3.47 \(\int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx\)
Optimal. Leaf size=89 \[ \frac {35 \text {sech}^3(a+b x)}{24 b}+\frac {35 \text {sech}(a+b x)}{8 b}-\frac {35 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {7 \text {csch}^2(a+b x) \text {sech}^3(a+b x)}{8 b} \]
[Out]
-35/8*arctanh(cosh(b*x+a))/b+35/8*sech(b*x+a)/b+35/24*sech(b*x+a)^3/b+7/8*csch(b*x+a)^2*sech(b*x+a)^3/b-1/4*cs
ch(b*x+a)^4*sech(b*x+a)^3/b
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Rubi [A] time = 0.06, 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 =
{2622, 288, 302, 207} \[ \frac {35 \text {sech}^3(a+b x)}{24 b}+\frac {35 \text {sech}(a+b x)}{8 b}-\frac {35 \tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {7 \text {csch}^2(a+b x) \text {sech}^3(a+b x)}{8 b} \]
Antiderivative was successfully verified.
[In]
Int[Csch[a + b*x]^5*Sech[a + b*x]^4,x]
[Out]
(-35*ArcTanh[Cosh[a + b*x]])/(8*b) + (35*Sech[a + b*x])/(8*b) + (35*Sech[a + b*x]^3)/(24*b) + (7*Csch[a + b*x]
^2*Sech[a + b*x]^3)/(8*b) - (Csch[a + b*x]^4*Sech[a + b*x]^3)/(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 2622
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Rubi steps
\begin {align*} \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {7 \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\text {sech}(a+b x)\right )}{4 b}\\ &=\frac {7 \text {csch}^2(a+b x) \text {sech}^3(a+b x)}{8 b}-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {35 \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\text {sech}(a+b x)\right )}{8 b}\\ &=\frac {7 \text {csch}^2(a+b x) \text {sech}^3(a+b x)}{8 b}-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {35 \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\text {sech}(a+b x)\right )}{8 b}\\ &=\frac {35 \text {sech}(a+b x)}{8 b}+\frac {35 \text {sech}^3(a+b x)}{24 b}+\frac {7 \text {csch}^2(a+b x) \text {sech}^3(a+b x)}{8 b}-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}+\frac {35 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(a+b x)\right )}{8 b}\\ &=-\frac {35 \tanh ^{-1}(\cosh (a+b x))}{8 b}+\frac {35 \text {sech}(a+b x)}{8 b}+\frac {35 \text {sech}^3(a+b x)}{24 b}+\frac {7 \text {csch}^2(a+b x) \text {sech}^3(a+b x)}{8 b}-\frac {\text {csch}^4(a+b x) \text {sech}^3(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 121, normalized size = 1.36 \[ -\frac {\text {csch}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {11 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\text {sech}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\text {sech}^3(a+b x)}{3 b}+\frac {11 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {3 \text {sech}(a+b x)}{b}+\frac {35 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[a + b*x]^5*Sech[a + b*x]^4,x]
[Out]
(11*Csch[(a + b*x)/2]^2)/(32*b) - Csch[(a + b*x)/2]^4/(64*b) + (35*Log[Tanh[(a + b*x)/2]])/(8*b) + (11*Sech[(a
+ b*x)/2]^2)/(32*b) + Sech[(a + b*x)/2]^4/(64*b) + (3*Sech[a + b*x])/b + Sech[a + b*x]^3/(3*b)
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fricas [B] time = 0.46, size = 2802, normalized size = 31.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(b*x+a)^5*sech(b*x+a)^4,x, algorithm="fricas")
[Out]
1/24*(210*cosh(b*x + a)^13 + 2730*cosh(b*x + a)*sinh(b*x + a)^12 + 210*sinh(b*x + a)^13 + 140*(117*cosh(b*x +
a)^2 - 1)*sinh(b*x + a)^11 - 140*cosh(b*x + a)^11 + 1540*(39*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^10
+ 14*(10725*cosh(b*x + a)^4 - 550*cosh(b*x + a)^2 - 47)*sinh(b*x + a)^9 - 658*cosh(b*x + a)^9 + 42*(6435*cosh
(b*x + a)^5 - 550*cosh(b*x + a)^3 - 141*cosh(b*x + a))*sinh(b*x + a)^8 + 24*(15015*cosh(b*x + a)^6 - 1925*cosh
(b*x + a)^4 - 987*cosh(b*x + a)^2 + 17)*sinh(b*x + a)^7 + 408*cosh(b*x + a)^7 + 168*(2145*cosh(b*x + a)^7 - 38
5*cosh(b*x + a)^5 - 329*cosh(b*x + a)^3 + 17*cosh(b*x + a))*sinh(b*x + a)^6 + 14*(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 - 658*cosh(b*x + a)^5 + 14
*(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))*sinh(b*x + a)^4 + 28*(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 - 140*cosh(b*x + a)^3 + 28*(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)*sin
h(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 + cosh(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 - 37
8*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*cos
h(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)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 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 + cosh(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)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 14*(195*cosh(b*x + a)^12 - 110*cosh(b*x + a)^10 - 423*c
osh(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) + 210*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*c
osh(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*cosh(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*cosh(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*cosh(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*cos
h(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*cosh(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 = 0.14, size = 152, normalized size = 1.71 \[ \frac {\frac {12 \, {\left (11 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 52 \, e^{\left (b x + a\right )} - 52 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}^{2}} + \frac {32 \, {\left (9 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3}} - 105 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 105 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(b*x+a)^5*sech(b*x+a)^4,x, algorithm="giac")
[Out]
1/48*(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 - 105*log(e^(b*x + a) + e^
(-b*x - a) + 2) + 105*log(e^(b*x + a) + e^(-b*x - a) - 2))/b
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maple [A] time = 0.22, size = 71, normalized size = 0.80 \[ \frac {-\frac {1}{4 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{3}}+\frac {7}{8 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}+\frac {35}{24 \cosh \left (b x +a \right )^{3}}+\frac {35}{8 \cosh \left (b x +a \right )}-\frac {35 \arctanh \left ({\mathrm e}^{b x +a}\right )}{4}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(b*x+a)^5*sech(b*x+a)^4,x)
[Out]
1/b*(-1/4/sinh(b*x+a)^4/cosh(b*x+a)^3+7/8/sinh(b*x+a)^2/cosh(b*x+a)^3+35/24/cosh(b*x+a)^3+35/8/cosh(b*x+a)-35/
4*arctanh(exp(b*x+a)))
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maxima [B] time = 0.34, size = 195, normalized size = 2.19 \[ -\frac {35 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac {35 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} - \frac {105 \, e^{\left (-b x - a\right )} - 70 \, e^{\left (-3 \, b x - 3 \, a\right )} - 329 \, e^{\left (-5 \, b x - 5 \, a\right )} + 204 \, e^{\left (-7 \, b x - 7 \, a\right )} - 329 \, e^{\left (-9 \, b x - 9 \, a\right )} - 70 \, e^{\left (-11 \, b x - 11 \, a\right )} + 105 \, e^{\left (-13 \, b x - 13 \, a\right )}}{12 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + 3 \, e^{\left (-10 \, b x - 10 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - e^{\left (-14 \, b x - 14 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(b*x+a)^5*sech(b*x+a)^4,x, algorithm="maxima")
[Out]
-35/8*log(e^(-b*x - a) + 1)/b + 35/8*log(e^(-b*x - a) - 1)/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^(-14*b*x - 14*a) - 1))
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mupad [B] time = 0.10, size = 295, normalized size = 3.31 \[ \frac {7\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {35\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{4\,\sqrt {-b^2}}+\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}+\frac {11\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}+\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(cosh(a + b*x)^4*sinh(a + b*x)^5),x)
[Out]
(7*exp(a + b*x))/(2*b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (35*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b
))/(4*(-b^2)^(1/2)) + (8*exp(a + b*x))/(3*b*(2*exp(2*a + 2*b*x) + exp(4*a + 4*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)) - (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*(6*exp(4*a + 4*b*x) - 4*exp(2*a + 2*b*x)
- 4*exp(6*a + 6*b*x) + exp(8*a + 8*b*x) + 1)) + (11*exp(a + b*x))/(4*b*(exp(2*a + 2*b*x) - 1)) + (6*exp(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 \[ \int \operatorname {csch}^{5}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(b*x+a)**5*sech(b*x+a)**4,x)
[Out]
Integral(csch(a + b*x)**5*sech(a + b*x)**4, x)
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