Integrand size = 23, antiderivative size = 102 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {x}{b^2}-\frac {\sqrt {a} (2 a-3 b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 (a-b)^{3/2} b^2 d}-\frac {a \tanh (c+d x)}{2 (a-b) b d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
x/b^2-1/2*(2*a-3*b)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))*a^(1/2)/(a-b) ^(3/2)/b^2/d-1/2*a*tanh(d*x+c)/(a-b)/b/d/(a-(a-b)*tanh(d*x+c)^2)
Time = 11.42 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {-2 (c+d x)+\frac {\sqrt {a} (2 a-3 b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{(a-b)^{3/2}}+\frac {a b \sinh (2 (c+d x))}{(a-b) (2 a-b+b \cosh (2 (c+d x)))}}{2 b^2 d} \]
-1/2*(-2*(c + d*x) + (Sqrt[a]*(2*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d* x])/Sqrt[a]])/(a - b)^(3/2) + (a*b*Sinh[2*(c + d*x)])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(b^2*d)
Time = 0.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3666, 372, 397, 219, 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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3666 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\int \frac {(a-2 b) \tanh ^2(c+d x)+a}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{2 b (a-b)}-\frac {a \tanh (c+d x)}{2 b (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {2 (a-b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{b}-\frac {a (2 a-3 b) \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{2 b (a-b)}-\frac {a \tanh (c+d x)}{2 b (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {2 (a-b) \text {arctanh}(\tanh (c+d x))}{b}-\frac {a (2 a-3 b) \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{2 b (a-b)}-\frac {a \tanh (c+d x)}{2 b (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {2 (a-b) \text {arctanh}(\tanh (c+d x))}{b}-\frac {\sqrt {a} (2 a-3 b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{b \sqrt {a-b}}}{2 b (a-b)}-\frac {a \tanh (c+d x)}{2 b (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
(((2*(a - b)*ArcTanh[Tanh[c + d*x]])/b - (Sqrt[a]*(2*a - 3*b)*ArcTanh[(Sqr t[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a - b]*b))/(2*(a - b)*b) - (a*Tanh [c + d*x])/(2*(a - b)*b*(a - (a - b)*Tanh[c + d*x]^2)))/d
3.1.42.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((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)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1)) , x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] & & IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(90)=180\).
Time = 0.27 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.09
| method | result | size |
| risch | \(\frac {x}{b^{2}}+\frac {a \left (2 a \,{\mathrm e}^{2 d x +2 c}-b \,{\mathrm e}^{2 d x +2 c}+b \right )}{b^{2} \left (a -b \right ) d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {\sqrt {\left (a -b \right ) a}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {\left (a -b \right ) a}-b}{b}\right ) a}{2 \left (a -b \right )^{2} d \,b^{2}}-\frac {3 \sqrt {\left (a -b \right ) a}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {\left (a -b \right ) a}-b}{b}\right )}{4 \left (a -b \right )^{2} d b}-\frac {\sqrt {\left (a -b \right ) a}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {\left (a -b \right ) a}+b}{b}\right ) a}{2 \left (a -b \right )^{2} d \,b^{2}}+\frac {3 \sqrt {\left (a -b \right ) a}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {\left (a -b \right ) a}+b}{b}\right )}{4 \left (a -b \right )^{2} d b}\) | \(315\) |
| derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2}}+\frac {2 a \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (2 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a -2 b}\right )}{b^{2}}}{d}\) | \(319\) |
| default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2}}+\frac {2 a \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (2 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a -2 b}\right )}{b^{2}}}{d}\) | \(319\) |
x/b^2+a*(2*a*exp(2*d*x+2*c)-b*exp(2*d*x+2*c)+b)/b^2/(a-b)/d/(b*exp(4*d*x+4 *c)+4*a*exp(2*d*x+2*c)-2*b*exp(2*d*x+2*c)+b)+1/2*((a-b)*a)^(1/2)/(a-b)^2/d /b^2*ln(exp(2*d*x+2*c)+(2*a+2*((a-b)*a)^(1/2)-b)/b)*a-3/4*((a-b)*a)^(1/2)/ (a-b)^2/d/b*ln(exp(2*d*x+2*c)+(2*a+2*((a-b)*a)^(1/2)-b)/b)-1/2*((a-b)*a)^( 1/2)/(a-b)^2/d/b^2*ln(exp(2*d*x+2*c)-(-2*a+2*((a-b)*a)^(1/2)+b)/b)*a+3/4*( (a-b)*a)^(1/2)/(a-b)^2/d/b*ln(exp(2*d*x+2*c)-(-2*a+2*((a-b)*a)^(1/2)+b)/b)
Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (91) = 182\).
Time = 0.31 (sec) , antiderivative size = 1772, normalized size of antiderivative = 17.37 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/4*(4*(a*b - b^2)*d*x*cosh(d*x + c)^4 + 16*(a*b - b^2)*d*x*cosh(d*x + c) *sinh(d*x + c)^3 + 4*(a*b - b^2)*d*x*sinh(d*x + c)^4 + 4*(a*b - b^2)*d*x + 4*(2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^2 - a*b)*cosh(d*x + c)^2 + 4*(6*(a*b - b^2)*d*x*cosh(d*x + c)^2 + 2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^2 - a*b)*s inh(d*x + c)^2 + ((2*a*b - 3*b^2)*cosh(d*x + c)^4 + 4*(2*a*b - 3*b^2)*cosh (d*x + c)*sinh(d*x + c)^3 + (2*a*b - 3*b^2)*sinh(d*x + c)^4 + 2*(4*a^2 - 8 *a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b - 3*b^2)*cosh(d*x + c)^2 + 4*a ^2 - 8*a*b + 3*b^2)*sinh(d*x + c)^2 + 2*a*b - 3*b^2 + 4*((2*a*b - 3*b^2)*c osh(d*x + c)^3 + (4*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqr t(a/(a - b))*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^ 3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh( d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*c osh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*((a*b - b^ 2)*cosh(d*x + c)^2 + 2*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b - b^ 2)*sinh(d*x + c)^2 + 2*a^2 - 3*a*b + b^2)*sqrt(a/(a - b)))/(b*cosh(d*x + c )^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)* cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b *cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 4*a*b + 8*(2*(a*b - b^2)*d*x*cosh(d*x + c)^3 + (2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^ 2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/((a*b^3 - b^4)*d*cosh(d*x + c)^4...
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]