Integrand size = 23, antiderivative size = 76 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b} d}+\frac {(a-b) \sinh (c+d x)}{a^2 d}+\frac {\sinh ^3(c+d x)}{3 a d} \]
(a-b)*sinh(d*x+c)/a^2/d+1/3*sinh(d*x+c)^3/a/d+b^2*arctan(sinh(d*x+c)*a^(1/ 2)/(a+b)^(1/2))/a^(5/2)/d/(a+b)^(1/2)
Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {-\frac {12 b^2 \arctan \left (\frac {\sqrt {a+b} \text {csch}(c+d x)}{\sqrt {a}}\right )}{\sqrt {a+b}}+3 \sqrt {a} (3 a-4 b) \sinh (c+d x)+a^{3/2} \sinh (3 (c+d x))}{12 a^{5/2} d} \]
((-12*b^2*ArcTan[(Sqrt[a + b]*Csch[c + d*x])/Sqrt[a]])/Sqrt[a + b] + 3*Sqr t[a]*(3*a - 4*b)*Sinh[c + d*x] + a^(3/2)*Sinh[3*(c + d*x)])/(12*a^(5/2)*d)
Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4635, 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.
| \(\displaystyle \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^2}{a \sinh ^2(c+d x)+a+b}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\frac {b^2}{a^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\sinh ^2(c+d x)}{a}+\frac {a-b}{a^2}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {(a-b) \sinh (c+d x)}{a^2}+\frac {\sinh ^3(c+d x)}{3 a}}{d}\) |
((b^2*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]) + ((a - b)*Sinh[c + d*x])/a^2 + Sinh[c + d*x]^3/(3*a))/d
3.1.74.3.1 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_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b + 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] && In tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(66)=132\).
Time = 0.85 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.75
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a -b}{a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a -b}{a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 b^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{a^{2}}}{d}\) | \(209\) |
| default | \(\frac {-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a -b}{a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a -b}{a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 b^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{a^{2}}}{d}\) | \(209\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 d a}+\frac {3 \,{\mathrm e}^{d x +c}}{8 a d}-\frac {{\mathrm e}^{d x +c} b}{2 a^{2} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a d}+\frac {{\mathrm e}^{-d x -c} b}{2 a^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 d a}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, d \,a^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, d \,a^{2}}\) | \(216\) |
1/d*(-1/3/a/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/a/(tanh(1/2*d*x+1/2*c)-1)^2-(a-b )/a^2/(tanh(1/2*d*x+1/2*c)-1)-1/3/a/(1+tanh(1/2*d*x+1/2*c))^3+1/2/a/(1+tan h(1/2*d*x+1/2*c))^2-(a-b)/a^2/(1+tanh(1/2*d*x+1/2*c))+2*b^2/a^2*(1/2/(a+b) ^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/a^ (1/2))+1/2/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2* c)+2*b^(1/2))/a^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (66) = 132\).
Time = 0.29 (sec) , antiderivative size = 1616, normalized size of antiderivative = 21.26 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Too large to display} \]
[1/24*((a^3 + a^2*b)*cosh(d*x + c)^6 + 6*(a^3 + a^2*b)*cosh(d*x + c)*sinh( d*x + c)^5 + (a^3 + a^2*b)*sinh(d*x + c)^6 + 3*(3*a^3 - a^2*b - 4*a*b^2)*c osh(d*x + c)^4 + 3*(3*a^3 - a^2*b - 4*a*b^2 + 5*(a^3 + a^2*b)*cosh(d*x + c )^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + a^2*b)*cosh(d*x + c)^3 + 3*(3*a^3 - a^2 *b - 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - a^2*b - 3*(3*a^3 - a^ 2*b - 4*a*b^2)*cosh(d*x + c)^2 + 3*(5*(a^3 + a^2*b)*cosh(d*x + c)^4 - 3*a^ 3 + a^2*b + 4*a*b^2 + 6*(3*a^3 - a^2*b - 4*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*sinh(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*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2 *b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*si nh(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)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a *sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c)) *sinh(d*x + c) + a)) + 6*((a^3 + a^2*b)*cosh(d*x + c)^5 + 2*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^3 - (3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c))*sin h(d*x + c))/((a^4 + a^3*b)*d*cosh(d*x + c)^3 + 3*(a^4 + a^3*b)*d*cosh(d...
\[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
1/24*(3*(3*a*e^(4*c) - 4*b*e^(4*c))*e^(4*d*x) - 3*(3*a*e^(2*c) - 4*b*e^(2* c))*e^(2*d*x) + a*e^(6*d*x + 6*c) - a)*e^(-3*d*x - 3*c)/(a^2*d) + 1/8*inte grate(16*(b^2*e^(3*d*x + 3*c) + b^2*e^(d*x + c))/(a^3*e^(4*d*x + 4*c) + a^ 3 + 2*(a^3*e^(2*c) + 2*a^2*b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.59 (sec) , antiderivative size = 332, normalized size of antiderivative = 4.37 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,a\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,a\,d}-\frac {\sqrt {b^4}\,\left (2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,b^2}{a^8\,d\,{\left (a+b\right )}^2\,\sqrt {b^4}}-\frac {4\,\left (2\,a^2\,b^4\,d\,\sqrt {b^4}+2\,a^3\,b^3\,d\,\sqrt {b^4}\right )}{a^6\,b^5\,\left (a+b\right )\,\sqrt {a^6\,d^2+b\,a^5\,d^2}\,\sqrt {a^5\,d^2\,\left (a+b\right )}}\right )-\frac {2\,b^2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{a^8\,d\,{\left (a+b\right )}^2\,\sqrt {b^4}}\right )\,\left (\frac {a^7\,\sqrt {a^6\,d^2+b\,a^5\,d^2}}{4}+\frac {a^6\,b\,\sqrt {a^6\,d^2+b\,a^5\,d^2}}{4}\right )\right )-2\,\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^5\,d^2\,\left (a+b\right )}}{2\,a^2\,d\,\left (a+b\right )\,\sqrt {b^4}}\right )\right )}{2\,\sqrt {a^6\,d^2+b\,a^5\,d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a-4\,b\right )}{8\,a^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a-4\,b\right )}{8\,a^2\,d} \]
exp(3*c + 3*d*x)/(24*a*d) - exp(- 3*c - 3*d*x)/(24*a*d) - ((b^4)^(1/2)*(2* atan((exp(d*x)*exp(c)*((2*b^2)/(a^8*d*(a + b)^2*(b^4)^(1/2)) - (4*(2*a^2*b ^4*d*(b^4)^(1/2) + 2*a^3*b^3*d*(b^4)^(1/2)))/(a^6*b^5*(a + b)*(a^6*d^2 + a ^5*b*d^2)^(1/2)*(a^5*d^2*(a + b))^(1/2))) - (2*b^2*exp(3*c)*exp(3*d*x))/(a ^8*d*(a + b)^2*(b^4)^(1/2)))*((a^7*(a^6*d^2 + a^5*b*d^2)^(1/2))/4 + (a^6*b *(a^6*d^2 + a^5*b*d^2)^(1/2))/4)) - 2*atan((b^2*exp(d*x)*exp(c)*(a^5*d^2*( a + b))^(1/2))/(2*a^2*d*(a + b)*(b^4)^(1/2)))))/(2*(a^6*d^2 + a^5*b*d^2)^( 1/2)) + (exp(c + d*x)*(3*a - 4*b))/(8*a^2*d) - (exp(- c - d*x)*(3*a - 4*b) )/(8*a^2*d)