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*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)
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Rubi [A] time = 0.0478115, antiderivative size = 89, normalized size of antiderivative = 1.,
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 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])
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 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])
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*}
Mathematica [C] time = 0.017379, 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]
-(Csch[a + b*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, -Sinh[a + b*x]^2])/(3*b)
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Maple [A] time = 0.024, size = 92, normalized size = 1. \begin{align*} -{\frac{1}{3\,b \left ( \sinh \left ( bx+a \right ) \right ) ^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}}+{\frac{7}{3\,b\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}}+{\frac{35\, \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}\tanh \left ( bx+a \right ) }{12\,b}}+{\frac{35\,{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{8\,b}}+{\frac{35\,\arctan \left ({{\rm e}^{bx+a}} \right ) }{4\,b}} \end{align*}
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 = 1.56036, size = 240, normalized size = 2.7 \begin{align*} -\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 )}} \end{align*}
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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Fricas [B] time = 2.62474, size = 6041, normalized size = 67.88 \begin{align*} \text{result too large to display} \end{align*}
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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}^{4}{\left (a + b x \right )} \operatorname{sech}^{5}{\left (a + b x \right )}\, dx \end{align*}
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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Giac [A] time = 1.19093, size = 207, normalized size = 2.33 \begin{align*} \frac{35 \,{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )\right )}}{16 \, b} + \frac{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 )}}{4 \,{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2} b} + \frac{2 \,{\left (9 \,{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}}{3 \, b{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3}} \end{align*}
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]
35/16*(pi + 2*arctan(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)))/b + 1/4*(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*b) + 2/3*(9*(e^(b*x + a) - e^(-b*x - a)
)^2 - 4)/(b*(e^(b*x + a) - e^(-b*x - a))^3)