3.213 \(\int \frac{\tanh ^4(x)}{(a+b \text{sech}^2(x))^{5/2}} \, dx\)
Optimal. Leaf size=90 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a-b \tanh ^2(x)+b}}-\frac{(a+b) \tanh (x)}{3 a b \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - ((a + b)*Tanh[x])/(3*a*b*(a + b - b*Tanh[x]^2)^
(3/2)) + ((a - 3*b)*Tanh[x])/(3*a^2*b*Sqrt[a + b - b*Tanh[x]^2])
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Rubi [A] time = 0.262349, antiderivative size = 90, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used =
{4141, 1975, 470, 527, 12, 377, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a-b \tanh ^2(x)+b}}\right )}{a^{5/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a-b \tanh ^2(x)+b}}-\frac{(a+b) \tanh (x)}{3 a b \left (a-b \tanh ^2(x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[x]^4/(a + b*Sech[x]^2)^(5/2),x]
[Out]
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - ((a + b)*Tanh[x])/(3*a*b*(a + b - b*Tanh[x]^2)^
(3/2)) + ((a - 3*b)*Tanh[x])/(3*a^2*b*Sqrt[a + b - b*Tanh[x]^2])
Rule 4141
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((d*ff*x)^m*(a + b*(1 + ff^2*x^2)^(n/2))^p)/(1 + ff^
2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])
Rule 1975
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&& !BinomialMatchQ[{u, v}, x]
Rule 470
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 527
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 377
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{\tanh ^4(x)}{\left (a+b \text{sech}^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=-\frac{(a+b) \tanh (x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{a+b+(-a+2 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 a b}\\ &=-\frac{(a+b) \tanh (x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a+b-b \tanh ^2(x)}}-\frac{\operatorname{Subst}\left (\int -\frac{3 b (a+b)}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 b (a+b)}\\ &=-\frac{(a+b) \tanh (x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a+b-b \tanh ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (x)\right )}{a^2}\\ &=-\frac{(a+b) \tanh (x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a+b-b \tanh ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a+b-b \tanh ^2(x)}}\right )}{a^2}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b-b \tanh ^2(x)}}\right )}{a^{5/2}}-\frac{(a+b) \tanh (x)}{3 a b \left (a+b-b \tanh ^2(x)\right )^{3/2}}+\frac{(a-3 b) \tanh (x)}{3 a^2 b \sqrt{a+b-b \tanh ^2(x)}}\\ \end{align*}
Mathematica [B] time = 2.01579, size = 290, normalized size = 3.22 \[ \frac{\text{sech}^4(x) \left (\frac{\sqrt{2} \text{csch}(x) \text{sech}(x) \left (-\frac{16 \left (a \sinh ^2(x)+a+b\right ) \left (\frac{a \sinh ^2(x)}{a+b}+1\right ) \left (\frac{a^2 (a+b) \sinh ^4(x)}{\left (a \sinh ^2(x)+a+b\right )^2}+\frac{3 a (a+b) \sinh ^2(x)}{a \sinh ^2(x)+a+b}-\frac{3 \sqrt{a} \sqrt{a+b} \sinh (x) \sinh ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{a+b}}\right )}{\sqrt{\frac{a \sinh ^2(x)+a+b}{a+b}}}\right )}{a^3}+\frac{12 \sinh ^4(x)}{a+b}+\frac{2 \sinh ^2(x) \left (a \sinh ^2(x)+a+b\right )}{(a+b)^2}+\frac{\sinh ^2(x)}{a+b}\right ) (a \cosh (2 x)+a+2 b)^{5/2}}{\left (a \sinh ^2(x)+a+b\right )^{3/2}}+\frac{8 \tanh (x) (a \cosh (2 x)+2 a+3 b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}-\frac{12 \tanh (x) ((3 a+2 b) \cosh (2 x)+b) (a \cosh (2 x)+a+2 b)}{(a+b)^2}\right )}{384 \left (a+b \text{sech}^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[x]^4/(a + b*Sech[x]^2)^(5/2),x]
[Out]
(Sech[x]^4*((Sqrt[2]*(a + 2*b + a*Cosh[2*x])^(5/2)*Csch[x]*Sech[x]*(Sinh[x]^2/(a + b) + (12*Sinh[x]^4)/(a + b)
+ (2*Sinh[x]^2*(a + b + a*Sinh[x]^2))/(a + b)^2 - (16*(a + b + a*Sinh[x]^2)*(1 + (a*Sinh[x]^2)/(a + b))*((a^2
*(a + b)*Sinh[x]^4)/(a + b + a*Sinh[x]^2)^2 + (3*a*(a + b)*Sinh[x]^2)/(a + b + a*Sinh[x]^2) - (3*Sqrt[a]*Sqrt[
a + b]*ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]]*Sinh[x])/Sqrt[(a + b + a*Sinh[x]^2)/(a + b)]))/a^3))/(a + b + a*
Sinh[x]^2)^(3/2) + (8*(a + 2*b + a*Cosh[2*x])*(2*a + 3*b + a*Cosh[2*x])*Tanh[x])/(a + b)^2 - (12*(a + 2*b + a*
Cosh[2*x])*(b + (3*a + 2*b)*Cosh[2*x])*Tanh[x])/(a + b)^2))/(384*(a + b*Sech[x]^2)^(5/2))
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tanh \left ( x \right ) \right ) ^{4} \left ( a+b \left ({\rm sech} \left (x\right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^4/(a+b*sech(x)^2)^(5/2),x)
[Out]
int(tanh(x)^4/(a+b*sech(x)^2)^(5/2),x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )^{4}}{{\left (b \operatorname{sech}\left (x\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^4/(a+b*sech(x)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(tanh(x)^4/(b*sech(x)^2 + a)^(5/2), x)
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Fricas [B] time = 4.46926, size = 9694, normalized size = 107.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^4/(a+b*sech(x)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*(a^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 + 4*(a^2 + 2*a*b)*cosh(x)^6 + 4*(7*a^2*cosh(
x)^2 + a^2 + 2*a*b)*sinh(x)^6 + 8*(7*a^2*cosh(x)^3 + 3*(a^2 + 2*a*b)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 8*a*b + 8
*b^2)*cosh(x)^4 + 2*(35*a^2*cosh(x)^4 + 30*(a^2 + 2*a*b)*cosh(x)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(x)^4 + 8*(7*a
^2*cosh(x)^5 + 10*(a^2 + 2*a*b)*cosh(x)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x))*sinh(x)^3 + 4*(a^2 + 2*a*b)*cosh(
x)^2 + 4*(7*a^2*cosh(x)^6 + 15*(a^2 + 2*a*b)*cosh(x)^4 + 3*(3*a^2 + 8*a*b + 8*b^2)*cosh(x)^2 + a^2 + 2*a*b)*si
nh(x)^2 + a^2 + 8*(a^2*cosh(x)^7 + 3*(a^2 + 2*a*b)*cosh(x)^5 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x)^3 + (a^2 + 2*a*
b)*cosh(x))*sinh(x))*sqrt(a)*log((a*b^2*cosh(x)^8 + 8*a*b^2*cosh(x)*sinh(x)^7 + a*b^2*sinh(x)^8 - 2*(a*b^2 - b
^3)*cosh(x)^6 + 2*(14*a*b^2*cosh(x)^2 - a*b^2 + b^3)*sinh(x)^6 + 4*(14*a*b^2*cosh(x)^3 - 3*(a*b^2 - b^3)*cosh(
x))*sinh(x)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^4 + (70*a*b^2*cosh(x)^4 + a^3 + 4*a^2*b + 9*a*b^2 - 30*(a*b^
2 - b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*a*b^2*cosh(x)^5 - 10*(a*b^2 - b^3)*cosh(x)^3 + (a^3 + 4*a^2*b + 9*a*b^2)
*cosh(x))*sinh(x)^3 + a^3 + 2*(a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*a*b^2*cosh(x)^6 - 15*(a*b^2 - b^3)*cosh(x)^4 +
a^3 + 3*a^2*b + 3*(a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sin
h(x)^5 + b^2*sinh(x)^6 - 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 - 3*b^2*co
sh(x))*sinh(x)^3 - (a^2 + 4*a*b)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cosh(x)^2 - a^2 - 4*a*b)*sinh(x)^2 - a
^2 + 2*(3*b^2*cosh(x)^5 - 6*b^2*cosh(x)^3 - (a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh
(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(2*a*b^2*cosh(x)^7 - 3*(a*b^2 - b^3)*cosh(x)
^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(x)^3 + (a^3 + 3*a^2*b)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x)
+ 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6))
+ 3*(a^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 + 4*(a^2 + 2*a*b)*cosh(x)^6 + 4*(7*a^2*cosh(x)^2
+ a^2 + 2*a*b)*sinh(x)^6 + 8*(7*a^2*cosh(x)^3 + 3*(a^2 + 2*a*b)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 8*a*b + 8*b^2
)*cosh(x)^4 + 2*(35*a^2*cosh(x)^4 + 30*(a^2 + 2*a*b)*cosh(x)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(x)^4 + 8*(7*a^2*c
osh(x)^5 + 10*(a^2 + 2*a*b)*cosh(x)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x))*sinh(x)^3 + 4*(a^2 + 2*a*b)*cosh(x)^2
+ 4*(7*a^2*cosh(x)^6 + 15*(a^2 + 2*a*b)*cosh(x)^4 + 3*(3*a^2 + 8*a*b + 8*b^2)*cosh(x)^2 + a^2 + 2*a*b)*sinh(x
)^2 + a^2 + 8*(a^2*cosh(x)^7 + 3*(a^2 + 2*a*b)*cosh(x)^5 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x)^3 + (a^2 + 2*a*b)*c
osh(x))*sinh(x))*sqrt(a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*(a + b)*cosh(x)^2 + 2*(3*
a*cosh(x)^2 + a + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a)*sqrt((a*cosh(
x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(a*cosh(x)^3 + (a + b)*cosh(x))
*sinh(x) + a)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 16*sqrt(2)*(a^2*cosh(x)^6 + 6*a^2*cosh(x)*sinh(x)
^5 + a^2*sinh(x)^6 + 3*a*b*cosh(x)^4 + 3*(5*a^2*cosh(x)^2 + a*b)*sinh(x)^4 - 3*a*b*cosh(x)^2 + 4*(5*a^2*cosh(x
)^3 + 3*a*b*cosh(x))*sinh(x)^3 + 3*(5*a^2*cosh(x)^4 + 6*a*b*cosh(x)^2 - a*b)*sinh(x)^2 - a^2 + 6*(a^2*cosh(x)^
5 + 2*a*b*cosh(x)^3 - a*b*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*
sinh(x) + sinh(x)^2)))/(a^5*cosh(x)^8 + 8*a^5*cosh(x)*sinh(x)^7 + a^5*sinh(x)^8 + 4*(a^5 + 2*a^4*b)*cosh(x)^6
+ 4*(7*a^5*cosh(x)^2 + a^5 + 2*a^4*b)*sinh(x)^6 + 8*(7*a^5*cosh(x)^3 + 3*(a^5 + 2*a^4*b)*cosh(x))*sinh(x)^5 +
a^5 + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x)^4 + 2*(35*a^5*cosh(x)^4 + 3*a^5 + 8*a^4*b + 8*a^3*b^2 + 30*(a^5
+ 2*a^4*b)*cosh(x)^2)*sinh(x)^4 + 8*(7*a^5*cosh(x)^5 + 10*(a^5 + 2*a^4*b)*cosh(x)^3 + (3*a^5 + 8*a^4*b + 8*a^3
*b^2)*cosh(x))*sinh(x)^3 + 4*(a^5 + 2*a^4*b)*cosh(x)^2 + 4*(7*a^5*cosh(x)^6 + a^5 + 2*a^4*b + 15*(a^5 + 2*a^4*
b)*cosh(x)^4 + 3*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x)^2)*sinh(x)^2 + 8*(a^5*cosh(x)^7 + 3*(a^5 + 2*a^4*b)*cos
h(x)^5 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x)^3 + (a^5 + 2*a^4*b)*cosh(x))*sinh(x)), -1/6*(3*(a^2*cosh(x)^8 +
8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 + 4*(a^2 + 2*a*b)*cosh(x)^6 + 4*(7*a^2*cosh(x)^2 + a^2 + 2*a*b)*sinh(
x)^6 + 8*(7*a^2*cosh(x)^3 + 3*(a^2 + 2*a*b)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 8*a*b + 8*b^2)*cosh(x)^4 + 2*(35*a
^2*cosh(x)^4 + 30*(a^2 + 2*a*b)*cosh(x)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(x)^4 + 8*(7*a^2*cosh(x)^5 + 10*(a^2 +
2*a*b)*cosh(x)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x))*sinh(x)^3 + 4*(a^2 + 2*a*b)*cosh(x)^2 + 4*(7*a^2*cosh(x)^6
+ 15*(a^2 + 2*a*b)*cosh(x)^4 + 3*(3*a^2 + 8*a*b + 8*b^2)*cosh(x)^2 + a^2 + 2*a*b)*sinh(x)^2 + a^2 + 8*(a^2*co
sh(x)^7 + 3*(a^2 + 2*a*b)*cosh(x)^5 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x)^3 + (a^2 + 2*a*b)*cosh(x))*sinh(x))*sqrt
(-a)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + a)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(
x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*b*cosh(x)^4 + 4*a*b*cosh(x)*sinh(x)^3 + a*b*si
nh(x)^4 - (a^2 + 3*a*b)*cosh(x)^2 + (6*a*b*cosh(x)^2 - a^2 - 3*a*b)*sinh(x)^2 - a^2 + 2*(2*a*b*cosh(x)^3 - (a^
2 + 3*a*b)*cosh(x))*sinh(x))) + 3*(a^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 + 4*(a^2 + 2*a*b)*c
osh(x)^6 + 4*(7*a^2*cosh(x)^2 + a^2 + 2*a*b)*sinh(x)^6 + 8*(7*a^2*cosh(x)^3 + 3*(a^2 + 2*a*b)*cosh(x))*sinh(x)
^5 + 2*(3*a^2 + 8*a*b + 8*b^2)*cosh(x)^4 + 2*(35*a^2*cosh(x)^4 + 30*(a^2 + 2*a*b)*cosh(x)^2 + 3*a^2 + 8*a*b +
8*b^2)*sinh(x)^4 + 8*(7*a^2*cosh(x)^5 + 10*(a^2 + 2*a*b)*cosh(x)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(x))*sinh(x)^
3 + 4*(a^2 + 2*a*b)*cosh(x)^2 + 4*(7*a^2*cosh(x)^6 + 15*(a^2 + 2*a*b)*cosh(x)^4 + 3*(3*a^2 + 8*a*b + 8*b^2)*co
sh(x)^2 + a^2 + 2*a*b)*sinh(x)^2 + a^2 + 8*(a^2*cosh(x)^7 + 3*(a^2 + 2*a*b)*cosh(x)^5 + (3*a^2 + 8*a*b + 8*b^2
)*cosh(x)^3 + (a^2 + 2*a*b)*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2
+ a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)
) + 8*sqrt(2)*(a^2*cosh(x)^6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 + 3*a*b*cosh(x)^4 + 3*(5*a^2*cosh(x)^2
+ a*b)*sinh(x)^4 - 3*a*b*cosh(x)^2 + 4*(5*a^2*cosh(x)^3 + 3*a*b*cosh(x))*sinh(x)^3 + 3*(5*a^2*cosh(x)^4 + 6*a*
b*cosh(x)^2 - a*b)*sinh(x)^2 - a^2 + 6*(a^2*cosh(x)^5 + 2*a*b*cosh(x)^3 - a*b*cosh(x))*sinh(x))*sqrt((a*cosh(x
)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(a^5*cosh(x)^8 + 8*a^5*cosh(x)*sinh
(x)^7 + a^5*sinh(x)^8 + 4*(a^5 + 2*a^4*b)*cosh(x)^6 + 4*(7*a^5*cosh(x)^2 + a^5 + 2*a^4*b)*sinh(x)^6 + 8*(7*a^5
*cosh(x)^3 + 3*(a^5 + 2*a^4*b)*cosh(x))*sinh(x)^5 + a^5 + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x)^4 + 2*(35*a^
5*cosh(x)^4 + 3*a^5 + 8*a^4*b + 8*a^3*b^2 + 30*(a^5 + 2*a^4*b)*cosh(x)^2)*sinh(x)^4 + 8*(7*a^5*cosh(x)^5 + 10*
(a^5 + 2*a^4*b)*cosh(x)^3 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x))*sinh(x)^3 + 4*(a^5 + 2*a^4*b)*cosh(x)^2 + 4
*(7*a^5*cosh(x)^6 + a^5 + 2*a^4*b + 15*(a^5 + 2*a^4*b)*cosh(x)^4 + 3*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x)^2)*
sinh(x)^2 + 8*(a^5*cosh(x)^7 + 3*(a^5 + 2*a^4*b)*cosh(x)^5 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(x)^3 + (a^5 +
2*a^4*b)*cosh(x))*sinh(x))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{4}{\left (x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)**4/(a+b*sech(x)**2)**(5/2),x)
[Out]
Integral(tanh(x)**4/(a + b*sech(x)**2)**(5/2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )^{4}}{{\left (b \operatorname{sech}\left (x\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(x)^4/(a+b*sech(x)^2)^(5/2),x, algorithm="giac")
[Out]
integrate(tanh(x)^4/(b*sech(x)^2 + a)^(5/2), x)