Integrand size = 25, antiderivative size = 275 \[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\sqrt {a} \sqrt {b} (7 a+b) \cosh (e+f x) E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}+\frac {(3 a+5 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {4 a \tanh (e+f x)}{3 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]
-1/3*(7*a+b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)^2/a))^(1/2)*(1+b*sinh(f*x+e)^ 2/a)^(1/2)*EllipticE(sinh(f*x+e)*b^(1/2)/a^(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/ 2),(1-a/b)^(1/2))*a^(1/2)*b^(1/2)/(a-b)^3/f/(a*cosh(f*x+e)^2/(a+b*sinh(f*x +e)^2))^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)+1/3*(3*a+5*b)*(1/(1+sinh(f*x+e)^2) )^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1 /2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/(a-b)^3/f/(sech(f *x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-4/3*a*tanh(f*x+e)/(a-b)^2/f/(a+b*sinh (f*x+e)^2)^(1/2)+1/3*sech(f*x+e)^2*tanh(f*x+e)/(a-b)/f/(a+b*sinh(f*x+e)^2) ^(1/2)
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.77 \[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {-2 i a (7 a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )-\frac {\left (8 a^2+21 a b-5 b^2+4 \left (4 a^2+3 a b+b^2\right ) \cosh (2 (e+f x))+b (7 a+b) \cosh (4 (e+f x))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{2 \sqrt {2}}}{6 (a-b)^3 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
((-2*I)*a*(7*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (8*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*El lipticF[I*(e + f*x), b/a] - ((8*a^2 + 21*a*b - 5*b^2 + 4*(4*a^2 + 3*a*b + b^2)*Cosh[2*(e + f*x)] + b*(7*a + b)*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Ta nh[e + f*x])/(2*Sqrt[2]))/(6*(a - b)^3*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x) ]])
Time = 0.46 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3675, 372, 27, 402, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (i e+i f x)^4}{\left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\sinh ^4(e+f x)}{\left (\sinh ^2(e+f x)+1\right )^{5/2} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int \frac {a \left (1-3 \sinh ^2(e+f x)\right )}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 (a-b)}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {a \int \frac {1-3 \sinh ^2(e+f x)}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 (a-b)}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {a \left (\frac {4 \sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int \frac {-4 b \sinh ^2(e+f x)+3 a+b}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}\right )}{3 (a-b)}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {a \left (\frac {4 \sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\frac {(3 a+5 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {b (7 a+b) \int \frac {\sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}}{a-b}\right )}{3 (a-b)}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {a \left (\frac {4 \sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\frac {(3 a+5 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {\sqrt {b} (7 a+b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}}{a-b}\right )}{3 (a-b)}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {a \left (\frac {4 \sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\frac {(3 a+5 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a (a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {\sqrt {b} (7 a+b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}}{a-b}\right )}{3 (a-b)}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(Sinh[e + f*x]/(3*(a - b)*(1 + Sinh[e + f*x]^2)^(3/2)*Sqrt[a + b*Sinh[e + f*x]^2]) - (a*((4*Sinh[e + f*x])/((a - b)*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2]) - (-((Sqrt[b]* (7*a + b)*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b]*Sqrt [1 + Sinh[e + f*x]^2])/(Sqrt[a]*(a - b)*Sqrt[(a*(1 + Sinh[e + f*x]^2))/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2])) + ((3*a + 5*b)*Ellipti cF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b) *Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f *x]^2))]))/(a - b)))/(3*(a - b))))/f
3.5.96.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 5.52 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.29
| method | result | size |
| default | \(-\frac {\left (7 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (4 \sqrt {-\frac {b}{a}}\, a^{2}-4 \sqrt {-\frac {b}{a}}\, a b \right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )-\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \left (3 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-2 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -\operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+7 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +\operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \cosh \left (f x +e \right )^{2}+\left (-\sqrt {-\frac {b}{a}}\, a^{2}+2 \sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{3} \left (a -b \right )^{3} \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(354\) |
| risch | \(\text {Expression too large to display}\) | \(1037296\) |
-1/3*((7*(-b/a)^(1/2)*a*b+(-b/a)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(4*( -b/a)^(1/2)*a^2-4*(-b/a)^(1/2)*a*b)*cosh(f*x+e)^2*sinh(f*x+e)-(cosh(f*x+e) ^2)^(1/2)*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(3*EllipticF(sinh(f*x+e)*(-b/a )^(1/2),(a/b)^(1/2))*a^2-2*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2)) *a*b-EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2+7*EllipticE(sinh( f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b+EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a /b)^(1/2))*b^2)*cosh(f*x+e)^2+(-(-b/a)^(1/2)*a^2+2*(-b/a)^(1/2)*a*b-(-b/a) ^(1/2)*b^2)*sinh(f*x+e))/(-b/a)^(1/2)/cosh(f*x+e)^3/(a-b)^3/(a+b*sinh(f*x+ e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 7400 vs. \(2 (279) = 558\).
Time = 0.26 (sec) , antiderivative size = 7400, normalized size of antiderivative = 26.91 \[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\tanh ^{4}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\tanh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\tanh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]