3.43 \(\int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx\)

Optimal. Leaf size=89 \[ -\frac {35 \text {csch}^3(a+b x)}{24 b}+\frac {35 \text {csch}(a+b x)}{8 b}+\frac {35 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b} \]

[Out]

35/8*arctan(sinh(b*x+a))/b+35/8*csch(b*x+a)/b-35/24*csch(b*x+a)^3/b+7/8*csch(b*x+a)^3*sech(b*x+a)^2/b+1/4*csch
(b*x+a)^3*sech(b*x+a)^4/b

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Rubi [A]  time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2621, 288, 302, 207} \[ -\frac {35 \text {csch}^3(a+b x)}{24 b}+\frac {35 \text {csch}(a+b x)}{8 b}+\frac {35 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x]^4*Sech[a + b*x]^5,x]

[Out]

(35*ArcTan[Sinh[a + b*x]])/(8*b) + (35*Csch[a + b*x])/(8*b) - (35*Csch[a + b*x]^3)/(24*b) + (7*Csch[a + b*x]^3
*Sech[a + b*x]^2)/(8*b) + (Csch[a + b*x]^3*Sech[a + b*x]^4)/(4*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}^5(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,-i \text {csch}(a+b x)\right )}{b}\\ &=\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {(7 i) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,-i \text {csch}(a+b x)\right )}{4 b}\\ &=\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {(35 i) \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{8 b}\\ &=\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {(35 i) \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,-i \text {csch}(a+b x)\right )}{8 b}\\ &=\frac {35 \text {csch}(a+b x)}{8 b}-\frac {35 \text {csch}^3(a+b x)}{24 b}+\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {(35 i) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{8 b}\\ &=\frac {35 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac {35 \text {csch}(a+b x)}{8 b}-\frac {35 \text {csch}^3(a+b x)}{24 b}+\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 33, normalized size = 0.37 \[ -\frac {\text {csch}^3(a+b x) \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};-\sinh ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x]^4*Sech[a + b*x]^5,x]

[Out]

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

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fricas [B]  time = 0.42, size = 2092, normalized size = 23.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^5,x, algorithm="fricas")

[Out]

1/12*(105*cosh(b*x + a)^13 + 1365*cosh(b*x + a)*sinh(b*x + a)^12 + 105*sinh(b*x + a)^13 + 70*(117*cosh(b*x + a
)^2 + 1)*sinh(b*x + a)^11 + 70*cosh(b*x + a)^11 + 770*(39*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^10 +
7*(10725*cosh(b*x + a)^4 + 550*cosh(b*x + a)^2 - 47)*sinh(b*x + a)^9 - 329*cosh(b*x + a)^9 + 21*(6435*cosh(b*x
+ a)^5 + 550*cosh(b*x + a)^3 - 141*cosh(b*x + a))*sinh(b*x + a)^8 + 12*(15015*cosh(b*x + a)^6 + 1925*cosh(b*x
+ a)^4 - 987*cosh(b*x + a)^2 - 17)*sinh(b*x + a)^7 - 204*cosh(b*x + a)^7 + 84*(2145*cosh(b*x + a)^7 + 385*cos
h(b*x + a)^5 - 329*cosh(b*x + a)^3 - 17*cosh(b*x + a))*sinh(b*x + a)^6 + 7*(19305*cosh(b*x + a)^8 + 4620*cosh(
b*x + a)^6 - 5922*cosh(b*x + a)^4 - 612*cosh(b*x + a)^2 - 47)*sinh(b*x + a)^5 - 329*cosh(b*x + a)^5 + 7*(10725
*cosh(b*x + a)^9 + 3300*cosh(b*x + a)^7 - 5922*cosh(b*x + a)^5 - 1020*cosh(b*x + a)^3 - 235*cosh(b*x + a))*sin
h(b*x + a)^4 + 14*(2145*cosh(b*x + a)^10 + 825*cosh(b*x + a)^8 - 1974*cosh(b*x + a)^6 - 510*cosh(b*x + a)^4 -
235*cosh(b*x + a)^2 + 5)*sinh(b*x + a)^3 + 70*cosh(b*x + a)^3 + 14*(585*cosh(b*x + a)^11 + 275*cosh(b*x + a)^9
- 846*cosh(b*x + a)^7 - 306*cosh(b*x + a)^5 - 235*cosh(b*x + a)^3 + 15*cosh(b*x + a))*sinh(b*x + a)^2 + 105*(
cosh(b*x + a)^14 + 14*cosh(b*x + a)*sinh(b*x + a)^13 + sinh(b*x + a)^14 + (91*cosh(b*x + a)^2 + 1)*sinh(b*x +
a)^12 + cosh(b*x + a)^12 + 4*(91*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^11 + (1001*cosh(b*x + a)^4 +
66*cosh(b*x + a)^2 - 3)*sinh(b*x + a)^10 - 3*cosh(b*x + a)^10 + 2*(1001*cosh(b*x + a)^5 + 110*cosh(b*x + a)^3
- 15*cosh(b*x + a))*sinh(b*x + a)^9 + 3*(1001*cosh(b*x + a)^6 + 165*cosh(b*x + a)^4 - 45*cosh(b*x + a)^2 - 1)
*sinh(b*x + a)^8 - 3*cosh(b*x + a)^8 + 24*(143*cosh(b*x + a)^7 + 33*cosh(b*x + a)^5 - 15*cosh(b*x + a)^3 - cos
h(b*x + a))*sinh(b*x + a)^7 + 3*(1001*cosh(b*x + a)^8 + 308*cosh(b*x + a)^6 - 210*cosh(b*x + a)^4 - 28*cosh(b*
x + a)^2 + 1)*sinh(b*x + a)^6 + 3*cosh(b*x + a)^6 + 2*(1001*cosh(b*x + a)^9 + 396*cosh(b*x + a)^7 - 378*cosh(b
*x + a)^5 - 84*cosh(b*x + a)^3 + 9*cosh(b*x + a))*sinh(b*x + a)^5 + (1001*cosh(b*x + a)^10 + 495*cosh(b*x + a)
^8 - 630*cosh(b*x + a)^6 - 210*cosh(b*x + a)^4 + 45*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^4 + 3*cosh(b*x + a)^4 +
4*(91*cosh(b*x + a)^11 + 55*cosh(b*x + a)^9 - 90*cosh(b*x + a)^7 - 42*cosh(b*x + a)^5 + 15*cosh(b*x + a)^3 +
3*cosh(b*x + a))*sinh(b*x + a)^3 + (91*cosh(b*x + a)^12 + 66*cosh(b*x + a)^10 - 135*cosh(b*x + a)^8 - 84*cosh(
b*x + a)^6 + 45*cosh(b*x + a)^4 + 18*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(7*cosh(b*x +
a)^13 + 6*cosh(b*x + a)^11 - 15*cosh(b*x + a)^9 - 12*cosh(b*x + a)^7 + 9*cosh(b*x + a)^5 + 6*cosh(b*x + a)^3 -
cosh(b*x + a))*sinh(b*x + a) - 1)*arctan(cosh(b*x + a) + sinh(b*x + a)) + 7*(195*cosh(b*x + a)^12 + 110*cosh(
b*x + a)^10 - 423*cosh(b*x + a)^8 - 204*cosh(b*x + a)^6 - 235*cosh(b*x + a)^4 + 30*cosh(b*x + a)^2 + 15)*sinh(
b*x + a) + 105*cosh(b*x + a))/(b*cosh(b*x + a)^14 + 14*b*cosh(b*x + a)*sinh(b*x + a)^13 + b*sinh(b*x + a)^14 +
b*cosh(b*x + a)^12 + (91*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^12 + 4*(91*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a
))*sinh(b*x + a)^11 - 3*b*cosh(b*x + a)^10 + (1001*b*cosh(b*x + a)^4 + 66*b*cosh(b*x + a)^2 - 3*b)*sinh(b*x +
a)^10 + 2*(1001*b*cosh(b*x + a)^5 + 110*b*cosh(b*x + a)^3 - 15*b*cosh(b*x + a))*sinh(b*x + a)^9 - 3*b*cosh(b*x
+ a)^8 + 3*(1001*b*cosh(b*x + a)^6 + 165*b*cosh(b*x + a)^4 - 45*b*cosh(b*x + a)^2 - b)*sinh(b*x + a)^8 + 24*(
143*b*cosh(b*x + a)^7 + 33*b*cosh(b*x + a)^5 - 15*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a)^7 + 3*b*c
osh(b*x + a)^6 + 3*(1001*b*cosh(b*x + a)^8 + 308*b*cosh(b*x + a)^6 - 210*b*cosh(b*x + a)^4 - 28*b*cosh(b*x + a
)^2 + b)*sinh(b*x + a)^6 + 2*(1001*b*cosh(b*x + a)^9 + 396*b*cosh(b*x + a)^7 - 378*b*cosh(b*x + a)^5 - 84*b*co
sh(b*x + a)^3 + 9*b*cosh(b*x + a))*sinh(b*x + a)^5 + 3*b*cosh(b*x + a)^4 + (1001*b*cosh(b*x + a)^10 + 495*b*co
sh(b*x + a)^8 - 630*b*cosh(b*x + a)^6 - 210*b*cosh(b*x + a)^4 + 45*b*cosh(b*x + a)^2 + 3*b)*sinh(b*x + a)^4 +
4*(91*b*cosh(b*x + a)^11 + 55*b*cosh(b*x + a)^9 - 90*b*cosh(b*x + a)^7 - 42*b*cosh(b*x + a)^5 + 15*b*cosh(b*x
+ a)^3 + 3*b*cosh(b*x + a))*sinh(b*x + a)^3 - b*cosh(b*x + a)^2 + (91*b*cosh(b*x + a)^12 + 66*b*cosh(b*x + a)^
10 - 135*b*cosh(b*x + a)^8 - 84*b*cosh(b*x + a)^6 + 45*b*cosh(b*x + a)^4 + 18*b*cosh(b*x + a)^2 - b)*sinh(b*x
+ a)^2 + 2*(7*b*cosh(b*x + a)^13 + 6*b*cosh(b*x + a)^11 - 15*b*cosh(b*x + a)^9 - 12*b*cosh(b*x + a)^7 + 9*b*co
sh(b*x + a)^5 + 6*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a) - b)

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giac [A]  time = 0.15, size = 148, normalized size = 1.66 \[ \frac {105 \, \pi + \frac {12 \, {\left (11 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 52 \, e^{\left (b x + a\right )} - 52 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2}} + \frac {32 \, {\left (9 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3}} + 210 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^5,x, algorithm="giac")

[Out]

1/48*(105*pi + 12*(11*(e^(b*x + a) - e^(-b*x - a))^3 + 52*e^(b*x + a) - 52*e^(-b*x - a))/((e^(b*x + a) - e^(-b
*x - a))^2 + 4)^2 + 32*(9*(e^(b*x + a) - e^(-b*x - a))^2 - 4)/(e^(b*x + a) - e^(-b*x - a))^3 + 210*arctan(1/2*
(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)))/b

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maple [A]  time = 0.36, size = 92, normalized size = 1.03 \[ -\frac {1}{3 b \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{4}}+\frac {7}{3 b \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}+\frac {35 \mathrm {sech}\left (b x +a \right )^{3} \tanh \left (b x +a \right )}{12 b}+\frac {35 \,\mathrm {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{8 b}+\frac {35 \arctan \left ({\mathrm e}^{b x +a}\right )}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^4*sech(b*x+a)^5,x)

[Out]

-1/3/b/sinh(b*x+a)^3/cosh(b*x+a)^4+7/3/b/sinh(b*x+a)/cosh(b*x+a)^4+35/12*sech(b*x+a)^3*tanh(b*x+a)/b+35/8*sech
(b*x+a)*tanh(b*x+a)/b+35/4*arctan(exp(b*x+a))/b

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maxima [B]  time = 0.51, size = 178, normalized size = 2.00 \[ -\frac {35 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac {105 \, e^{\left (-b x - a\right )} + 70 \, e^{\left (-3 \, b x - 3 \, a\right )} - 329 \, e^{\left (-5 \, b x - 5 \, a\right )} - 204 \, e^{\left (-7 \, b x - 7 \, a\right )} - 329 \, e^{\left (-9 \, b x - 9 \, a\right )} + 70 \, e^{\left (-11 \, b x - 11 \, a\right )} + 105 \, e^{\left (-13 \, b x - 13 \, a\right )}}{12 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-6 \, b x - 6 \, a\right )} + 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + 3 \, e^{\left (-10 \, b x - 10 \, a\right )} - e^{\left (-12 \, b x - 12 \, a\right )} - e^{\left (-14 \, b x - 14 \, a\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^4*sech(b*x+a)^5,x, algorithm="maxima")

[Out]

-35/4*arctan(e^(-b*x - a))/b + 1/12*(105*e^(-b*x - a) + 70*e^(-3*b*x - 3*a) - 329*e^(-5*b*x - 5*a) - 204*e^(-7
*b*x - 7*a) - 329*e^(-9*b*x - 9*a) + 70*e^(-11*b*x - 11*a) + 105*e^(-13*b*x - 13*a))/(b*(e^(-2*b*x - 2*a) - 3*
e^(-4*b*x - 4*a) - 3*e^(-6*b*x - 6*a) + 3*e^(-8*b*x - 8*a) + 3*e^(-10*b*x - 10*a) - e^(-12*b*x - 12*a) - e^(-1
4*b*x - 14*a) + 1))

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mupad [B]  time = 1.47, size = 291, normalized size = 3.27 \[ \frac {35\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{4\,\sqrt {b^2}}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {7\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}+\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (4\,{\mathrm {e}}^{2\,a+2\,b\,x}+6\,{\mathrm {e}}^{4\,a+4\,b\,x}+4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}+\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}+\frac {11\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(a + b*x)^5*sinh(a + b*x)^4),x)

[Out]

(35*atan((exp(b*x)*exp(a)*(b^2)^(1/2))/b))/(4*(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)) - (7*exp(a + b*x))/(2*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)) - (6*exp(a + b*x))/(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*(4*exp(2*a + 2*b*x) + 6*exp(4*a + 4*b*x) +
4*exp(6*a + 6*b*x) + exp(8*a + 8*b*x) + 1)) + (6*exp(a + b*x))/(b*(exp(2*a + 2*b*x) - 1)) + (11*exp(a + b*x))/
(4*b*(exp(2*a + 2*b*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{4}{\left (a + b x \right )} \operatorname {sech}^{5}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**4*sech(b*x+a)**5,x)

[Out]

Integral(csch(a + b*x)**4*sech(a + b*x)**5, x)

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