Integrand size = 17, antiderivative size = 90 \[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=-\frac {(a+3 b) \arctan \left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{2 b^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b-b \tanh ^2(x)}}\right )}{\sqrt {a}}+\frac {\tanh (x) \sqrt {a+b-b \tanh ^2(x)}}{2 b} \]
-1/2*(a+3*b)*arctan(b^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/b^(3/2)+arcta nh(a^(1/2)*tanh(x)/(a+b-b*tanh(x)^2)^(1/2))/a^(1/2)+1/2*(a+b-b*tanh(x)^2)^ (1/2)*tanh(x)/b
Time = 0.45 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.88 \[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=\frac {\text {sech}(x) \left (2 \sqrt {2} b^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sinh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right ) \sqrt {a+2 b+a \cosh (2 x)}+\sqrt {a} \left (-\sqrt {2} (a+3 b) \arctan \left (\frac {\sqrt {2} \sqrt {b} \sinh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right ) \sqrt {a+2 b+a \cosh (2 x)}+\sqrt {b} (a+2 b+a \cosh (2 x)) \text {sech}(x) \tanh (x)\right )\right )}{4 \sqrt {a} b^{3/2} \sqrt {a+b \text {sech}^2(x)}} \]
(Sech[x]*(2*Sqrt[2]*b^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2*b + a*Cosh[2*x]] + Sqrt[a]*(-(Sqrt[2]*(a + 3*b )*ArcTan[(Sqrt[2]*Sqrt[b]*Sinh[x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Sqrt[a + 2 *b + a*Cosh[2*x]]) + Sqrt[b]*(a + 2*b + a*Cosh[2*x])*Sech[x]*Tanh[x])))/(4 *Sqrt[a]*b^(3/2)*Sqrt[a + b*Sech[x]^2])
Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 4629, 2075, 381, 398, 224, 216, 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.
| \(\displaystyle \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (i x)^4}{\sqrt {a+b \sec (i x)^2}}dx\) |
| \(\Big \downarrow \) 4629 |
| \(\displaystyle \int \frac {\tanh ^4(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {a+b \left (1-\tanh ^2(x)\right )}}d\tanh (x)\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \int \frac {\tanh ^4(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {a-b \tanh ^2(x)+b}}d\tanh (x)\) |
| \(\Big \downarrow \) 381 |
| \(\displaystyle \frac {\tanh (x) \sqrt {a-b \tanh ^2(x)+b}}{2 b}-\frac {\int \frac {-\left ((a+3 b) \tanh ^2(x)\right )+a+b}{\left (1-\tanh ^2(x)\right ) \sqrt {-b \tanh ^2(x)+a+b}}d\tanh (x)}{2 b}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\tanh (x) \sqrt {a-b \tanh ^2(x)+b}}{2 b}-\frac {(a+3 b) \int \frac {1}{\sqrt {-b \tanh ^2(x)+a+b}}d\tanh (x)-2 b \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {-b \tanh ^2(x)+a+b}}d\tanh (x)}{2 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\tanh (x) \sqrt {a-b \tanh ^2(x)+b}}{2 b}-\frac {(a+3 b) \int \frac {1}{\frac {b \tanh ^2(x)}{-b \tanh ^2(x)+a+b}+1}d\frac {\tanh (x)}{\sqrt {-b \tanh ^2(x)+a+b}}-2 b \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {-b \tanh ^2(x)+a+b}}d\tanh (x)}{2 b}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\tanh (x) \sqrt {a-b \tanh ^2(x)+b}}{2 b}-\frac {\frac {(a+3 b) \arctan \left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{\sqrt {b}}-2 b \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {-b \tanh ^2(x)+a+b}}d\tanh (x)}{2 b}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\tanh (x) \sqrt {a-b \tanh ^2(x)+b}}{2 b}-\frac {\frac {(a+3 b) \arctan \left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{\sqrt {b}}-2 b \int \frac {1}{1-\frac {a \tanh ^2(x)}{-b \tanh ^2(x)+a+b}}d\frac {\tanh (x)}{\sqrt {-b \tanh ^2(x)+a+b}}}{2 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\tanh (x) \sqrt {a-b \tanh ^2(x)+b}}{2 b}-\frac {\frac {(a+3 b) \arctan \left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{\sqrt {b}}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-b \tanh ^2(x)+b}}\right )}{\sqrt {a}}}{2 b}\) |
-1/2*(((a + 3*b)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/Sqrt [b] - (2*b*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/Sqrt[a])/ b + (Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/(2*b)
3.2.95.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 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^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q }, x] && NeQ[b*c - a*d, 0] && GtQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2 , p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , 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])
\[\int \frac {\tanh \left (x \right )^{4}}{\sqrt {a +\operatorname {sech}\left (x \right )^{2} b}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (72) = 144\).
Time = 0.40 (sec) , antiderivative size = 4569, normalized size of antiderivative = 50.77 \[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=\text {Too large to display} \]
[1/4*((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*b^2*cos h(x)^2 + 2*(3*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x)^3 + b^ 2*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)*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))*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)) - ((a^2 +...
\[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=\int \frac {\tanh ^{4}{\left (x \right )}}{\sqrt {a + b \operatorname {sech}^{2}{\left (x \right )}}}\, dx \]
\[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=\int { \frac {\tanh \left (x\right )^{4}}{\sqrt {b \operatorname {sech}\left (x\right )^{2} + a}} \,d x } \]
Exception generated. \[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\tanh ^4(x)}{\sqrt {a+b \text {sech}^2(x)}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^4}{\sqrt {a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \]