3.41 \(\int \text {csch}^4(a+b x) \text {sech}^3(a+b x) \, dx\)

Optimal. Leaf size=66 \[ -\frac {5 \text {csch}^3(a+b x)}{6 b}+\frac {5 \text {csch}(a+b x)}{2 b}+\frac {5 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{2 b} \]

[Out]

5/2*arctan(sinh(b*x+a))/b+5/2*csch(b*x+a)/b-5/6*csch(b*x+a)^3/b+1/2*csch(b*x+a)^3*sech(b*x+a)^2/b

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Rubi [A]  time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2621, 288, 302, 207} \[ -\frac {5 \text {csch}^3(a+b x)}{6 b}+\frac {5 \text {csch}(a+b x)}{2 b}+\frac {5 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^4*Sech[a + b*x]^3,x]

[Out]

(5*ArcTan[Sinh[a + b*x]])/(2*b) + (5*Csch[a + b*x])/(2*b) - (5*Csch[a + b*x]^3)/(6*b) + (Csch[a + b*x]^3*Sech[
a + b*x]^2)/(2*b)

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \text {csch}^4(a+b x) \text {sech}^3(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,-i \text {csch}(a+b x)\right )}{b}\\ &=\frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{2 b}+\frac {(5 i) \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{2 b}\\ &=\frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{2 b}+\frac {(5 i) \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,-i \text {csch}(a+b x)\right )}{2 b}\\ &=\frac {5 \text {csch}(a+b x)}{2 b}-\frac {5 \text {csch}^3(a+b x)}{6 b}+\frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{2 b}+\frac {(5 i) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{2 b}\\ &=\frac {5 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac {5 \text {csch}(a+b x)}{2 b}-\frac {5 \text {csch}^3(a+b x)}{6 b}+\frac {\text {csch}^3(a+b x) \text {sech}^2(a+b x)}{2 b}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 33, normalized size = 0.50 \[ -\frac {\text {csch}^3(a+b x) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\sinh ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^4*Sech[a + b*x]^3,x]

[Out]

-1/3*(Csch[a + b*x]^3*Hypergeometric2F1[-3/2, 2, -1/2, -Sinh[a + b*x]^2])/b

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fricas [B]  time = 0.44, size = 1176, normalized size = 17.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^3,x, algorithm="fricas")

[Out]

1/3*(15*cosh(b*x + a)^9 + 135*cosh(b*x + a)*sinh(b*x + a)^8 + 15*sinh(b*x + a)^9 + 20*(27*cosh(b*x + a)^2 - 1)
*sinh(b*x + a)^7 - 20*cosh(b*x + a)^7 + 140*(9*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^6 + 2*(945*cosh(
b*x + a)^4 - 210*cosh(b*x + a)^2 - 11)*sinh(b*x + a)^5 - 22*cosh(b*x + a)^5 + 10*(189*cosh(b*x + a)^5 - 70*cos
h(b*x + a)^3 - 11*cosh(b*x + a))*sinh(b*x + a)^4 + 20*(63*cosh(b*x + a)^6 - 35*cosh(b*x + a)^4 - 11*cosh(b*x +
a)^2 - 1)*sinh(b*x + a)^3 - 20*cosh(b*x + a)^3 + 20*(27*cosh(b*x + a)^7 - 21*cosh(b*x + a)^5 - 11*cosh(b*x +
a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^2 + 15*(cosh(b*x + a)^10 + 10*cosh(b*x + a)*sinh(b*x + a)^9 + sinh(b*x +
a)^10 + (45*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^8 - cosh(b*x + a)^8 + 8*(15*cosh(b*x + a)^3 - cosh(b*x + a))*s
inh(b*x + a)^7 + 2*(105*cosh(b*x + a)^4 - 14*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^6 - 2*cosh(b*x + a)^6 + 4*(63*
cosh(b*x + a)^5 - 14*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^5 + 2*(105*cosh(b*x + a)^6 - 35*cosh(b*x
+ a)^4 - 15*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^4 + 2*cosh(b*x + a)^4 + 8*(15*cosh(b*x + a)^7 - 7*cosh(b*x + a
)^5 - 5*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^3 + (45*cosh(b*x + a)^8 - 28*cosh(b*x + a)^6 - 30*cosh(
b*x + a)^4 + 12*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + cosh(b*x + a)^2 + 2*(5*cosh(b*x + a)^9 - 4*cosh(b*x + a
)^7 - 6*cosh(b*x + a)^5 + 4*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 1)*arctan(cosh(b*x + a) + sinh(b*
x + a)) + 5*(27*cosh(b*x + a)^8 - 28*cosh(b*x + a)^6 - 22*cosh(b*x + a)^4 - 12*cosh(b*x + a)^2 + 3)*sinh(b*x +
a) + 15*cosh(b*x + a))/(b*cosh(b*x + a)^10 + 10*b*cosh(b*x + a)*sinh(b*x + a)^9 + b*sinh(b*x + a)^10 - b*cosh
(b*x + a)^8 + (45*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^8 + 8*(15*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x
+ a)^7 - 2*b*cosh(b*x + a)^6 + 2*(105*b*cosh(b*x + a)^4 - 14*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^6 + 4*(63*b
*cosh(b*x + a)^5 - 14*b*cosh(b*x + a)^3 - 3*b*cosh(b*x + a))*sinh(b*x + a)^5 + 2*b*cosh(b*x + a)^4 + 2*(105*b*
cosh(b*x + a)^6 - 35*b*cosh(b*x + a)^4 - 15*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^4 + 8*(15*b*cosh(b*x + a)^7 -
7*b*cosh(b*x + a)^5 - 5*b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a)^3 + b*cosh(b*x + a)^2 + (45*b*cosh
(b*x + a)^8 - 28*b*cosh(b*x + a)^6 - 30*b*cosh(b*x + a)^4 + 12*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^2 + 2*(5*b
*cosh(b*x + a)^9 - 4*b*cosh(b*x + a)^7 - 6*b*cosh(b*x + a)^5 + 4*b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x
+ a) - b)

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giac [B]  time = 0.16, size = 124, normalized size = 1.88 \[ \frac {15 \, \pi + \frac {12 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + \frac {16 \, {\left (3 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} - 2\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3}} + 30 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^3,x, algorithm="giac")

[Out]

1/12*(15*pi + 12*(e^(b*x + a) - e^(-b*x - a))/((e^(b*x + a) - e^(-b*x - a))^2 + 4) + 16*(3*(e^(b*x + a) - e^(-
b*x - a))^2 - 2)/(e^(b*x + a) - e^(-b*x - a))^3 + 30*arctan(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)))/b

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maple [A]  time = 0.34, size = 73, normalized size = 1.11 \[ -\frac {1}{3 b \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{2}}+\frac {5}{3 b \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}+\frac {5 \,\mathrm {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2 b}+\frac {5 \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^4*sech(b*x+a)^3,x)

[Out]

-1/3/b/sinh(b*x+a)^3/cosh(b*x+a)^2+5/3/b/sinh(b*x+a)/cosh(b*x+a)^2+5/2*sech(b*x+a)*tanh(b*x+a)/b+5*arctan(exp(
b*x+a))/b

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maxima [B]  time = 0.42, size = 132, normalized size = 2.00 \[ -\frac {5 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {15 \, e^{\left (-b x - a\right )} - 20 \, e^{\left (-3 \, b x - 3 \, a\right )} - 22 \, e^{\left (-5 \, b x - 5 \, a\right )} - 20 \, e^{\left (-7 \, b x - 7 \, a\right )} + 15 \, e^{\left (-9 \, b x - 9 \, a\right )}}{3 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, e^{\left (-4 \, b x - 4 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^3,x, algorithm="maxima")

[Out]

-5*arctan(e^(-b*x - a))/b - 1/3*(15*e^(-b*x - a) - 20*e^(-3*b*x - 3*a) - 22*e^(-5*b*x - 5*a) - 20*e^(-7*b*x -
7*a) + 15*e^(-9*b*x - 9*a))/(b*(e^(-2*b*x - 2*a) + 2*e^(-4*b*x - 4*a) - 2*e^(-6*b*x - 6*a) - e^(-8*b*x - 8*a)
+ e^(-10*b*x - 10*a) - 1))

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mupad [B]  time = 1.56, size = 187, normalized size = 2.83 \[ \frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}+\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}+\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^3*sinh(a + b*x)^4),x)

[Out]

(5*atan((exp(b*x)*exp(a)*(b^2)^(1/2))/b))/(b^2)^(1/2) - (8*exp(a + b*x))/(3*b*(exp(4*a + 4*b*x) - 2*exp(2*a +
2*b*x) + 1)) - (2*exp(a + b*x))/(b*(2*exp(2*a + 2*b*x) + exp(4*a + 4*b*x) + 1)) - (8*exp(a + b*x))/(3*b*(3*exp
(2*a + 2*b*x) - 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1)) + (4*exp(a + b*x))/(b*(exp(2*a + 2*b*x) - 1)) + ex
p(a + b*x)/(b*(exp(2*a + 2*b*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{4}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**4*sech(b*x+a)**3,x)

[Out]

Integral(csch(a + b*x)**4*sech(a + b*x)**3, x)

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