3.2.88 \(\int (a+b \text {sech}^2(x))^{3/2} \tanh ^2(x) \, dx\) [188]
Optimal. Leaf size=125 \[ -\frac {\left (3 a^2-6 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{8 \sqrt {b}}+a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )-\frac {1}{8} (5 a+b) \tanh (x) \sqrt {a+b-b \tanh ^2(x)}+\frac {1}{4} b \tanh ^3(x) \sqrt {a+b-b \tanh ^2(x)} \]
[Out]
a^(3/2)*arctanh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))-1/8*(3*a^2-6*a*b-b^2)*arctan(b^(1/2)*tanh(x)/(a+b-b*t
anh(x)^2)^(1/2))/b^(1/2)-1/8*(5*a+b)*(a+b-b*tanh(x)^2)^(1/2)*tanh(x)+1/4*b*(a+b-b*tanh(x)^2)^(1/2)*tanh(x)^3
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Rubi [A]
time = 0.24, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {4226, 2000,
488, 596, 537, 223, 209, 385, 212} \begin {gather*} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )-\frac {\left (3 a^2-6 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{8 \sqrt {b}}-\frac {1}{8} (5 a+b) \tanh (x) \sqrt {a-b \tanh ^2(x)+b}+\frac {1}{4} b \tanh ^3(x) \sqrt {a-b \tanh ^2(x)+b} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*Sech[x]^2)^(3/2)*Tanh[x]^2,x]
[Out]
-1/8*((3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/Sqrt[b] + a^(3/2)*ArcTanh[(Sq
rt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] - ((5*a + b)*Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/8 + (b*Tanh[x]^3*Sqr
t[a + b - b*Tanh[x]^2])/4
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
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 223
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 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 488
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 537
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]
Rule 596
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
Rule 2000
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 4226
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])
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^2(x) \, dx &=\text {Subst}\left (\int \frac {x^2 \left (a+b \left (1-x^2\right )\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\text {Subst}\left (\int \frac {x^2 \left (a+b-b x^2\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{4} b \tanh ^3(x) \sqrt {a+b-b \tanh ^2(x)}-\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (-(a+b) (4 a+b)+b (5 a+b) x^2\right )}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{8} (5 a+b) \tanh (x) \sqrt {a+b-b \tanh ^2(x)}+\frac {1}{4} b \tanh ^3(x) \sqrt {a+b-b \tanh ^2(x)}+\frac {\text {Subst}\left (\int \frac {b (a+b) (5 a+b)+b \left (3 a^2-6 a b-b^2\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )}{8 b}\\ &=-\frac {1}{8} (5 a+b) \tanh (x) \sqrt {a+b-b \tanh ^2(x)}+\frac {1}{4} b \tanh ^3(x) \sqrt {a+b-b \tanh ^2(x)}+a^2 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )+\frac {1}{8} \left (-3 a^2+6 a b+b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{8} (5 a+b) \tanh (x) \sqrt {a+b-b \tanh ^2(x)}+\frac {1}{4} b \tanh ^3(x) \sqrt {a+b-b \tanh ^2(x)}+a^2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )+\frac {1}{8} \left (-3 a^2+6 a b+b^2\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )\\ &=-\frac {\left (3 a^2-6 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{8 \sqrt {b}}+a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )-\frac {1}{8} (5 a+b) \tanh (x) \sqrt {a+b-b \tanh ^2(x)}+\frac {1}{4} b \tanh ^3(x) \sqrt {a+b-b \tanh ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 197, normalized size = 1.58 \begin {gather*} -\frac {\cosh ^3(x) \left (a+b \text {sech}^2(x)\right )^{3/2} \left (\sqrt {2} \left (3 a^2-6 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {b} \sinh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right )-8 \sqrt {2} a^{3/2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sinh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right )+(5 a-b) \sqrt {b} \sqrt {a+2 b+a \cosh (2 x)} \text {sech}(x) \tanh (x)+2 b^{3/2} \sqrt {a+2 b+a \cosh (2 x)} \text {sech}^3(x) \tanh (x)\right )}{4 \sqrt {b} (a+2 b+a \cosh (2 x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sech[x]^2)^(3/2)*Tanh[x]^2,x]
[Out]
-1/4*(Cosh[x]^3*(a + b*Sech[x]^2)^(3/2)*(Sqrt[2]*(3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a
+ 2*b + a*Cosh[2*x]]] - 8*Sqrt[2]*a^(3/2)*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x
]]] + (5*a - b)*Sqrt[b]*Sqrt[a + 2*b + a*Cosh[2*x]]*Sech[x]*Tanh[x] + 2*b^(3/2)*Sqrt[a + 2*b + a*Cosh[2*x]]*Se
ch[x]^3*Tanh[x]))/(Sqrt[b]*(a + 2*b + a*Cosh[2*x])^(3/2))
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Maple [F]
time = 0.92, size = 0, normalized size = 0.00 \[\int \left (a +b \mathrm {sech}\left (x \right )^{2}\right )^{\frac {3}{2}} \left (\tanh ^{2}\left (x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sech(x)^2)^(3/2)*tanh(x)^2,x)
[Out]
int((a+b*sech(x)^2)^(3/2)*tanh(x)^2,x)
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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((a+b*sech(x)^2)^(3/2)*tanh(x)^2,x, algorithm="maxima")
[Out]
integrate((b*sech(x)^2 + a)^(3/2)*tanh(x)^2, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1826 vs.
\(2 (103) = 206\).
time = 0.69, size = 8582, normalized size = 68.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(x)^2)^(3/2)*tanh(x)^2,x, algorithm="fricas")
[Out]
[1/16*(4*(a*b*cosh(x)^8 + 8*a*b*cosh(x)*sinh(x)^7 + a*b*sinh(x)^8 + 4*a*b*cosh(x)^6 + 4*(7*a*b*cosh(x)^2 + a*b
)*sinh(x)^6 + 6*a*b*cosh(x)^4 + 8*(7*a*b*cosh(x)^3 + 3*a*b*cosh(x))*sinh(x)^5 + 2*(35*a*b*cosh(x)^4 + 30*a*b*c
osh(x)^2 + 3*a*b)*sinh(x)^4 + 4*a*b*cosh(x)^2 + 8*(7*a*b*cosh(x)^5 + 10*a*b*cosh(x)^3 + 3*a*b*cosh(x))*sinh(x)
^3 + 4*(7*a*b*cosh(x)^6 + 15*a*b*cosh(x)^4 + 9*a*b*cosh(x)^2 + a*b)*sinh(x)^2 + a*b + 8*(a*b*cosh(x)^7 + 3*a*b
*cosh(x)^5 + 3*a*b*cosh(x)^3 + 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*c
osh(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)*sinh(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*cosh(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))*sq
rt(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*cos
h(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 - 6*a*b - b^2)*cosh(x)^8 + 8*(3*a^2 - 6*a*b - b^2)*cosh(x)*sinh(x)^
7 + (3*a^2 - 6*a*b - b^2)*sinh(x)^8 + 4*(3*a^2 - 6*a*b - b^2)*cosh(x)^6 + 4*(7*(3*a^2 - 6*a*b - b^2)*cosh(x)^2
+ 3*a^2 - 6*a*b - b^2)*sinh(x)^6 + 8*(7*(3*a^2 - 6*a*b - b^2)*cosh(x)^3 + 3*(3*a^2 - 6*a*b - b^2)*cosh(x))*si
nh(x)^5 + 6*(3*a^2 - 6*a*b - b^2)*cosh(x)^4 + 2*(35*(3*a^2 - 6*a*b - b^2)*cosh(x)^4 + 30*(3*a^2 - 6*a*b - b^2)
*cosh(x)^2 + 9*a^2 - 18*a*b - 3*b^2)*sinh(x)^4 + 8*(7*(3*a^2 - 6*a*b - b^2)*cosh(x)^5 + 10*(3*a^2 - 6*a*b - b^
2)*cosh(x)^3 + 3*(3*a^2 - 6*a*b - b^2)*cosh(x))*sinh(x)^3 + 4*(3*a^2 - 6*a*b - b^2)*cosh(x)^2 + 4*(7*(3*a^2 -
6*a*b - b^2)*cosh(x)^6 + 15*(3*a^2 - 6*a*b - b^2)*cosh(x)^4 + 9*(3*a^2 - 6*a*b - b^2)*cosh(x)^2 + 3*a^2 - 6*a*
b - b^2)*sinh(x)^2 + 3*a^2 - 6*a*b - b^2 + 8*((3*a^2 - 6*a*b - b^2)*cosh(x)^7 + 3*(3*a^2 - 6*a*b - b^2)*cosh(x
)^5 + 3*(3*a^2 - 6*a*b - b^2)*cosh(x)^3 + (3*a^2 - 6*a*b - b^2)*cosh(x))*sinh(x))*sqrt(-b)*log(-((a - b)*cosh(
x)^4 + 4*(a - b)*cosh(x)*sinh(x)^3 + (a - b)*sinh(x)^4 + 2*(a + 3*b)*cosh(x)^2 + 2*(3*(a - b)*cosh(x)^2 + a +
3*b)*sinh(x)^2 + 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-b)*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 - b)*cosh(x)^3 + (a + 3*b)*cosh(x))*sinh
(x) + a - b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(c
osh(x)^3 + cosh(x))*sinh(x) + 1)) + 4*(a*b*cosh(x)^8 + 8*a*b*cosh(x)*sinh(x)^7 + a*b*sinh(x)^8 + 4*a*b*cosh(x)
^6 + 4*(7*a*b*cosh(x)^2 + a*b)*sinh(x)^6 + 6*a*b*cosh(x)^4 + 8*(7*a*b*cosh(x)^3 + 3*a*b*cosh(x))*sinh(x)^5 + 2
*(35*a*b*cosh(x)^4 + 30*a*b*cosh(x)^2 + 3*a*b)*sinh(x)^4 + 4*a*b*cosh(x)^2 + 8*(7*a*b*cosh(x)^5 + 10*a*b*cosh(
x)^3 + 3*a*b*cosh(x))*sinh(x)^3 + 4*(7*a*b*cosh(x)^6 + 15*a*b*cosh(x)^4 + 9*a*b*cosh(x)^2 + a*b)*sinh(x)^2 + a
*b + 8*(a*b*cosh(x)^7 + 3*a*b*cosh(x)^5 + 3*a*b*cosh(x)^3 + a*b*cosh(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)*(cos
h(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)) - 2*sqrt(2)*((5*a*b - b^2)*cosh(x)^6 + 6*(5*a*b - b^2)*cosh(x)*sinh(x)^5 + (5*a*b - b^2)*sinh(x)
^6 + (5*a*b + 7*b^2)*cosh(x)^4 + (15*(5*a*b - b^2)*cosh(x)^2 + 5*a*b + 7*b^2)*sinh(x)^4 + 4*(5*(5*a*b - b^2)*c
osh(x)^3 + (5*a*b + 7*b^2)*cosh(x))*sinh(x)^3 - (5*a*b + 7*b^2)*cosh(x)^2 + (15*(5*a*b - b^2)*cosh(x)^4 + 6*(5
*a*b + 7*b^2)*cosh(x)^2 - 5*a*b - 7*b^2)*sinh(x)^2 - 5*a*b + b^2 + 2*(3*(5*a*b - b^2)*cosh(x)^5 + 2*(5*a*b + 7
*b^2)*cosh(x)^3 - (5*a*b + 7*b^2)*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)))/(b*cosh(x)^8 + 8*b*cosh(x)*sinh(x)^7 + b*sinh(x)^8 + 4*b*cosh(x)^6 + 4*(7*b*cos
h(x)^2 + b)*sinh(x)^6 + 8*(7*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^5 + 6*b*cosh(x)^4 + 2*(35*b*cosh(x)^4 + 30*b*c
osh(x)^2 + 3*b)*sinh(x)^4 + 8*(7*b*cosh(x)^5 + 10*b*cosh(x)^3 + 3*b*cosh(x))*sinh(x)^3 + 4*b*cosh(x)^2 + 4*(7*
b*cosh(x)^6 + 15*b*cosh(x)^4 + 9*b*cosh(x)^2 + ...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}} \tanh ^{2}{\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(x)**2)**(3/2)*tanh(x)**2,x)
[Out]
Integral((a + b*sech(x)**2)**(3/2)*tanh(x)**2, x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(x)^2)^(3/2)*tanh(x)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tanh}\left (x\right )}^2\,{\left (a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(x)^2*(a + b/cosh(x)^2)^(3/2),x)
[Out]
int(tanh(x)^2*(a + b/cosh(x)^2)^(3/2), x)
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