f^2*x^2 - 18*I*a*d^3*e^2*f*
x - 6*I*a*d^3*e^3)*e^(3*d*x + 3*c) - (-6*I*a*d^3*f^3*x^3 - 18*I*a*d^3*e*f^2*x^2 - 18*I*a*d^3*e^2*f*x - 6*I*a*d
^3*e^3)*e^(d*x + c))*integral(-1/3*(4*d^2*f^3*x^2 + 8*d^2*e*f^2*x + 4*d^2*e^2*f - 6*f^3 - (I*d^2*f^3*x^2 + 2*I
*d^2*e*f^2*x + I*d^2*e^2*f - 6*I*f^3)*e^(d*x + c))/(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 + (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^(2*d*x + 2*c)), x))/(3*a*d^3*f^3*x^3 + 9*a*d^3*e*f^2*x^2 + 9*a*d^3*e^2*f*x + 3*a*d^3*e^3 - 3*(a*d^3*
f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*e^(4*d*x + 4*c) - (-6*I*a*d^3*f^3*x^3 - 18*I*a*d^3*
e*f^2*x^2 - 18*I*a*d^3*e^2*f*x - 6*I*a*d^3*e^3)*e^(3*d*x + 3*c) - (-6*I*a*d^3*f^3*x^3 - 18*I*a*d^3*e*f^2*x^2 -
18*I*a*d^3*e^2*f*x - 6*I*a*d^3*e^3)*e^(d*x + c))

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

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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maple [A]  time = 0.39, 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 \[ -4 i \, f \int \frac {1}{8 i \, a d f^{2} x^{2} + 16 i \, a d e f x + 8 i \, a d e^{2} + 8 \, {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac {4 \, {\left (4 \, d^{2} f^{2} x^{2} + 8 \, d^{2} e f x + 4 \, d^{2} e^{2} - 2 \, f^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f^{2} + {\left (i \, d f^{2} x e^{\left (3 \, c\right )} + {\left (i \, d e f - 2 i \, f^{2}\right )} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (8 i \, d^{2} f^{2} x^{2} e^{c} + {\left (16 i \, d^{2} e f + i \, d f^{2}\right )} x e^{c} + {\left (8 i \, d^{2} e^{2} + i \, d e f - 2 i \, f^{2}\right )} e^{c}\right )} e^{\left (d x\right )}\right )}}{12 \, a d^{3} f^{3} x^{3} + 36 \, a d^{3} e f^{2} x^{2} + 36 \, a d^{3} e^{2} f x + 12 \, a d^{3} e^{3} - 12 \, {\left (a d^{3} f^{3} x^{3} e^{\left (4 \, c\right )} + 3 \, a d^{3} e f^{2} x^{2} e^{\left (4 \, c\right )} + 3 \, a d^{3} e^{2} f x e^{\left (4 \, c\right )} + a d^{3} e^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - {\left (-24 i \, a d^{3} f^{3} x^{3} e^{\left (3 \, c\right )} - 72 i \, a d^{3} e f^{2} x^{2} e^{\left (3 \, c\right )} - 72 i \, a d^{3} e^{2} f x e^{\left (3 \, c\right )} - 24 i \, a d^{3} e^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (-24 i \, a d^{3} f^{3} x^{3} e^{c} - 72 i \, a d^{3} e f^{2} x^{2} e^{c} - 72 i \, a d^{3} e^{2} f x e^{c} - 24 i \, a d^{3} e^{3} e^{c}\right )} e^{\left (d x\right )}} - 4 \, \int \frac {5 \, d^{2} f^{3} x^{2} + 10 \, d^{2} e f^{2} x + 5 \, d^{2} e^{2} f - 12 \, f^{3}}{24 \, a d^{3} f^{4} x^{4} + 96 \, a d^{3} e f^{3} x^{3} + 144 \, a d^{3} e^{2} f^{2} x^{2} + 96 \, a d^{3} e^{3} f x + 24 \, a d^{3} e^{4} + {\left (24 i \, a d^{3} f^{4} x^{4} e^{c} + 96 i \, a d^{3} e f^{3} x^{3} e^{c} + 144 i \, a d^{3} e^{2} f^{2} x^{2} e^{c} + 96 i \, a d^{3} e^{3} f x e^{c} + 24 i \, a d^{3} e^{4} e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2/(f*x+e)/(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*e*f*x + 8*I*a*d*e^2 + 8*(a*d*f^2*x^2*e^c + 2*a*d*e*f*x*e^c + a*
d*e^2*e^c)*e^(d*x)), x) - 4*(4*d^2*f^2*x^2 + 8*d^2*e*f*x + 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) + (I*d*e*f - 2*I*f^2)*e^(3*c))*e^(3*d*x) + (8*I*d^2*f^2*x^2*e^c + (16*I*d^2*e*f + I*d*f^2)*x*e^c +
(8*I*d^2*e^2 + I*d*e*f - 2*I*f^2)*e^c)*e^(d*x))/(12*a*d^3*f^3*x^3 + 36*a*d^3*e*f^2*x^2 + 36*a*d^3*e^2*f*x + 1
2*a*d^3*e^3 - 12*(a*d^3*f^3*x^3*e^(4*c) + 3*a*d^3*e*f^2*x^2*e^(4*c) + 3*a*d^3*e^2*f*x*e^(4*c) + a*d^3*e^3*e^(4
*c))*e^(4*d*x) - (-24*I*a*d^3*f^3*x^3*e^(3*c) - 72*I*a*d^3*e*f^2*x^2*e^(3*c) - 72*I*a*d^3*e^2*f*x*e^(3*c) - 24
*I*a*d^3*e^3*e^(3*c))*e^(3*d*x) - (-24*I*a*d^3*f^3*x^3*e^c - 72*I*a*d^3*e*f^2*x^2*e^c - 72*I*a*d^3*e^2*f*x*e^c
- 24*I*a*d^3*e^3*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 - 12*f^3)/(24*a*d^
3*f^4*x^4 + 96*a*d^3*e*f^3*x^3 + 144*a*d^3*e^2*f^2*x^2 + 96*a*d^3*e^3*f*x + 24*a*d^3*e^4 + (24*I*a*d^3*f^4*x^4
*e^c + 96*I*a*d^3*e*f^3*x^3*e^c + 144*I*a*d^3*e^2*f^2*x^2*e^c + 96*I*a*d^3*e^3*f*x*e^c + 24*I*a*d^3*e^4*e^c)*e
^(d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^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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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {\operatorname {sech}^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]

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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