Integrand size = 13, antiderivative size = 224 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 a^5 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {8 a^3 b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2} \] Output:
-2*a^5*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)+8*a^3*b^ 2*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)-4*a^3*b*sech( x)/(a^2+b^2)^3+2/3*a*b*sech(x)^3/(a^2+b^2)^2-a^4*b*cosh(x)/(a^2+b^2)^3/(a+ b*sinh(x))+(a^2-b^2)*tanh(x)/(a^2+b^2)^2-(2*a^4-3*a^2*b^2-b^4)*tanh(x)/(a^ 2+b^2)^3-1/3*(a^2-b^2)*tanh(x)^3/(a^2+b^2)^2
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.64 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {6 a^3 \left (a^2-4 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-12 a^3 b \text {sech}(x)-\frac {3 a^4 b \cosh (x)}{a+b \sinh (x)}+\left (a^2+b^2\right ) \text {sech}^3(x) \left (2 a b+\left (a^2-b^2\right ) \sinh (x)\right )+\left (-4 a^4+9 a^2 b^2+b^4\right ) \tanh (x)}{3 \left (a^2+b^2\right )^3} \] Input:
Integrate[Tanh[x]^4/(a + b*Sinh[x])^2,x]
Output:
((6*a^3*(a^2 - 4*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^ 2 - b^2] - 12*a^3*b*Sech[x] - (3*a^4*b*Cosh[x])/(a + b*Sinh[x]) + (a^2 + b ^2)*Sech[x]^3*(2*a*b + (a^2 - b^2)*Sinh[x]) + (-4*a^4 + 9*a^2*b^2 + b^4)*T anh[x])/(3*(a^2 + b^2)^3)
Time = 0.73 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 3210, 2009}
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.
| \(\displaystyle \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (i x)^4}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3210 |
| \(\displaystyle \int \left (\frac {\text {sech}^4(x) \left (a^2 \left (1-\frac {b^2}{a^2}\right )-2 a b \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {a^4}{\left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {4 a^3 b^2}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (4 a^3 b \sinh (x)-2 a^4 \left (1-\frac {3 a^2 b^2+b^4}{2 a^4}\right )\right )}{\left (a^2+b^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a^2-b^2\right ) \tanh ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac {2 a b \text {sech}^3(x)}{3 \left (a^2+b^2\right )^2}-\frac {2 a^5 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {a^4 b \cosh (x)}{\left (a^2+b^2\right )^3 (a+b \sinh (x))}-\frac {\left (2 a^4-3 a^2 b^2-b^4\right ) \tanh (x)}{\left (a^2+b^2\right )^3}+\frac {8 a^3 b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {4 a^3 b \text {sech}(x)}{\left (a^2+b^2\right )^3}\) |
Input:
Int[Tanh[x]^4/(a + b*Sinh[x])^2,x]
Output:
(-2*a^5*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (8 *a^3*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - ( 4*a^3*b*Sech[x])/(a^2 + b^2)^3 + (2*a*b*Sech[x]^3)/(3*(a^2 + b^2)^2) - (a^ 4*b*Cosh[x])/((a^2 + b^2)^3*(a + b*Sinh[x])) + ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2 - ((2*a^4 - 3*a^2*b^2 - b^4)*Tanh[x])/(a^2 + b^2)^3 - ((a^2 - b^2 )*Tanh[x]^3)/(3*(a^2 + b^2)^2)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^m/ (1 - Sin[e + f*x]^2)^(p/2)), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, p/2]
Time = 3.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(-\frac {2 a^{3} \left (\frac {-b^{2} \tanh \left (\frac {x}{2}\right )-a b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}-4 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {2 \left (-a^{4}+3 a^{2} b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{5}+2 \left (-2 a^{3} b +2 a \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{4}+2 \left (-\frac {10}{3} a^{4}+6 a^{2} b^{2}+\frac {4}{3} b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}-16 a^{3} b \tanh \left (\frac {x}{2}\right )^{2}+2 \left (-a^{4}+3 a^{2} b^{2}\right ) \tanh \left (\frac {x}{2}\right )-\frac {20 a^{3} b}{3}+\frac {4 a \,b^{3}}{3}}{\left (a^{2}+b^{2}\right )^{3} \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}\) | \(245\) |
| risch | \(\frac {2 a^{5} {\mathrm e}^{7 x}-8 a^{3} b^{2} {\mathrm e}^{7 x}-14 a^{4} b \,{\mathrm e}^{6 x}-6 a^{2} b^{3} {\mathrm e}^{6 x}-2 b^{5} {\mathrm e}^{6 x}+14 a^{5} {\mathrm e}^{5 x}-\frac {44 a^{3} b^{2} {\mathrm e}^{5 x}}{3}+\frac {4 a \,b^{4} {\mathrm e}^{5 x}}{3}-\frac {82 a^{4} b \,{\mathrm e}^{4 x}}{3}+\frac {14 a^{2} b^{3} {\mathrm e}^{4 x}}{3}+2 b^{5} {\mathrm e}^{4 x}+14 a^{5} {\mathrm e}^{3 x}-\frac {64 a^{3} b^{2} {\mathrm e}^{3 x}}{3}-\frac {16 a \,b^{4} {\mathrm e}^{3 x}}{3}-\frac {70 a^{4} b \,{\mathrm e}^{2 x}}{3}+6 a^{2} b^{3} {\mathrm e}^{2 x}-\frac {2 b^{5} {\mathrm e}^{2 x}}{3}+\frac {22 a^{5} {\mathrm e}^{x}}{3}-4 a^{3} b^{2} {\mathrm e}^{x}-\frac {4 b^{4} {\mathrm e}^{x} a}{3}-\frac {14 a^{4} b}{3}+6 a^{2} b^{3}+\frac {2 b^{5}}{3}}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 x}+1\right )^{3} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {a^{5} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {4 a^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) b^{2}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {a^{5} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {4 a^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) b^{2}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\) | \(579\) |
Input:
int(tanh(x)^4/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-2*a^3/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-b^2*tanh(1/2*x)-a*b)/(tanh(1/2*x)^ 2*a-2*b*tanh(1/2*x)-a)-(a^2-4*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1 /2*x)-2*b)/(a^2+b^2)^(1/2)))+2/(a^2+b^2)^3*((-a^4+3*a^2*b^2)*tanh(1/2*x)^5 +(-2*a^3*b+2*a*b^3)*tanh(1/2*x)^4+(-10/3*a^4+6*a^2*b^2+4/3*b^4)*tanh(1/2*x )^3-8*a^3*b*tanh(1/2*x)^2+(-a^4+3*a^2*b^2)*tanh(1/2*x)-10/3*a^3*b+2/3*a*b^ 3)/(1+tanh(1/2*x)^2)^3
Leaf count of result is larger than twice the leaf count of optimal. 3534 vs. \(2 (212) = 424\).
Time = 0.16 (sec) , antiderivative size = 3534, normalized size of antiderivative = 15.78 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\tanh ^{4}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(tanh(x)**4/(a+b*sinh(x))**2,x)
Output:
Integral(tanh(x)**4/(a + b*sinh(x))**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (212) = 424\).
Time = 0.14 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.33 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{2} - 4 \, b^{2}\right )} a^{3} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (7 \, a^{4} b - 9 \, a^{2} b^{3} - b^{5} + {\left (11 \, a^{5} - 6 \, a^{3} b^{2} - 2 \, a b^{4}\right )} e^{\left (-x\right )} + {\left (35 \, a^{4} b - 9 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )} + {\left (21 \, a^{5} - 32 \, a^{3} b^{2} - 8 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + {\left (41 \, a^{4} b - 7 \, a^{2} b^{3} - 3 \, b^{5}\right )} e^{\left (-4 \, x\right )} + {\left (21 \, a^{5} - 22 \, a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (-5 \, x\right )} + 3 \, {\left (7 \, a^{4} b + 3 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-6 \, x\right )} + 3 \, {\left (a^{5} - 4 \, a^{3} b^{2}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-3 \, x\right )} + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-5 \, x\right )} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-6 \, x\right )} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-7 \, x\right )} - {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-8 \, x\right )}\right )}} \] Input:
integrate(tanh(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
(a^2 - 4*b^2)*a^3*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqr t(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 2/3 *(7*a^4*b - 9*a^2*b^3 - b^5 + (11*a^5 - 6*a^3*b^2 - 2*a*b^4)*e^(-x) + (35* a^4*b - 9*a^2*b^3 + b^5)*e^(-2*x) + (21*a^5 - 32*a^3*b^2 - 8*a*b^4)*e^(-3* x) + (41*a^4*b - 7*a^2*b^3 - 3*b^5)*e^(-4*x) + (21*a^5 - 22*a^3*b^2 + 2*a* b^4)*e^(-5*x) + 3*(7*a^4*b + 3*a^2*b^3 + b^5)*e^(-6*x) + 3*(a^5 - 4*a^3*b^ 2)*e^(-7*x))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 + 2*(a^7 + 3*a^5*b^2 + 3 *a^3*b^4 + a*b^6)*e^(-x) + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*e^(-2*x ) + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e^(-3*x) + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e^(-5*x) - 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*e^ (-6*x) + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*e^(-7*x) - (a^6*b + 3*a^4 *b^3 + 3*a^2*b^5 + b^7)*e^(-8*x))
Time = 0.15 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.30 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{5} - 4 \, a^{3} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{5} e^{x} - a^{4} b\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac {2 \, {\left (12 \, a^{3} b e^{\left (5 \, x\right )} - 6 \, a^{4} e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 3 \, b^{4} e^{\left (4 \, x\right )} + 16 \, a^{3} b e^{\left (3 \, x\right )} - 8 \, a b^{3} e^{\left (3 \, x\right )} - 6 \, a^{4} e^{\left (2 \, x\right )} + 18 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 12 \, a^{3} b e^{x} - 4 \, a^{4} + 9 \, a^{2} b^{2} + b^{4}\right )}}{3 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \] Input:
integrate(tanh(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
(a^5 - 4*a^3*b^2)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(a^5*e^x - a^4*b)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(b*e^(2* x) + 2*a*e^x - b)) - 2/3*(12*a^3*b*e^(5*x) - 6*a^4*e^(4*x) + 9*a^2*b^2*e^( 4*x) + 3*b^4*e^(4*x) + 16*a^3*b*e^(3*x) - 8*a*b^3*e^(3*x) - 6*a^4*e^(2*x) + 18*a^2*b^2*e^(2*x) + 12*a^3*b*e^x - 4*a^4 + 9*a^2*b^2 + b^4)/((a^6 + 3*a ^4*b^2 + 3*a^2*b^4 + b^6)*(e^(2*x) + 1)^3)
Time = 2.14 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.42 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {8\,\left (a^2-b^2\right )}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {16\,a\,b\,{\mathrm {e}}^x}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,\left (a^6+a^4\,b^2-a^2\,b^4-b^6\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}-\frac {16\,{\mathrm {e}}^x\,\left (a^5\,b+2\,a^3\,b^3+a\,b^5\right )}{3\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {2\,\left (a^6\,b^5+a^4\,b^7\right )}{b^3\,\left (a^2\,b+b^3\right )\,{\left (a^2+b^2\right )}^3}-\frac {2\,{\mathrm {e}}^x\,\left (a^7\,b^5+a^5\,b^7\right )}{b^4\,\left (a^2\,b+b^3\right )\,{\left (a^2+b^2\right )}^3}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {\frac {2\,\left (-2\,a^6+a^4\,b^2+4\,a^2\,b^4+b^6\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}+\frac {8\,{\mathrm {e}}^x\,\left (a^5\,b+a^3\,b^3\right )}{{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (a^5-4\,a^3\,b^2\right )}{b\,{\left (a^2+b^2\right )}^3}-\frac {2\,\left (a^5-4\,a^3\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{7/2}}\right )\,\left (a^5-4\,a^3\,b^2\right )}{{\left (a^2+b^2\right )}^{7/2}}+\frac {\ln \left (\frac {2\,\left (a^5-4\,a^3\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{7/2}}-\frac {2\,{\mathrm {e}}^x\,\left (a^5-4\,a^3\,b^2\right )}{b\,{\left (a^2+b^2\right )}^3}\right )\,\left (a^5-4\,a^3\,b^2\right )}{{\left (a^2+b^2\right )}^{7/2}} \] Input:
int(tanh(x)^4/(a + b*sinh(x))^2,x)
Output:
((8*(a^2 - b^2))/(3*(a^4 + b^4 + 2*a^2*b^2)) - (16*a*b*exp(x))/(3*(a^4 + b ^4 + 2*a^2*b^2)))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((4*(a^6 - b^ 6 - a^2*b^4 + a^4*b^2))/(a^4 + b^4 + 2*a^2*b^2)^2 - (16*exp(x)*(a*b^5 + a^ 5*b + 2*a^3*b^3))/(3*(a^4 + b^4 + 2*a^2*b^2)^2))/(2*exp(2*x) + exp(4*x) + 1) - ((2*(a^4*b^7 + a^6*b^5))/(b^3*(a^2*b + b^3)*(a^2 + b^2)^3) - (2*exp(x )*(a^5*b^7 + a^7*b^5))/(b^4*(a^2*b + b^3)*(a^2 + b^2)^3))/(2*a*exp(x) - b + b*exp(2*x)) - ((2*(b^6 - 2*a^6 + 4*a^2*b^4 + a^4*b^2))/(a^4 + b^4 + 2*a^ 2*b^2)^2 + (8*exp(x)*(a^5*b + a^3*b^3))/(a^4 + b^4 + 2*a^2*b^2)^2)/(exp(2* x) + 1) - (log(- (2*exp(x)*(a^5 - 4*a^3*b^2))/(b*(a^2 + b^2)^3) - (2*(a^5 - 4*a^3*b^2)*(b - a*exp(x)))/(b*(a^2 + b^2)^(7/2)))*(a^5 - 4*a^3*b^2))/(a^ 2 + b^2)^(7/2) + (log((2*(a^5 - 4*a^3*b^2)*(b - a*exp(x)))/(b*(a^2 + b^2)^ (7/2)) - (2*exp(x)*(a^5 - 4*a^3*b^2))/(b*(a^2 + b^2)^3))*(a^5 - 4*a^3*b^2) )/(a^2 + b^2)^(7/2)
Time = 0.17 (sec) , antiderivative size = 1445, normalized size of antiderivative = 6.45 \[ \int \frac {\tanh ^4(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:
int(tanh(x)^4/(a+b*sinh(x))^2,x)
Output:
(6*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a** 5*b*i - 24*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b* *2))*a**3*b**3*i + 12*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqr t(a**2 + b**2))*a**6*i - 48*e**(7*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a* i)/sqrt(a**2 + b**2))*a**4*b**2*i + 12*e**(6*x)*sqrt(a**2 + b**2)*atan((e* *x*b*i + a*i)/sqrt(a**2 + b**2))*a**5*b*i - 48*e**(6*x)*sqrt(a**2 + b**2)* atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**3*i + 36*e**(5*x)*sqrt(a* *2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**6*i - 144*e**(5*x)* sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**4*b**2*i + 3 6*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**6 *i - 144*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2 ))*a**4*b**2*i - 12*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt( a**2 + b**2))*a**5*b*i + 48*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a* i)/sqrt(a**2 + b**2))*a**3*b**3*i + 12*e**x*sqrt(a**2 + b**2)*atan((e**x*b *i + a*i)/sqrt(a**2 + b**2))*a**6*i - 48*e**x*sqrt(a**2 + b**2)*atan((e**x *b*i + a*i)/sqrt(a**2 + b**2))*a**4*b**2*i - 6*sqrt(a**2 + b**2)*atan((e** x*b*i + a*i)/sqrt(a**2 + b**2))*a**5*b*i + 24*sqrt(a**2 + b**2)*atan((e**x *b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**3*i - 3*e**(8*x)*a**6*b + 9*e**(8*x )*a**4*b**3 + 12*e**(8*x)*a**2*b**5 - 48*e**(6*x)*a**6*b - 42*e**(6*x)*a** 4*b**3 - 6*e**(6*x)*b**7 + 24*e**(5*x)*a**7 + 52*e**(5*x)*a**5*b**2 + 3...