Integrand size = 23, antiderivative size = 80 \[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{5/2} d}+\frac {(a+2 b) \sinh (c+d x)}{(a+b)^2 d}+\frac {\sinh ^3(c+d x)}{3 (a+b) d} \]
(a+2*b)*sinh(d*x+c)/(a+b)^2/d+1/3*sinh(d*x+c)^3/(a+b)/d+b^2*arctan(sinh(d* x+c)*(a+b)^(1/2)/a^(1/2))/(a+b)^(5/2)/d/a^(1/2)
Time = 0.56 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99 \[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {-\frac {12 b^2 \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}+\frac {3 (3 a+7 b) \sinh (c+d x)}{(a+b)^2}+\frac {\sinh (3 (c+d x))}{a+b}}{12 d} \]
((-12*b^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5 /2)) + (3*(3*a + 7*b)*Sinh[c + d*x])/(a + b)^2 + Sinh[3*(c + d*x)]/(a + b) )/(12*d)
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4159, 300, 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.
| \(\displaystyle \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4159 |
| \(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^2}{(a+b) \sinh ^2(c+d x)+a}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\frac {b^2}{(a+b)^2 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sinh ^2(c+d x)}{a+b}+\frac {a+2 b}{(a+b)^2}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{5/2}}+\frac {\sinh ^3(c+d x)}{3 (a+b)}+\frac {(a+2 b) \sinh (c+d x)}{(a+b)^2}}{d}\) |
((b^2*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)) + ((a + 2*b)*Sinh[c + d*x])/(a + b)^2 + Sinh[c + d*x]^3/(3*(a + b)))/d
3.2.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2 *x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(70)=140\).
Time = 3.97 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.88
| method | result | size |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 d \left (a +b \right )}+\frac {3 \,{\mathrm e}^{d x +c} a}{8 \left (a +b \right )^{2} d}+\frac {7 \,{\mathrm e}^{d x +c} b}{8 \left (a +b \right )^{2} d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 \left (a +b \right )^{2} d}-\frac {7 \,{\mathrm e}^{-d x -c} b}{8 \left (a +b \right )^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 d \left (a +b \right )}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}\) | \(230\) |
| derivativedivides | \(\frac {-\frac {2}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (2 a +2 b \right )}-\frac {1}{\left (2 a +2 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +2 b}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{3 \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (2 a +2 b \right )}+\frac {1}{\left (2 a +2 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a +2 b}{\left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 b^{2} a \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{\left (a +b \right )^{2}}}{d}\) | \(307\) |
| default | \(\frac {-\frac {2}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (2 a +2 b \right )}-\frac {1}{\left (2 a +2 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +2 b}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{3 \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (2 a +2 b \right )}+\frac {1}{\left (2 a +2 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a +2 b}{\left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 b^{2} a \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{\left (a +b \right )^{2}}}{d}\) | \(307\) |
1/24/d/(a+b)*exp(3*d*x+3*c)+3/8/(a+b)^2/d*exp(d*x+c)*a+7/8/(a+b)^2/d*exp(d *x+c)*b-3/8/(a+b)^2/d*exp(-d*x-c)*a-7/8/(a+b)^2/d*exp(-d*x-c)*b-1/24/d/(a+ b)*exp(-3*d*x-3*c)-1/2/(-a^2-a*b)^(1/2)*b^2/(a+b)^2/d*ln(exp(2*d*x+2*c)-2* a/(-a^2-a*b)^(1/2)*exp(d*x+c)-1)+1/2/(-a^2-a*b)^(1/2)*b^2/(a+b)^2/d*ln(exp (2*d*x+2*c)+2*a/(-a^2-a*b)^(1/2)*exp(d*x+c)-1)
Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (70) = 140\).
Time = 0.30 (sec) , antiderivative size = 1850, normalized size of antiderivative = 23.12 \[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
[1/24*((a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2) *cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 + 2*a^2*b + a*b^2)*sinh(d*x + c)^6 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 10*a^2*b + 7* a*b^2 + 5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5* (a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^3 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*c osh(d*x + c))*sinh(d*x + c)^3 - a^3 - 2*a^2*b - a*b^2 - 3*(3*a^3 + 10*a^2* b + 7*a*b^2)*cosh(d*x + c)^2 + 3*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^ 4 - 3*a^3 - 10*a^2*b - 7*a*b^2 + 6*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 12*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*s inh(d*x + c) + 3*b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)* sqrt(-a^2 - a*b)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*si nh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(3*a + b)*cosh(d*x + c)^2 + 2* (3*(a + b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d* x + c)^3 - (3*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3 *cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1) *sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a + b)/((a + b)*cosh(d* x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c) ^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sin h(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d* x + c) + a + b)) + 6*((a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 2*(3*a^...
\[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\cosh ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
1/24*((a*e^(6*c) + b*e^(6*c))*e^(6*d*x) + 3*(3*a*e^(4*c) + 7*b*e^(4*c))*e^ (4*d*x) - 3*(3*a*e^(2*c) + 7*b*e^(2*c))*e^(2*d*x) - a - b)*e^(-3*d*x)/(a^2 *d*e^(3*c) + 2*a*b*d*e^(3*c) + b^2*d*e^(3*c)) + 1/8*integrate(16*(b^2*e^(3 *d*x + 3*c) + b^2*e^(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + (a^3*e^(4* c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a^3*e ^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 3.44 (sec) , antiderivative size = 2194, normalized size of antiderivative = 27.42 \[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
exp(3*c + 3*d*x)/(24*d*(a + b)) - exp(- 3*c - 3*d*x)/(24*d*(a + b)) + ((b^ 4)^(1/2)*(2*atan((exp(d*x)*exp(c)*((4*(10*a^2*d*(b^4)^(5/2) + 12*a^6*d*(b^ 4)^(3/2) + 2*a*b^9*d*(b^4)^(1/2) + 10*a^3*b^3*d*(b^4)^(3/2) + 2*a^2*b^8*d* (b^4)^(1/2) + 20*a^3*b^7*d*(b^4)^(1/2) + 40*a^4*b^6*d*(b^4)^(1/2) + 30*a^5 *b^5*d*(b^4)^(1/2) + 2*a^7*b^3*d*(b^4)^(1/2)))/(a*b^5*(a + b)^5*(a*d^2*(a + b)^5)^(1/2)*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)*(5*a*b^4 + 5*a^4 *b + a^5 + b^5 + 10*a^2*b^3 + 10*a^3*b^2)*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d ^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)) - (2*(b^9*(a^ 6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4* b^2*d^2)^(1/2) + 4*a*b^8*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^ 2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + 6*a^2*b^7*(a^6*d^2 + a*b^5*d^ 2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + 4*a^3*b^6*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3 *d^2 + 10*a^4*b^2*d^2)^(1/2) + a^4*b^5*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)))/(a^2*b^3*d*(a + b)^7*(b^4)^(1/2)*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)*(5*a*b^4 + 5 *a^4*b + a^5 + b^5 + 10*a^2*b^3 + 10*a^3*b^2)*(a^6*d^2 + a*b^5*d^2 + 5*a^5 *b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))) + (2*exp (3*c)*exp(3*d*x)*(b^9*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + 4*a*b^8*(a^6*d^2 + a*b^5*d^2 ...