Integrand size = 25, antiderivative size = 167 \[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=-\frac {\left (8 a^2+8 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{3/2} f}+\frac {\left (8 a^2+8 a b-b^2\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^2 f}-\frac {(8 a-b) \text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 a f} \]
-1/8*(8*a^2+8*a*b-b^2)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/ f-1/8*(8*a-b)*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(3/2)/a^2/f-1/4*csch(f*x+e )^4*(a+b*sinh(f*x+e)^2)^(3/2)/a/f+1/8*(8*a^2+8*a*b-b^2)*(a+b*sinh(f*x+e)^2 )^(1/2)/a^2/f
Time = 0.41 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.61 \[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {\left (-8 a^2-8 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )-\sqrt {a} \left (-8 a+(8 a+b) \text {csch}^2(e+f x)+2 a \text {csch}^4(e+f x)\right ) \sqrt {a+b \sinh ^2(e+f x)}}{8 a^{3/2} f} \]
((-8*a^2 - 8*a*b + b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] - Sqr t[a]*(-8*a + (8*a + b)*Csch[e + f*x]^2 + 2*a*Csch[e + f*x]^4)*Sqrt[a + b*S inh[e + f*x]^2])/(8*a^(3/2)*f)
Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 26, 3673, 100, 27, 87, 60, 73, 221}
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.
\(\displaystyle \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sqrt {a-b \sin (i e+i f x)^2}}{\tan (i e+i f x)^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sqrt {a-b \sin (i e+i f x)^2}}{\tan (i e+i f x)^5}dx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {\int \text {csch}^6(e+f x) \left (\sinh ^2(e+f x)+1\right )^2 \sqrt {b \sinh ^2(e+f x)+a}d\sinh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\frac {\int \frac {1}{2} \text {csch}^4(e+f x) \left (4 a \sinh ^2(e+f x)+8 a-b\right ) \sqrt {b \sinh ^2(e+f x)+a}d\sinh ^2(e+f x)}{2 a}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a}}{2 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \text {csch}^4(e+f x) \left (4 a \sinh ^2(e+f x)+8 a-b\right ) \sqrt {b \sinh ^2(e+f x)+a}d\sinh ^2(e+f x)}{4 a}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a}}{2 f}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\frac {\frac {\left (8 a^2+8 a b-b^2\right ) \int \text {csch}^2(e+f x) \sqrt {b \sinh ^2(e+f x)+a}d\sinh ^2(e+f x)}{2 a}-\frac {(8 a-b) \text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{a}}{4 a}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a}}{2 f}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\frac {\frac {\left (8 a^2+8 a b-b^2\right ) \left (a \int \frac {\text {csch}^2(e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh ^2(e+f x)+2 \sqrt {a+b \sinh ^2(e+f x)}\right )}{2 a}-\frac {(8 a-b) \text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{a}}{4 a}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {\frac {\left (8 a^2+8 a b-b^2\right ) \left (\frac {2 a \int \frac {1}{\frac {\sinh ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \sinh ^2(e+f x)+a}}{b}+2 \sqrt {a+b \sinh ^2(e+f x)}\right )}{2 a}-\frac {(8 a-b) \text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{a}}{4 a}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a}}{2 f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\frac {\left (8 a^2+8 a b-b^2\right ) \left (2 \sqrt {a+b \sinh ^2(e+f x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )\right )}{2 a}-\frac {(8 a-b) \text {csch}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{a}}{4 a}-\frac {\text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{2 a}}{2 f}\) |
(-1/2*(Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2))/a + (-(((8*a - b)*Cs ch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2))/a) + ((8*a^2 + 8*a*b - b^2)*( -2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]] + 2*Sqrt[a + b*Sin h[e + f*x]^2]))/(2*a))/(4*a))/(2*f)
3.5.62.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1)/2)/(2*f) Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m + 1 )/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && Integ erQ[(m - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {\operatorname {`\,int/indef0`\,}\left (\frac {\cosh \left (f x +e \right )^{4} \left (a -b +b \cosh \left (f x +e \right )^{2}\right )}{\sinh \left (f x +e \right ) \left (\cosh \left (f x +e \right )^{4}-2 \cosh \left (f x +e \right )^{2}+1\right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\) | \(80\) |
`int/indef0`(1/sinh(f*x+e)/(cosh(f*x+e)^4-2*cosh(f*x+e)^2+1)*cosh(f*x+e)^4 *(a-b+b*cosh(f*x+e)^2)/(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/f
Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (147) = 294\).
Time = 0.60 (sec) , antiderivative size = 3880, normalized size of antiderivative = 23.23 \[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Too large to display} \]
[-1/16*(((8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^9 + 9*(8*a^2 + 8*a*b - b^2)*c osh(f*x + e)*sinh(f*x + e)^8 + (8*a^2 + 8*a*b - b^2)*sinh(f*x + e)^9 - 4*( 8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^7 + 4*(9*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^2 - 8*a^2 - 8*a*b + b^2)*sinh(f*x + e)^7 + 28*(3*(8*a^2 + 8*a*b - b^ 2)*cosh(f*x + e)^3 - (8*a^2 + 8*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^6 + 6*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^5 + 6*(21*(8*a^2 + 8*a*b - b^2)*co sh(f*x + e)^4 - 14*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^2 + 8*a^2 + 8*a*b - b^2)*sinh(f*x + e)^5 + 2*(63*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^5 - 70*( 8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^3 + 15*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^4 - 4*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^3 + 4*(21*(8* a^2 + 8*a*b - b^2)*cosh(f*x + e)^6 - 35*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e )^4 + 15*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^2 - 8*a^2 - 8*a*b + b^2)*sinh (f*x + e)^3 + 12*(3*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^7 - 7*(8*a^2 + 8*a *b - b^2)*cosh(f*x + e)^5 + 5*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^3 - (8*a ^2 + 8*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^2 + (8*a^2 + 8*a*b - b^2)*c osh(f*x + e) + (9*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^8 - 28*(8*a^2 + 8*a* b - b^2)*cosh(f*x + e)^6 + 30*(8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^4 - 12*( 8*a^2 + 8*a*b - b^2)*cosh(f*x + e)^2 + 8*a^2 + 8*a*b - b^2)*sinh(f*x + e)) *sqrt(a)*log((b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*si nh(f*x + e)^4 + 2*(4*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + ...
\[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \coth ^{5}{\left (e + f x \right )}\, dx \]
\[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int { \sqrt {b \sinh \left (f x + e\right )^{2} + a} \coth \left (f x + e\right )^{5} \,d x } \]
Exception generated. \[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage3:=type(sage2):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \coth ^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int {\mathrm {coth}\left (e+f\,x\right )}^5\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]