3.2.19 \(\int \frac {\text {csch}(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [119]
Optimal. Leaf size=136 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{a^{5/2} f}-\frac {b \cosh (e+f x)}{3 a (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}-\frac {(5 a-3 b) b \cosh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}} \]
[Out]
-arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(5/2)/f-1/3*b*cosh(f*x+e)/a/(a-b)/f/(a-b+b*cosh(f*
x+e)^2)^(3/2)-1/3*(5*a-3*b)*b*cosh(f*x+e)/a^2/(a-b)^2/f/(a-b+b*cosh(f*x+e)^2)^(1/2)
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Rubi [A]
time = 0.12, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3265, 425, 541,
12, 385, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{a^{5/2} f}-\frac {b (5 a-3 b) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt {a+b \cosh ^2(e+f x)-b}}-\frac {b \cosh (e+f x)}{3 a f (a-b) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csch[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]
[Out]
-(ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(a^(5/2)*f)) - (b*Cosh[e + f*x])/(3*a*(a -
b)*f*(a - b + b*Cosh[e + f*x]^2)^(3/2)) - ((5*a - 3*b)*b*Cosh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a - b + b*Cosh
[e + f*x]^2])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 425
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]
Rule 541
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Rule 3265
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac {b \cosh (e+f x)}{3 a (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-3 a+b+2 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\cosh (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac {b \cosh (e+f x)}{3 a (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}-\frac {(5 a-3 b) b \cosh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {3 (a-b)^2}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac {b \cosh (e+f x)}{3 a (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}-\frac {(5 a-3 b) b \cosh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{a^2 f}\\ &=-\frac {b \cosh (e+f x)}{3 a (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}-\frac {(5 a-3 b) b \cosh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{a^2 f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{a^{5/2} f}-\frac {b \cosh (e+f x)}{3 a (a-b) f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}-\frac {(5 a-3 b) b \cosh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 130, normalized size = 0.96 \begin {gather*} \frac {-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )}{a^{5/2}}+\frac {\sqrt {2} b \cosh (e+f x) \left (-12 a^2+13 a b-3 b^2+b (-5 a+3 b) \cosh (2 (e+f x))\right )}{3 a^2 (a-b)^2 (2 a-b+b \cosh (2 (e+f x)))^{3/2}}}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csch[e + f*x]/(a + b*Sinh[e + f*x]^2)^(5/2),x]
[Out]
(-(ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/a^(5/2)) + (Sqrt[2]*b*Cosh[e +
f*x]*(-12*a^2 + 13*a*b - 3*b^2 + b*(-5*a + 3*b)*Cosh[2*(e + f*x)]))/(3*a^2*(a - b)^2*(2*a - b + b*Cosh[2*(e +
f*x)])^(3/2)))/f
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Maple [A]
time = 2.12, size = 236, normalized size = 1.74
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a^{2} \left (a -b \right ) \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +\left (a +b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )+2 \sqrt {a}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{\sinh \left (f x +e \right )^{2}}\right )}{2 a^{\frac {5}{2}}}-\frac {b \left (2 b \left (\sinh ^{2}\left (f x +e \right )\right )+3 a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{3 a \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\) |
\(236\) |
| risch |
\(\text {Expression too large to display}\) |
\(657915\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-b/a^2*cosh(f*x+e)^2/(a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2
)-1/2/a^(5/2)*ln((2*a+(a+b)*sinh(f*x+e)^2+2*a^(1/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/sinh(f*x+e)^2)-
1/3/a*b*(2*b*sinh(f*x+e)^2+3*a-b)*cosh(f*x+e)^2/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)/(a+b*sinh(f*x+e)^2)/
(a^2-2*a*b+b^2))/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
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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(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(csch(f*x + e)/(b*sinh(f*x + e)^2 + a)^(5/2), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2620 vs.
\(2 (122) = 244\).
time = 0.66, size = 5342, normalized size = 39.28 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/6*(3*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)*sinh(f*x + e)^7
+ (a^2*b^2 - 2*a*b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^6 + 4*(2*
a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b
^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e))*sinh(f*x + e)^5 +
2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f
*x + e)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f
*x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^5 + 10*(2*
a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^3 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f
*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 +
b^4)*cosh(f*x + e)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b
^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b^2
- 2*a*b^3 + b^4)*cosh(f*x + e)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^5 + (8*a^4 - 24*a^3*
b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + (2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e))*sinh
(f*x + e))*sqrt(a)*log(-((a + b)*cosh(f*x + e)^4 + 4*(a + b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a + b)*sinh(f*x
+ e)^4 + 2*(3*a - b)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 + 3*a - b)*sinh(f*x + e)^2 - 2*sqrt(2)*(co
sh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(
f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a + b)*cosh(f
*x + e)^3 + (3*a - b)*cosh(f*x + e))*sinh(f*x + e) + a + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3
+ sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 - 1)*sinh(f*x + e)^2 - 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 - cosh
(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*((5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^6 + 6*(5*a^2*b^2 - 3*a*b^3)*co
sh(f*x + e)*sinh(f*x + e)^5 + (5*a^2*b^2 - 3*a*b^3)*sinh(f*x + e)^6 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x
+ e)^4 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3 + 5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 5*a^2*b^
2 - 3*a*b^3 + 4*(5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^3 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e))*sinh
(f*x + e)^3 + 3*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^2 + 3*(5*(5*a^2*b^2 - 3*a*b^3)*cosh(f*x + e)^4 + 8
*a^3*b - 7*a^2*b^2 + a*b^3 + 6*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 6*((5*a^2*b^2
- 3*a*b^3)*cosh(f*x + e)^5 + 2*(8*a^3*b - 7*a^2*b^2 + a*b^3)*cosh(f*x + e)^3 + (8*a^3*b - 7*a^2*b^2 + a*b^3)*c
osh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(
f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^8 + 8*(a^5*b^2 -
2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*sinh(f*x + e)^8 + 4*(
2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^6 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x
+ e)^2 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f)*sinh(f*x + e)^6 + 2*(8*a^7 - 24*a^6*b + 27*a^5*b^2 -
14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^4 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^3 + 3*(2*a^6*
b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*
f*cosh(f*x + e)^4 + 30*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^2 + (8*a^7 - 24*a^6*b + 27*
a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f)*sinh(f*x + e)^4 + 4*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*
x + e)^2 + 8*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^5 + 10*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*
b^4)*f*cosh(f*x + e)^3 + (8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e))*sinh(f*x +
e)^3 + 4*(7*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f*cosh(f*x + e)^6 + 15*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)
*f*cosh(f*x + e)^4 + 3*(8*a^7 - 24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^2 + (2*a^6*b -
5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f)*sinh(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f + 8*((a^5*b^2 - 2*a^4
*b^3 + a^3*b^4)*f*cosh(f*x + e)^7 + 3*(2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^4)*f*cosh(f*x + e)^5 + (8*a^7 -
24*a^6*b + 27*a^5*b^2 - 14*a^4*b^3 + 3*a^3*b^4)*f*cosh(f*x + e)^3 + (2*a^6*b - 5*a^5*b^2 + 4*a^4*b^3 - a^3*b^
4)*f*cosh(f*x + e))*sinh(f*x + e)), 1/3*(3*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a^2*b^2 - 2*a*b^3 +
b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a
*b^3 - b^4)*cosh(f*x + e)^6 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x +
e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b^2 - 2*a*b^3...
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(f*x+e)/(a+b*sinh(f*x+e)**2)**(5/2),x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 5007 deep
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs.
\(2 (122) = 244\).
time = 0.66, size = 404, normalized size = 2.97 \begin {gather*} -\frac {{\left (\frac {{\left ({\left (\frac {{\left (5 \, a^{9} b^{2} e^{\left (12 \, e\right )} - 3 \, a^{8} b^{3} e^{\left (12 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}} + \frac {3 \, {\left (8 \, a^{10} b e^{\left (10 \, e\right )} - 7 \, a^{9} b^{2} e^{\left (10 \, e\right )} + a^{8} b^{3} e^{\left (10 \, e\right )}\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}}\right )} e^{\left (2 \, f x\right )} + \frac {3 \, {\left (8 \, a^{10} b e^{\left (8 \, e\right )} - 7 \, a^{9} b^{2} e^{\left (8 \, e\right )} + a^{8} b^{3} e^{\left (8 \, e\right )}\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}}\right )} e^{\left (2 \, f x\right )} + \frac {5 \, a^{9} b^{2} e^{\left (6 \, e\right )} - 3 \, a^{8} b^{3} e^{\left (6 \, e\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}}}{{\left (b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b\right )}^{\frac {3}{2}}} - \frac {6 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt {b}}{2 \, \sqrt {-a}}\right ) e^{\left (-6 \, e\right )}}{\sqrt {-a} a^{2}}\right )} e^{\left (6 \, e\right )}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")
[Out]
-1/3*(((((5*a^9*b^2*e^(12*e) - 3*a^8*b^3*e^(12*e))*e^(2*f*x)/(a^12*e^(12*e) - 2*a^11*b*e^(12*e) + a^10*b^2*e^(
12*e)) + 3*(8*a^10*b*e^(10*e) - 7*a^9*b^2*e^(10*e) + a^8*b^3*e^(10*e))/(a^12*e^(12*e) - 2*a^11*b*e^(12*e) + a^
10*b^2*e^(12*e)))*e^(2*f*x) + 3*(8*a^10*b*e^(8*e) - 7*a^9*b^2*e^(8*e) + a^8*b^3*e^(8*e))/(a^12*e^(12*e) - 2*a^
11*b*e^(12*e) + a^10*b^2*e^(12*e)))*e^(2*f*x) + (5*a^9*b^2*e^(6*e) - 3*a^8*b^3*e^(6*e))/(a^12*e^(12*e) - 2*a^1
1*b*e^(12*e) + a^10*b^2*e^(12*e)))/(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b)^(3/2) -
6*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) +
b) - sqrt(b))/sqrt(-a))*e^(-6*e)/(sqrt(-a)*a^2))*e^(6*e)/f
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\mathrm {sinh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sinh(e + f*x)*(a + b*sinh(e + f*x)^2)^(5/2)),x)
[Out]
int(1/(sinh(e + f*x)*(a + b*sinh(e + f*x)^2)^(5/2)), x)
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