3.9.82 \(\int e^{c+d x} \text {csch}^2(a+b x) \, dx\) [882]

Optimal. Leaf size=54 \[ \frac {4 e^{c+d x+2 (a+b x)} \, _2F_1\left (2,1+\frac {d}{2 b};2+\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b+d} \]

[Out]

4*exp(2*b*x+d*x+2*a+c)*hypergeom([2, 1+1/2*d/b],[2+1/2*d/b],exp(2*b*x+2*a))/(2*b+d)

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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5601} \begin {gather*} \frac {4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Csch[a + b*x]^2,x]

[Out]

(4*E^(c + d*x + 2*(a + b*x))*Hypergeometric2F1[2, 1 + d/(2*b), 2 + d/(2*b), E^(2*(a + b*x))])/(2*b + d)

Rule 5601

Int[Csch[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-2)^n*E^(n*(d + e*x))
*(F^(c*(a + b*x))/(e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 + b*c*(Log[F]/(2*e)), 1 + n/2 + b*c*(Log[F]/(2*
e)), E^(2*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int e^{c+d x} \text {csch}^2(a+b x) \, dx &=\frac {4 e^{c+d x+2 (a+b x)} \, _2F_1\left (2,1+\frac {d}{2 b};2+\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b+d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(54)=108\).
time = 2.25, size = 131, normalized size = 2.43 \begin {gather*} \frac {e^c \left (-\frac {2 d e^{2 a} \left (\frac {e^{d x} \, _2F_1\left (1,\frac {d}{2 b};1+\frac {d}{2 b};e^{2 (a+b x)}\right )}{d}-\frac {e^{(2 b+d) x} \, _2F_1\left (1,1+\frac {d}{2 b};2+\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b+d}\right )}{-1+e^{2 a}}+e^{d x} \text {csch}(a) \text {csch}(a+b x) \sinh (b x)\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x)*Csch[a + b*x]^2,x]

[Out]

(E^c*((-2*d*E^(2*a)*((E^(d*x)*Hypergeometric2F1[1, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d - (E^((2*b + d)*x
)*Hypergeometric2F1[1, 1 + d/(2*b), 2 + d/(2*b), E^(2*(a + b*x))])/(2*b + d)))/(-1 + E^(2*a)) + E^(d*x)*Csch[a
]*Csch[a + b*x]*Sinh[b*x]))/b

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Maple [F]
time = 2.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d x +c} \mathrm {csch}\left (b x +a \right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x+c)*csch(b*x+a)^2,x)

[Out]

int(exp(d*x+c)*csch(b*x+a)^2,x)

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Maxima [F]
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(exp(d*x+c)*csch(b*x+a)^2,x, algorithm="maxima")

[Out]

16*b*d*integrate(-e^(d*x + c)/(8*b^2 - 6*b*d + d^2 - (8*b^2 - 6*b*d + d^2)*e^(6*b*x + 6*a) + 3*(8*b^2 - 6*b*d
+ d^2)*e^(4*b*x + 4*a) - 3*(8*b^2 - 6*b*d + d^2)*e^(2*b*x + 2*a)), x) - 4*((4*b*e^c - d*e^c)*e^(2*b*x + 2*a) -
4*b*e^c)*e^(d*x)/(8*b^2 - 6*b*d + d^2 + (8*b^2 - 6*b*d + d^2)*e^(4*b*x + 4*a) - 2*(8*b^2 - 6*b*d + d^2)*e^(2*
b*x + 2*a))

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Fricas [F]
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(exp(d*x+c)*csch(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(csch(b*x + a)^2*e^(d*x + c), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int e^{d x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x+c)*csch(b*x+a)**2,x)

[Out]

exp(c)*Integral(exp(d*x)*csch(a + b*x)**2, x)

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Giac [F]
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(exp(d*x+c)*csch(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(csch(b*x + a)^2*e^(d*x + c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c + d*x)/sinh(a + b*x)^2,x)

[Out]

int(exp(c + d*x)/sinh(a + b*x)^2, x)

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