Integrand size = 23, antiderivative size = 121 \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {\left (8 a^2-20 a b+15 b^2\right ) x}{8 b^3}-\frac {(a-b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d} \] Output:
1/8*(8*a^2-20*a*b+15*b^2)*x/b^3-(a-b)^(5/2)*arctanh((a-b)^(1/2)*tanh(d*x+c )/a^(1/2))/a^(1/2)/b^3/d-1/8*(4*a-7*b)*cosh(d*x+c)*sinh(d*x+c)/b^2/d+1/4*c osh(d*x+c)^3*sinh(d*x+c)/b/d
Time = 0.41 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-32 (a-b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \left (4 \left (8 a^2-20 a b+15 b^2\right ) (c+d x)-8 (a-2 b) b \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))\right )}{32 \sqrt {a} b^3 d} \] Input:
Integrate[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]
Output:
(-32*(a - b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]* (4*(8*a^2 - 20*a*b + 15*b^2)*(c + d*x) - 8*(a - 2*b)*b*Sinh[2*(c + d*x)] + b^2*Sinh[4*(c + d*x)]))/(32*Sqrt[a]*b^3*d)
Time = 0.43 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.23, 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, 25, 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.
| \(\displaystyle \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i c+i d x)^6}{a-b \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 3670 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^3 \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int -\frac {3 (a-b) \tanh ^2(c+d x)+a-4 b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{4 b}+\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\int \frac {3 (a-b) \tanh ^2(c+d x)+a-4 b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{4 b}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {\int -\frac {4 a^2-9 b a+8 b^2+(4 a-7 b) (a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{2 b}+\frac {(4 a-7 b) \tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}}{4 b}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-7 b) \tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\int \frac {4 a^2-9 b a+8 b^2+(4 a-7 b) (a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )}d\tanh (c+d x)}{2 b}}{4 b}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-7 b) \tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {\left (8 a^2-20 a b+15 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{b}-\frac {8 (a-b)^3 \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{2 b}}{4 b}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-7 b) \tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {\left (8 a^2-20 a b+15 b^2\right ) \text {arctanh}(\tanh (c+d x))}{b}-\frac {8 (a-b)^3 \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{2 b}}{4 b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 b \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-7 b) \tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {\left (8 a^2-20 a b+15 b^2\right ) \text {arctanh}(\tanh (c+d x))}{b}-\frac {8 (a-b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b}}{2 b}}{4 b}}{d}\) |
Input:
Int[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]
Output:
(Tanh[c + d*x]/(4*b*(1 - Tanh[c + d*x]^2)^2) - (-1/2*(((8*a^2 - 20*a*b + 1 5*b^2)*ArcTanh[Tanh[c + d*x]])/b - (8*(a - b)^(5/2)*ArcTanh[(Sqrt[a - b]*T anh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b))/b + ((4*a - 7*b)*Tanh[c + d*x])/(2*b* (1 - Tanh[c + d*x]^2)))/(4*b))/d
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. \(430\) vs. \(2(107)=214\).
Time = 177.88 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.56
| method | result | size |
| risch | \(\frac {x \,a^{2}}{b^{3}}-\frac {5 a x}{2 b^{2}}+\frac {15 x}{8 b}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 b^{2} d}+\frac {{\mathrm e}^{2 d x +2 c}}{4 b d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 b^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c}}{4 b d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}+\frac {a \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{2 d \,b^{3}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{2 a d b}-\frac {a \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{2 d \,b^{3}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{2 a d b}\) | \(431\) |
| derivativedivides | \(\frac {-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a +11 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (8 a^{2}-20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a -11 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-8 a^{2}+20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{3}}+\frac {2 a \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right ) \left (-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{b^{3}}}{d}\) | \(438\) |
| default | \(\frac {-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a +11 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (8 a^{2}-20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a -11 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-8 a^{2}+20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{3}}+\frac {2 a \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right ) \left (-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{b^{3}}}{d}\) | \(438\) |
Input:
int(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
x/b^3*a^2-5/2*a*x/b^2+15/8*x/b+1/64/b/d*exp(4*d*x+4*c)-1/8/b^2/d*exp(2*d*x +2*c)*a+1/4/b/d*exp(2*d*x+2*c)+1/8/b^2/d*exp(-2*d*x-2*c)*a-1/4/b/d*exp(-2* d*x-2*c)-1/64/b/d*exp(-4*d*x-4*c)+1/2*a*(a*(a-b))^(1/2)/d/b^3*ln(exp(2*d*x +2*c)+(2*a+2*(a*(a-b))^(1/2)-b)/b)-(a*(a-b))^(1/2)/d/b^2*ln(exp(2*d*x+2*c) +(2*a+2*(a*(a-b))^(1/2)-b)/b)+1/2/a*(a*(a-b))^(1/2)/d/b*ln(exp(2*d*x+2*c)+ (2*a+2*(a*(a-b))^(1/2)-b)/b)-1/2*a*(a*(a-b))^(1/2)/d/b^3*ln(exp(2*d*x+2*c) -(-2*a+2*(a*(a-b))^(1/2)+b)/b)+(a*(a-b))^(1/2)/d/b^2*ln(exp(2*d*x+2*c)-(-2 *a+2*(a*(a-b))^(1/2)+b)/b)-1/2/a*(a*(a-b))^(1/2)/d/b*ln(exp(2*d*x+2*c)-(-2 *a+2*(a*(a-b))^(1/2)+b)/b)
Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (107) = 214\).
Time = 0.11 (sec) , antiderivative size = 1817, normalized size of antiderivative = 15.02 \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
Output:
[1/64*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^2*sin h(d*x + c)^8 + 8*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c)^4 - 8*(a*b - 2*b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 - 2*a*b + 4*b^2)*sinh(d* x + c)^6 + 8*(7*b^2*cosh(d*x + c)^3 - 6*(a*b - 2*b^2)*cosh(d*x + c))*sinh( d*x + c)^5 + 2*(35*b^2*cosh(d*x + c)^4 + 4*(8*a^2 - 20*a*b + 15*b^2)*d*x - 60*(a*b - 2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c )^5 + 4*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c) - 20*(a*b - 2*b^2)*cos h(d*x + c)^3)*sinh(d*x + c)^3 + 8*(a*b - 2*b^2)*cosh(d*x + c)^2 + 4*(7*b^2 *cosh(d*x + c)^6 + 12*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c)^2 - 30*( a*b - 2*b^2)*cosh(d*x + c)^4 + 2*a*b - 4*b^2)*sinh(d*x + c)^2 + 32*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^3*sinh (d*x + c) + 6*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - 2*a*b + b^2)*sinh(d *x + c)^4)*sqrt((a - b)/a)*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*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d* x + c)^2 + 2*a^2 - a*b)*sqrt((a - b)/a))/(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...
Timed out. \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)**6/(a+b*sinh(d*x+c)**2),x)
Output:
Timed out
Exception generated. \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (107) = 214\).
Time = 0.72 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.87 \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {\frac {8 \, {\left (8 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b e^{\left (2 \, d x + 2 \, c\right )}}{b^{2}} - \frac {{\left (48 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 120 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{b^{3}} - \frac {64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b^{3}}}{64 \, d} \] Input:
integrate(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="giac")
Output:
1/64*(8*(8*a^2 - 20*a*b + 15*b^2)*(d*x + c)/b^3 + (b*e^(4*d*x + 4*c) - 8*a *e^(2*d*x + 2*c) + 16*b*e^(2*d*x + 2*c))/b^2 - (48*a^2*e^(4*d*x + 4*c) - 1 20*a*b*e^(4*d*x + 4*c) + 90*b^2*e^(4*d*x + 4*c) - 8*a*b*e^(2*d*x + 2*c) + 16*b^2*e^(2*d*x + 2*c) + b^2)*e^(-4*d*x - 4*c)/b^3 - 64*(a^3 - 3*a^2*b + 3 *a*b^2 - b^3)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/( sqrt(-a^2 + a*b)*b^3))/d
Time = 2.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.18 \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {x\,\left (8\,a^2-20\,a\,b+15\,b^2\right )}{8\,b^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a-2\,b\right )}{8\,b^2\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a-2\,b\right )}{8\,b^2\,d}+\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a-b\right )}^3}{b^4}-\frac {2\,{\left (a-b\right )}^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^4}\right )\,{\left (a-b\right )}^{5/2}}{2\,\sqrt {a}\,b^3\,d}-\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a-b\right )}^3}{b^4}+\frac {2\,{\left (a-b\right )}^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^4}\right )\,{\left (a-b\right )}^{5/2}}{2\,\sqrt {a}\,b^3\,d} \] Input:
int(cosh(c + d*x)^6/(a + b*sinh(c + d*x)^2),x)
Output:
(x*(8*a^2 - 20*a*b + 15*b^2))/(8*b^3) - exp(- 4*c - 4*d*x)/(64*b*d) + exp( 4*c + 4*d*x)/(64*b*d) + (exp(- 2*c - 2*d*x)*(a - 2*b))/(8*b^2*d) - (exp(2* c + 2*d*x)*(a - 2*b))/(8*b^2*d) + (log((4*exp(2*c + 2*d*x)*(a - b)^3)/b^4 - (2*(a - b)^(5/2)*(b + 2*a*exp(2*c + 2*d*x) - b*exp(2*c + 2*d*x)))/(a^(1/ 2)*b^4))*(a - b)^(5/2))/(2*a^(1/2)*b^3*d) - (log((4*exp(2*c + 2*d*x)*(a - b)^3)/b^4 + (2*(a - b)^(5/2)*(b + 2*a*exp(2*c + 2*d*x) - b*exp(2*c + 2*d*x )))/(a^(1/2)*b^4))*(a - b)^(5/2))/(2*a^(1/2)*b^3*d)
Time = 0.19 (sec) , antiderivative size = 630, normalized size of antiderivative = 5.21 \[ \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-32 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a^{2}+64 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a b -32 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) b^{2}-32 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a^{2}+64 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a b -32 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) b^{2}+32 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a -b}+e^{2 d x +2 c} b +2 a -b \right ) a^{2}-64 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a -b}+e^{2 d x +2 c} b +2 a -b \right ) a b +32 e^{4 d x +4 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a -b}+e^{2 d x +2 c} b +2 a -b \right ) b^{2}+e^{8 d x +8 c} a \,b^{2}-8 e^{6 d x +6 c} a^{2} b +16 e^{6 d x +6 c} a \,b^{2}+64 e^{4 d x +4 c} a^{3} d x -160 e^{4 d x +4 c} a^{2} b d x +120 e^{4 d x +4 c} a \,b^{2} d x +8 e^{2 d x +2 c} a^{2} b -16 e^{2 d x +2 c} a \,b^{2}-a \,b^{2}}{64 e^{4 d x +4 c} a \,b^{3} d} \] Input:
int(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x)
Output:
( - 32*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a**2 + 64*e**(4*c + 4*d*x)*sqrt(a)* sqrt(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sq rt(b))*a*b - 32*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log( - sqrt(2*sqrt(a) *sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*b**2 - 32*e**(4*c + 4*d*x) *sqrt(a)*sqrt(a - b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d *x)*sqrt(b))*a**2 + 64*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log(sqrt(2*sqr t(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a*b - 32*e**(4*c + 4*d *x)*sqrt(a)*sqrt(a - b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*b**2 + 32*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log(2*sqrt( a)*sqrt(a - b) + e**(2*c + 2*d*x)*b + 2*a - b)*a**2 - 64*e**(4*c + 4*d*x)* sqrt(a)*sqrt(a - b)*log(2*sqrt(a)*sqrt(a - b) + e**(2*c + 2*d*x)*b + 2*a - b)*a*b + 32*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log(2*sqrt(a)*sqrt(a - b ) + e**(2*c + 2*d*x)*b + 2*a - b)*b**2 + e**(8*c + 8*d*x)*a*b**2 - 8*e**(6 *c + 6*d*x)*a**2*b + 16*e**(6*c + 6*d*x)*a*b**2 + 64*e**(4*c + 4*d*x)*a**3 *d*x - 160*e**(4*c + 4*d*x)*a**2*b*d*x + 120*e**(4*c + 4*d*x)*a*b**2*d*x + 8*e**(2*c + 2*d*x)*a**2*b - 16*e**(2*c + 2*d*x)*a*b**2 - a*b**2)/(64*e**( 4*c + 4*d*x)*a*b**3*d)