Integrand size = 25, antiderivative size = 91 \[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=-\frac {3 \arctan (\sinh (e+f x)) \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)}{2 f}+\frac {3 \sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{2 f}-\frac {\sqrt {a \cosh ^2(e+f x)} \tanh ^3(e+f x)}{2 f} \]
-3/2*arctan(sinh(f*x+e))*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f+3/2*(a*cosh (f*x+e)^2)^(1/2)*tanh(f*x+e)/f-1/2*(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)^3/f
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.60 \[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=\frac {a (-3 \arctan (\sinh (e+f x)) \cosh (e+f x)+(2+\cosh (2 (e+f x))) \tanh (e+f x))}{2 f \sqrt {a \cosh ^2(e+f x)}} \]
(a*(-3*ArcTan[Sinh[e + f*x]]*Cosh[e + f*x] + (2 + Cosh[2*(e + f*x)])*Tanh[ e + f*x]))/(2*f*Sqrt[a*Cosh[e + f*x]^2])
Time = 0.40 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3655, 3042, 3686, 3042, 3072, 252, 262, 216}
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.
| \(\displaystyle \int \tanh ^4(e+f x) \sqrt {a \sinh ^2(e+f x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (i e+i f x)^4 \sqrt {a-a \sin (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 3655 |
| \(\displaystyle \int \tanh ^4(e+f x) \sqrt {a \cosh ^2(e+f x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2}}{\tan \left (i e+i f x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int \sinh (e+f x) \tanh ^3(e+f x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int \sin (i e+i f x) \tan (i e+i f x)^3dx\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle \frac {\text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int \frac {\sinh ^4(e+f x)}{\left (\sinh ^2(e+f x)+1\right )^2}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \left (\frac {3}{2} \int \frac {\sinh ^2(e+f x)}{\sinh ^2(e+f x)+1}d\sinh (e+f x)-\frac {\sinh ^3(e+f x)}{2 \left (\sinh ^2(e+f x)+1\right )}\right )}{f}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \left (\frac {3}{2} \left (\sinh (e+f x)-\int \frac {1}{\sinh ^2(e+f x)+1}d\sinh (e+f x)\right )-\frac {\sinh ^3(e+f x)}{2 \left (\sinh ^2(e+f x)+1\right )}\right )}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \left (\frac {3}{2} (\sinh (e+f x)-\arctan (\sinh (e+f x)))-\frac {\sinh ^3(e+f x)}{2 \left (\sinh ^2(e+f x)+1\right )}\right )}{f}\) |
(Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x]*((3*(-ArcTan[Sinh[e + f*x]] + Sinh[ e + f*x]))/2 - Sinh[e + f*x]^3/(2*(1 + Sinh[e + f*x]^2))))/f
3.5.32.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.55
| method | result | size |
| default | \(\frac {\sqrt {a \sinh \left (f x +e \right )^{2}}\, \left (2 \sqrt {a \sinh \left (f x +e \right )^{2}}\, \cosh \left (f x +e \right )^{2} \sqrt {-a}+3 \ln \left (\frac {2 \sqrt {-a}\, \sqrt {a \sinh \left (f x +e \right )^{2}}-2 a}{\cosh \left (f x +e \right )}\right ) a \cosh \left (f x +e \right )^{2}+\sqrt {-a}\, \sqrt {a \sinh \left (f x +e \right )^{2}}\right )}{2 \cosh \left (f x +e \right ) \sqrt {-a}\, \sinh \left (f x +e \right ) \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(141\) |
| risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left (3 i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) {\mathrm e}^{5 f x +5 e}-3 i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) {\mathrm e}^{5 f x +5 e}+6 i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) {\mathrm e}^{3 f x +3 e}-6 i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) {\mathrm e}^{3 f x +3 e}+{\mathrm e}^{6 f x +6 e}+3 i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) {\mathrm e}^{f x +e}-3 i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) {\mathrm e}^{f x +e}+3 \,{\mathrm e}^{4 f x +4 e}-3 \,{\mathrm e}^{2 f x +2 e}-1\right )}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )^{3}}\) | \(222\) |
1/2/cosh(f*x+e)*(a*sinh(f*x+e)^2)^(1/2)*(2*(a*sinh(f*x+e)^2)^(1/2)*cosh(f* x+e)^2*(-a)^(1/2)+3*ln(2/cosh(f*x+e)*((-a)^(1/2)*(a*sinh(f*x+e)^2)^(1/2)-a ))*a*cosh(f*x+e)^2+(-a)^(1/2)*(a*sinh(f*x+e)^2)^(1/2))/(-a)^(1/2)/sinh(f*x +e)/(a*cosh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (79) = 158\).
Time = 0.26 (sec) , antiderivative size = 742, normalized size of antiderivative = 8.15 \[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=\frac {{\left (6 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{5} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{6} + 3 \, {\left (5 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, \cosh \left (f x + e\right )^{3} + 3 \, \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 3 \, {\left (5 \, \cosh \left (f x + e\right )^{4} + 6 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 6 \, {\left (\cosh \left (f x + e\right )^{5} + 2 \, \cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) - 6 \, {\left (5 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{5} + 2 \, {\left (5 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (5 \, \cosh \left (f x + e\right )^{3} + 3 \, \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + {\left (5 \, \cosh \left (f x + e\right )^{4} + 6 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (\cosh \left (f x + e\right )^{5} + 2 \, \cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )}\right )} \arctan \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + {\left (\cosh \left (f x + e\right )^{6} + 3 \, \cosh \left (f x + e\right )^{4} - 3 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )}\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{2 \, {\left (f \cosh \left (f x + e\right )^{5} + {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )^{5} + 5 \, {\left (f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{4} + 2 \, f \cosh \left (f x + e\right )^{3} + 2 \, {\left (5 \, f \cosh \left (f x + e\right )^{2} + {\left (5 \, f \cosh \left (f x + e\right )^{2} + f\right )} e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (5 \, f \cosh \left (f x + e\right )^{3} + 3 \, f \cosh \left (f x + e\right ) + {\left (5 \, f \cosh \left (f x + e\right )^{3} + 3 \, f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + f \cosh \left (f x + e\right ) + {\left (f \cosh \left (f x + e\right )^{5} + 2 \, f \cosh \left (f x + e\right )^{3} + f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (5 \, f \cosh \left (f x + e\right )^{4} + 6 \, f \cosh \left (f x + e\right )^{2} + {\left (5 \, f \cosh \left (f x + e\right )^{4} + 6 \, f \cosh \left (f x + e\right )^{2} + f\right )} e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )\right )}} \]
1/2*(6*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^5 + e^(f*x + e)*sinh(f*x + e)^6 + 3*(5*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^4 + 4*(5*cosh(f *x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^3 + 3*(5*cosh(f*x + e)^4 + 6*cosh(f*x + e)^2 - 1)*e^(f*x + e)*sinh(f*x + e)^2 + 6*(cosh(f*x + e)^5 + 2*cosh(f*x + e)^3 - cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) - 6*( 5*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^4 + e^(f*x + e)*sinh(f*x + e)^5 + 2*(5*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^3 + 2*(5*cosh(f*x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + (5*cosh(f*x + e)^4 + 6*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e) + (cosh(f*x + e)^5 + 2*c osh(f*x + e)^3 + cosh(f*x + e))*e^(f*x + e))*arctan(cosh(f*x + e) + sinh(f *x + e)) + (cosh(f*x + e)^6 + 3*cosh(f*x + e)^4 - 3*cosh(f*x + e)^2 - 1)*e ^(f*x + e))*sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e) /(f*cosh(f*x + e)^5 + (f*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^5 + 5*(f*cosh( f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e))*sinh(f*x + e)^4 + 2*f*cosh(f*x + e)^3 + 2*(5*f*cosh(f*x + e)^2 + (5*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2* e) + f)*sinh(f*x + e)^3 + 2*(5*f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e) + (5* f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^2 + f*cosh(f*x + e) + (f*cosh(f*x + e)^5 + 2*f*cosh(f*x + e)^3 + f*cosh(f*x + e))*e^(2*f*x + 2*e) + (5*f*cosh(f*x + e)^4 + 6*f*cosh(f*x + e)^2 + (5*f*co sh(f*x + e)^4 + 6*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x ...
\[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=\int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \tanh ^{4}{\left (e + f x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 387, normalized size of antiderivative = 4.25 \[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=\frac {15 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right )}{8 \, f} + \frac {3 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) + \frac {5 \, \sqrt {a} e^{\left (-f x - e\right )} + 3 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )}}{2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1}}{4 \, f} + \frac {3 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) - \frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} + 5 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )}}{2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1}}{4 \, f} - \frac {3 \, {\left (\sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) - \frac {\sqrt {a} e^{\left (-f x - e\right )} - \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )}}{2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1}\right )}}{8 \, f} + \frac {25 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 15 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 8 \, \sqrt {a}}{16 \, f {\left (e^{\left (-f x - e\right )} + 2 \, e^{\left (-3 \, f x - 3 \, e\right )} + e^{\left (-5 \, f x - 5 \, e\right )}\right )}} - \frac {15 \, \sqrt {a} e^{\left (-f x - e\right )} + 25 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 8 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{16 \, f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}} \]
15/8*sqrt(a)*arctan(e^(-f*x - e))/f + 1/4*(3*sqrt(a)*arctan(e^(-f*x - e)) + (5*sqrt(a)*e^(-f*x - e) + 3*sqrt(a)*e^(-3*f*x - 3*e))/(2*e^(-2*f*x - 2*e ) + e^(-4*f*x - 4*e) + 1))/f + 1/4*(3*sqrt(a)*arctan(e^(-f*x - e)) - (3*sq rt(a)*e^(-f*x - e) + 5*sqrt(a)*e^(-3*f*x - 3*e))/(2*e^(-2*f*x - 2*e) + e^( -4*f*x - 4*e) + 1))/f - 3/8*(sqrt(a)*arctan(e^(-f*x - e)) - (sqrt(a)*e^(-f *x - e) - sqrt(a)*e^(-3*f*x - 3*e))/(2*e^(-2*f*x - 2*e) + e^(-4*f*x - 4*e) + 1))/f + 1/16*(25*sqrt(a)*e^(-2*f*x - 2*e) + 15*sqrt(a)*e^(-4*f*x - 4*e) + 8*sqrt(a))/(f*(e^(-f*x - e) + 2*e^(-3*f*x - 3*e) + e^(-5*f*x - 5*e))) - 1/16*(15*sqrt(a)*e^(-f*x - e) + 25*sqrt(a)*e^(-3*f*x - 3*e) + 8*sqrt(a)*e ^(-5*f*x - 5*e))/(f*(2*e^(-2*f*x - 2*e) + e^(-4*f*x - 4*e) + 1))
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.10 \[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=-\frac {{\left (3 \, \pi - \frac {4 \, {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}}{{\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{2} + 4} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )} e^{\left (-f x - e\right )}\right ) - 2 \, e^{\left (f x + e\right )} + 2 \, e^{\left (-f x - e\right )}\right )} \sqrt {a}}{4 \, f} \]
-1/4*(3*pi - 4*(e^(f*x + e) - e^(-f*x - e))/((e^(f*x + e) - e^(-f*x - e))^ 2 + 4) + 6*arctan(1/2*(e^(2*f*x + 2*e) - 1)*e^(-f*x - e)) - 2*e^(f*x + e) + 2*e^(-f*x - e))*sqrt(a)/f
Timed out. \[ \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^4(e+f x) \, dx=\int {\mathrm {tanh}\left (e+f\,x\right )}^4\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]