Integrand size = 23, antiderivative size = 158 \[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {(4 a-5 b) x}{2 b^3}+\frac {(a-b)^{3/2} (4 a+b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^3 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {(a-b) (2 a-b) \tanh (c+d x)}{2 a b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
-1/2*(4*a-5*b)*x/b^3+1/2*(a-b)^(3/2)*(4*a+b)*arctanh((a-b)^(1/2)*tanh(d*x+ c)/a^(1/2))/a^(3/2)/b^3/d+1/2*cosh(d*x+c)*sinh(d*x+c)/b/d/(a-(a-b)*tanh(d* x+c)^2)+1/2*(a-b)*(2*a-b)*tanh(d*x+c)/a/b^2/d/(a-(a-b)*tanh(d*x+c)^2)
Time = 0.73 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75 \[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {2 (-4 a+5 b) (c+d x)+\frac {2 (a-b)^{3/2} (4 a+b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+b \sinh (2 (c+d x))+\frac {2 (a-b)^2 b \sinh (2 (c+d x))}{a (2 a-b+b \cosh (2 (c+d x)))}}{4 b^3 d} \]
(2*(-4*a + 5*b)*(c + d*x) + (2*(a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b ]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2) + b*Sinh[2*(c + d*x)] + (2*(a - b)^2*b* Sinh[2*(c + d*x)])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(4*b^3*d)
Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 3670, 316, 25, 402, 27, 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.
| \(\displaystyle \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i c+i d x)^6}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3670 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^2 \left (a-(a-b) \tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int -\frac {3 (a-b) \tanh ^2(c+d x)+a-2 b}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{2 b}+\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\int \frac {3 (a-b) \tanh ^2(c+d x)+a-2 b}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{2 b}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {-\frac {\int -\frac {2 \left (2 a^2-2 b a-b^2+(a-b) (2 a-b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{2 a b}-\frac {(a-b) (2 a-b) \tanh (c+d x)}{a b \left (a-(a-b) \tanh ^2(c+d x)\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\frac {\int \frac {2 a^2-2 b a-b^2+(a-b) (2 a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{a b}-\frac {(a-b) (2 a-b) \tanh (c+d x)}{a b \left (a-(a-b) \tanh ^2(c+d x)\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\frac {\frac {a (4 a-5 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{b}-\frac {(a-b)^2 (4 a+b) \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{a b}-\frac {(a-b) (2 a-b) \tanh (c+d x)}{a b \left (a-(a-b) \tanh ^2(c+d x)\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\frac {\frac {a (4 a-5 b) \text {arctanh}(\tanh (c+d x))}{b}-\frac {(a-b)^2 (4 a+b) \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{a b}-\frac {(a-b) (2 a-b) \tanh (c+d x)}{a b \left (a-(a-b) \tanh ^2(c+d x)\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\frac {\frac {a (4 a-5 b) \text {arctanh}(\tanh (c+d x))}{b}-\frac {(a-b)^{3/2} (4 a+b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b}}{a b}-\frac {(a-b) (2 a-b) \tanh (c+d x)}{a b \left (a-(a-b) \tanh ^2(c+d x)\right )}}{2 b}}{d}\) |
(Tanh[c + d*x]/(2*b*(1 - Tanh[c + d*x]^2)*(a - (a - b)*Tanh[c + d*x]^2)) - (((a*(4*a - 5*b)*ArcTanh[Tanh[c + d*x]])/b - ((a - b)^(3/2)*(4*a + b)*Arc Tanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b))/(a*b) - ((a - b)*( 2*a - b)*Tanh[c + d*x])/(a*b*(a - (a - b)*Tanh[c + d*x]^2)))/(2*b))/d
3.4.29.3.1 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[((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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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[((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[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Su bst[Int[(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. \(424\) vs. \(2(142)=284\).
Time = 0.17 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.69
\[\frac {-\frac {2 \left (\frac {-\frac {b \left (a^{2}-2 a b +b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (a^{2}-2 a b +b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (4 a^{3}-7 a^{2} b +2 a \,b^{2}+b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2}\right )}{b^{3}}-\frac {1}{2 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 b^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (-4 a +5 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b^{3}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 a -5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{3}}}{d}\]
1/d*(-2/b^3*((-1/2*b*(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c)^3-1/2*b*(a^2-2* a*b+b^2)/a*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/ 2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)+1/2*(4*a^3-7*a^2*b+2*a*b^2+b^3)*(1/2 *((-b*(a-b))^(1/2)+b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1 /2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/2 *((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1 /2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))))- 1/2/b^2/(1+tanh(1/2*d*x+1/2*c))^2+1/2/b^2/(1+tanh(1/2*d*x+1/2*c))+1/2/b^3* (-4*a+5*b)*ln(1+tanh(1/2*d*x+1/2*c))+1/2/b^2/(tanh(1/2*d*x+1/2*c)-1)^2+1/2 /b^2/(tanh(1/2*d*x+1/2*c)-1)+1/2*(4*a-5*b)/b^3*ln(tanh(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1682 vs. \(2 (144) = 288\).
Time = 0.32 (sec) , antiderivative size = 3629, normalized size of antiderivative = 22.97 \[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/8*(a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^ 2*sinh(d*x + c)^8 + 2*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d *x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + 2*a^2*b - a*b^2 - 2*(4*a^2*b - 5 *a*b^2)*d*x)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*(2*a ^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d*x + c)^4 + 2*(35*a*b^2*cosh(d*x + c)^4 - 8*a^3 + 20*a^2*b - 16*a*b^2 + 4*b^3 - 4*(8*a^3 - 14*a^2*b + 5*a*b^2)*d*x + 15*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a*b^2*cosh(d*x + c) ^5 + 5*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^3 - 4*( 2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d *x + c))*sinh(d*x + c)^3 - a*b^2 - 2*(6*a^2*b - 9*a*b^2 + 4*b^3 + 2*(4*a^2 *b - 5*a*b^2)*d*x)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(2*a ^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^4 - 6*a^2*b + 9*a* b^2 - 4*b^3 - 2*(4*a^2*b - 5*a*b^2)*d*x - 24*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*((4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(4*a^2*b - 3*a*b^2 - b^3 )*cosh(d*x + c)*sinh(d*x + c)^5 + (4*a^2*b - 3*a*b^2 - b^3)*sinh(d*x + c)^ 6 + 2*(8*a^3 - 10*a^2*b + a*b^2 + b^3)*cosh(d*x + c)^4 + (16*a^3 - 20*a^2* b + 2*a*b^2 + 2*b^3 + 15*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^2)*sin...
Timed out. \[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )^{6}}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\cosh ^6(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]