Integrand size = 17, antiderivative size = 51 \[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{a^{3/2}}-\frac {\tanh (x)}{a \sqrt {a+b-b \tanh ^2(x)}} \] Output:
arctanh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/a^(3/2)-tanh(x)/a/(a+b-b* tanh(x)^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(51)=102\).
Time = 0.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.51 \[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\frac {(a+2 b+a \cosh (2 x)) \text {sech}^2(x) \left (\text {arcsinh}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {a+b}}\right ) (a+2 b+a \cosh (2 x)) \text {sech}(x)-2 \sqrt {a} \sqrt {a+b} \sqrt {\frac {a+b+a \sinh ^2(x)}{a+b}} \tanh (x)\right )}{4 a^{3/2} \sqrt {a+b} \left (a+b \text {sech}^2(x)\right )^{3/2} \sqrt {\frac {a+b+a \sinh ^2(x)}{a+b}}} \] Input:
Integrate[Tanh[x]^2/(a + b*Sech[x]^2)^(3/2),x]
Output:
((a + 2*b + a*Cosh[2*x])*Sech[x]^2*(ArcSinh[(Sqrt[a]*Sinh[x])/Sqrt[a + b]] *(a + 2*b + a*Cosh[2*x])*Sech[x] - 2*Sqrt[a]*Sqrt[a + b]*Sqrt[(a + b + a*S inh[x]^2)/(a + b)]*Tanh[x]))/(4*a^(3/2)*Sqrt[a + b]*(a + b*Sech[x]^2)^(3/2 )*Sqrt[(a + b + a*Sinh[x]^2)/(a + b)])
Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 25, 4629, 25, 2075, 373, 291, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan (i x)^2}{\left (a+b \sec (i x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan (i x)^2}{\left (b \sec (i x)^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4629 |
\(\displaystyle -\int -\frac {\tanh ^2(x)}{\left (1-\tanh ^2(x)\right ) \left (a+b \left (1-\tanh ^2(x)\right )\right )^{3/2}}d\tanh (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\tanh ^2(x)}{\left (1-\tanh ^2(x)\right ) \left (a+b \left (1-\tanh ^2(x)\right )\right )^{3/2}}d\tanh (x)\) |
\(\Big \downarrow \) 2075 |
\(\displaystyle \int \frac {\tanh ^2(x)}{\left (1-\tanh ^2(x)\right ) \left (a-b \tanh ^2(x)+b\right )^{3/2}}d\tanh (x)\) |
\(\Big \downarrow \) 373 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {-b \tanh ^2(x)+a+b}}d\tanh (x)}{a}-\frac {\tanh (x)}{a \sqrt {a-b \tanh ^2(x)+b}}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {a \tanh ^2(x)}{-b \tanh ^2(x)+a+b}}d\frac {\tanh (x)}{\sqrt {-b \tanh ^2(x)+a+b}}}{a}-\frac {\tanh (x)}{a \sqrt {a-b \tanh ^2(x)+b}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{a^{3/2}}-\frac {\tanh (x)}{a \sqrt {a-b \tanh ^2(x)+b}}\) |
Input:
Int[Tanh[x]^2/(a + b*Sech[x]^2)^(3/2),x]
Output:
ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - Tanh[x]/(a* Sqrt[a + b - b*Tanh[x]^2])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
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]}, Sim p[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] && Inte gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
\[\int \frac {\tanh \left (x \right )^{2}}{\left (a +\operatorname {sech}\left (x \right )^{2} b \right )^{\frac {3}{2}}}d x\]
Input:
int(tanh(x)^2/(a+sech(x)^2*b)^(3/2),x)
Output:
int(tanh(x)^2/(a+sech(x)^2*b)^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (43) = 86\).
Time = 0.15 (sec) , antiderivative size = 1667, normalized size of antiderivative = 32.69 \[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^2/(a+b*sech(x)^2)^(3/2),x, algorithm="fricas")
Output:
[1/4*((a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*(a + 2*b)*cos h(x)^2 + 2*(3*a*cosh(x)^2 + a + 2*b)*sinh(x)^2 + 4*(a*cosh(x)^3 + (a + 2*b )*cosh(x))*sinh(x) + a)*sqrt(a)*log((a*b^2*cosh(x)^8 + 8*a*b^2*cosh(x)*sin h(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)*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))*sqr t(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sin h(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))/(cos h(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)) + (a*...
\[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(tanh(x)**2/(a+b*sech(x)**2)**(3/2),x)
Output:
Integral(tanh(x)**2/(a + b*sech(x)**2)**(3/2), x)
\[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\int { \frac {\tanh \left (x\right )^{2}}{{\left (b \operatorname {sech}\left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(tanh(x)^2/(a+b*sech(x)^2)^(3/2),x, algorithm="maxima")
Output:
integrate(tanh(x)^2/(b*sech(x)^2 + a)^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.69 \[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (a^{2} b + a b^{2}\right )} e^{\left (2 \, x\right )}}{a^{3} b + a^{2} b^{2}} - \frac {a^{2} b + a b^{2}}{a^{3} b + a^{2} b^{2}}}{\sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + 4 \, b e^{\left (2 \, x\right )} + a}} \] Input:
integrate(tanh(x)^2/(a+b*sech(x)^2)^(3/2),x, algorithm="giac")
Output:
-((a^2*b + a*b^2)*e^(2*x)/(a^3*b + a^2*b^2) - (a^2*b + a*b^2)/(a^3*b + a^2 *b^2))/sqrt(a*e^(4*x) + 2*a*e^(2*x) + 4*b*e^(2*x) + a)
Timed out. \[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^2}{{\left (a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2}} \,d x \] Input:
int(tanh(x)^2/(a + b/cosh(x)^2)^(3/2),x)
Output:
int(tanh(x)^2/(a + b/cosh(x)^2)^(3/2), x)
\[ \int \frac {\tanh ^2(x)}{\left (a+b \text {sech}^2(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (x \right )^{2} b +a}\, \tanh \left (x \right )^{2}}{\mathrm {sech}\left (x \right )^{4} b^{2}+2 \mathrm {sech}\left (x \right )^{2} a b +a^{2}}d x \] Input:
int(tanh(x)^2/(a+b*sech(x)^2)^(3/2),x)
Output:
int((sqrt(sech(x)**2*b + a)*tanh(x)**2)/(sech(x)**4*b**2 + 2*sech(x)**2*a* b + a**2),x)