Integrand size = 25, antiderivative size = 232 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\left (8 a^2+24 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{8 (a-b)^{9/2} f}+\frac {8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(8 a-b) \text {sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt {a+b \sinh ^2(e+f x)}} \]
-1/8*(8*a^2+24*a*b+3*b^2)*arctanh((a+b*sinh(f*x+e)^2)^(1/2)/(a-b)^(1/2))/( a-b)^(9/2)/f+1/24*(8*a^2+24*a*b+3*b^2)/(a-b)^3/f/(a+b*sinh(f*x+e)^2)^(3/2) +1/8*(8*a-b)*sech(f*x+e)^2/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(3/2)-1/4*sech(f* x+e)^4/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)+1/8*(8*a^2+24*a*b+3*b^2)/(a-b)^4/ f/(a+b*sinh(f*x+e)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.41 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.49 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 \left (8 a^2+24 a b+3 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \sinh ^2(e+f x)}{a-b}\right )+3 (a-b) (4 a+3 b+(8 a-b) \cosh (2 (e+f x))) \text {sech}^4(e+f x)}{48 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
(2*(8*a^2 + 24*a*b + 3*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sinh[e + f*x]^2)/(a - b)] + 3*(a - b)*(4*a + 3*b + (8*a - b)*Cosh[2*(e + f*x)])* Sech[e + f*x]^4)/(48*(a - b)^3*f*(a + b*Sinh[e + f*x]^2)^(3/2))
Time = 0.40 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 26, 3673, 100, 27, 87, 61, 61, 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.
\(\displaystyle \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i e+i f x)^5}{\left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i e+i f x)^5}{\left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {\int \frac {\sinh ^4(e+f x)}{\left (\sinh ^2(e+f x)+1\right )^3 \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\frac {\int -\frac {-4 (a-b) \sinh ^2(e+f x)+4 a+3 b}{2 \left (\sinh ^2(e+f x)+1\right )^2 \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh ^2(e+f x)}{2 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\int \frac {-4 (a-b) \sinh ^2(e+f x)+4 a+3 b}{\left (\sinh ^2(e+f x)+1\right )^2 \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh ^2(e+f x)}{4 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+24 a b+3 b^2\right ) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh ^2(e+f x)}{2 (a-b)}-\frac {8 a-b}{(a-b) \left (\sinh ^2(e+f x)+1\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{4 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+24 a b+3 b^2\right ) \left (\frac {\int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh ^2(e+f x)}{a-b}+\frac {2}{3 (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{2 (a-b)}-\frac {8 a-b}{(a-b) \left (\sinh ^2(e+f x)+1\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{4 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+24 a b+3 b^2\right ) \left (\frac {\frac {\int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh ^2(e+f x)}{a-b}+\frac {2}{(a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{a-b}+\frac {2}{3 (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{2 (a-b)}-\frac {8 a-b}{(a-b) \left (\sinh ^2(e+f x)+1\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{4 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+24 a b+3 b^2\right ) \left (\frac {\frac {2 \int \frac {1}{\frac {\sinh ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \sinh ^2(e+f x)+a}}{b (a-b)}+\frac {2}{(a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{a-b}+\frac {2}{3 (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{2 (a-b)}-\frac {8 a-b}{(a-b) \left (\sinh ^2(e+f x)+1\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{4 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+24 a b+3 b^2\right ) \left (\frac {\frac {2}{(a-b) \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}}{a-b}+\frac {2}{3 (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{2 (a-b)}-\frac {8 a-b}{(a-b) \left (\sinh ^2(e+f x)+1\right ) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{4 (a-b)}-\frac {1}{2 (a-b) \left (\sinh ^2(e+f x)+1\right )^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{2 f}\) |
(-1/2*1/((a - b)*(1 + Sinh[e + f*x]^2)^2*(a + b*Sinh[e + f*x]^2)^(3/2)) - (-((8*a - b)/((a - b)*(1 + Sinh[e + f*x]^2)*(a + b*Sinh[e + f*x]^2)^(3/2)) ) - ((8*a^2 + 24*a*b + 3*b^2)*(2/(3*(a - b)*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((-2*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(a - b)^(3/2) + 2/((a - b)*Sqrt[a + b*Sinh[e + f*x]^2]))/(a - b)))/(2*(a - b)))/(4*(a - b) ))/(2*f)
3.6.1.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 1.73 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\operatorname {`\,int/indef0`\,}\left (-\frac {\sinh \left (f x +e \right )^{5} \left (b^{2} \sinh \left (f x +e \right )^{4}+2 \sinh \left (f x +e \right )^{2} a b +a^{2}\right ) \cosh \left (f x +e \right )^{4}}{\left (-b^{4} \cosh \left (f x +e \right )^{18}+\left (-4 a \,b^{3}+4 b^{4}\right ) \cosh \left (f x +e \right )^{16}+\left (-6 a^{2} b^{2}+12 a \,b^{3}-6 b^{4}\right ) \cosh \left (f x +e \right )^{14}+\left (-4 a^{3} b +12 a^{2} b^{2}-12 a \,b^{3}+4 b^{4}\right ) \cosh \left (f x +e \right )^{12}+\left (-a^{4}+4 a^{3} b -6 a^{2} b^{2}+4 a \,b^{3}-b^{4}\right ) \cosh \left (f x +e \right )^{10}\right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\) | \(213\) |
risch | \(\text {Expression too large to display}\) | \(2629521\) |
`int/indef0`(-sinh(f*x+e)^5*(b^2*sinh(f*x+e)^4+2*sinh(f*x+e)^2*a*b+a^2)*co sh(f*x+e)^4/(-b^4*cosh(f*x+e)^18+(-4*a*b^3+4*b^4)*cosh(f*x+e)^16+(-6*a^2*b ^2+12*a*b^3-6*b^4)*cosh(f*x+e)^14+(-4*a^3*b+12*a^2*b^2-12*a*b^3+4*b^4)*cos h(f*x+e)^12+(-a^4+4*a^3*b-6*a^2*b^2+4*a*b^3-b^4)*cosh(f*x+e)^10)/(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. 10051 vs. \(2 (208) = 416\).
Time = 1.78 (sec) , antiderivative size = 20298, normalized size of antiderivative = 87.49 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\tanh ^{5}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\tanh \left (f x + e\right )^{5}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 3690 vs. \(2 (208) = 416\).
Time = 24.12 (sec) , antiderivative size = 3690, normalized size of antiderivative = 15.91 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
2/3*((3*(a^22*b^3*e^(21*e) - 14*a^21*b^4*e^(21*e) + 88*a^20*b^5*e^(21*e) - 320*a^19*b^6*e^(21*e) + 700*a^18*b^7*e^(21*e) - 728*a^17*b^8*e^(21*e) - 7 28*a^16*b^9*e^(21*e) + 4576*a^15*b^10*e^(21*e) - 10010*a^14*b^11*e^(21*e) + 14300*a^13*b^12*e^(21*e) - 14872*a^12*b^13*e^(21*e) + 11648*a^11*b^14*e^ (21*e) - 6916*a^10*b^15*e^(21*e) + 3080*a^9*b^16*e^(21*e) - 1000*a^8*b^17* e^(21*e) + 224*a^7*b^18*e^(21*e) - 31*a^6*b^19*e^(21*e) + 2*a^5*b^20*e^(21 *e))*e^(2*f*x)/(a^24*b^2*e^(16*e) - 20*a^23*b^3*e^(16*e) + 190*a^22*b^4*e^ (16*e) - 1140*a^21*b^5*e^(16*e) + 4845*a^20*b^6*e^(16*e) - 15504*a^19*b^7* e^(16*e) + 38760*a^18*b^8*e^(16*e) - 77520*a^17*b^9*e^(16*e) + 125970*a^16 *b^10*e^(16*e) - 167960*a^15*b^11*e^(16*e) + 184756*a^14*b^12*e^(16*e) - 1 67960*a^13*b^13*e^(16*e) + 125970*a^12*b^14*e^(16*e) - 77520*a^11*b^15*e^( 16*e) + 38760*a^10*b^16*e^(16*e) - 15504*a^9*b^17*e^(16*e) + 4845*a^8*b^18 *e^(16*e) - 1140*a^7*b^19*e^(16*e) + 190*a^6*b^20*e^(16*e) - 20*a^5*b^21*e ^(16*e) + a^4*b^22*e^(16*e)) + 2*(8*a^23*b^2*e^(19*e) - 121*a^22*b^3*e^(19 *e) + 842*a^21*b^4*e^(19*e) - 3544*a^20*b^5*e^(19*e) + 9920*a^19*b^6*e^(19 *e) - 18844*a^18*b^7*e^(19*e) + 22568*a^17*b^8*e^(19*e) - 9256*a^16*b^9*e^ (19*e) - 25168*a^15*b^10*e^(19*e) + 67210*a^14*b^11*e^(19*e) - 93236*a^13* b^12*e^(19*e) + 89752*a^12*b^13*e^(19*e) - 64064*a^11*b^14*e^(19*e) + 3446 8*a^10*b^15*e^(19*e) - 13880*a^9*b^16*e^(19*e) + 4072*a^8*b^17*e^(19*e) - 824*a^7*b^18*e^(19*e) + 103*a^6*b^19*e^(19*e) - 6*a^5*b^20*e^(19*e))/(a...
Timed out. \[ \int \frac {\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \]