Integrand size = 17, antiderivative size = 68 \[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {8 a^2 (5 A+3 B) \sinh (x)}{15 \sqrt {a+a \cosh (x)}}+\frac {2}{15} a (5 A+3 B) \sqrt {a+a \cosh (x)} \sinh (x)+\frac {2}{5} B (a+a \cosh (x))^{3/2} \sinh (x) \] Output:
8/15*a^2*(5*A+3*B)*sinh(x)/(a+a*cosh(x))^(1/2)+2/15*a*(5*A+3*B)*(a+a*cosh( x))^(1/2)*sinh(x)+2/5*B*(a+a*cosh(x))^(3/2)*sinh(x)
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {1}{15} a \sqrt {a (1+\cosh (x))} (50 A+39 B+2 (5 A+9 B) \cosh (x)+3 B \cosh (2 x)) \tanh \left (\frac {x}{2}\right ) \] Input:
Integrate[(a + a*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]
Output:
(a*Sqrt[a*(1 + Cosh[x])]*(50*A + 39*B + 2*(5*A + 9*B)*Cosh[x] + 3*B*Cosh[2 *x])*Tanh[x/2])/15
Time = 0.37 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 3230, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int (a \cosh (x)+a)^{3/2} (A+B \cosh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{5} (5 A+3 B) \int (\cosh (x) a+a)^{3/2}dx+\frac {2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {1}{5} (5 A+3 B) \int \left (\sin \left (i x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{5} (5 A+3 B) \left (\frac {4}{3} a \int \sqrt {\cosh (x) a+a}dx+\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)+a}\right )+\frac {2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {1}{5} (5 A+3 B) \left (\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)+a}+\frac {4}{3} a \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right ) a+a}dx\right )\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {1}{5} (5 A+3 B) \left (\frac {8 a^2 \sinh (x)}{3 \sqrt {a \cosh (x)+a}}+\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)+a}\right )+\frac {2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2}\) |
Input:
Int[(a + a*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]
Output:
(2*B*(a + a*Cosh[x])^(3/2)*Sinh[x])/5 + ((5*A + 3*B)*((8*a^2*Sinh[x])/(3*S qrt[a + a*Cosh[x]]) + (2*a*Sqrt[a + a*Cosh[x]]*Sinh[x])/3))/5
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 0.83 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {4 \cosh \left (\frac {x}{2}\right ) a^{2} \sinh \left (\frac {x}{2}\right ) \left (6 B \sinh \left (\frac {x}{2}\right )^{4}+\left (5 A +15 B \right ) \sinh \left (\frac {x}{2}\right )^{2}+15 A +15 B \right ) \sqrt {2}}{15 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(57\) |
| parts | \(\frac {4 A \,a^{2} \cosh \left (\frac {x}{2}\right ) \sinh \left (\frac {x}{2}\right ) \left (\cosh \left (\frac {x}{2}\right )^{2}+2\right ) \sqrt {2}}{3 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}+\frac {4 B \cosh \left (\frac {x}{2}\right ) a^{2} \sinh \left (\frac {x}{2}\right ) \left (2 \cosh \left (\frac {x}{2}\right )^{4}+\cosh \left (\frac {x}{2}\right )^{2}+2\right ) \sqrt {2}}{5 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(80\) |
Input:
int((a+cosh(x)*a)^(3/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
Output:
4/15*cosh(1/2*x)*a^2*sinh(1/2*x)*(6*B*sinh(1/2*x)^4+(5*A+15*B)*sinh(1/2*x) ^2+15*A+15*B)*2^(1/2)/(a*cosh(1/2*x)^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (56) = 112\).
Time = 0.11 (sec) , antiderivative size = 279, normalized size of antiderivative = 4.10 \[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {\sqrt {\frac {1}{2}} {\left (3 \, B a \cosh \left (x\right )^{5} + 3 \, B a \sinh \left (x\right )^{5} + 5 \, {\left (2 \, A + 3 \, B\right )} a \cosh \left (x\right )^{4} + 30 \, {\left (3 \, A + 2 \, B\right )} a \cosh \left (x\right )^{3} + 5 \, {\left (3 \, B a \cosh \left (x\right ) + {\left (2 \, A + 3 \, B\right )} a\right )} \sinh \left (x\right )^{4} - 30 \, {\left (3 \, A + 2 \, B\right )} a \cosh \left (x\right )^{2} + 10 \, {\left (3 \, B a \cosh \left (x\right )^{2} + 2 \, {\left (2 \, A + 3 \, B\right )} a \cosh \left (x\right ) + 3 \, {\left (3 \, A + 2 \, B\right )} a\right )} \sinh \left (x\right )^{3} - 5 \, {\left (2 \, A + 3 \, B\right )} a \cosh \left (x\right ) + 30 \, {\left (B a \cosh \left (x\right )^{3} + {\left (2 \, A + 3 \, B\right )} a \cosh \left (x\right )^{2} + 3 \, {\left (3 \, A + 2 \, B\right )} a \cosh \left (x\right ) - {\left (3 \, A + 2 \, B\right )} a\right )} \sinh \left (x\right )^{2} - 3 \, B a + 5 \, {\left (3 \, B a \cosh \left (x\right )^{4} + 4 \, {\left (2 \, A + 3 \, B\right )} a \cosh \left (x\right )^{3} + 18 \, {\left (3 \, A + 2 \, B\right )} a \cosh \left (x\right )^{2} - 12 \, {\left (3 \, A + 2 \, B\right )} a \cosh \left (x\right ) - {\left (2 \, A + 3 \, B\right )} a\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{30 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \] Input:
integrate((a+a*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="fricas")
Output:
1/30*sqrt(1/2)*(3*B*a*cosh(x)^5 + 3*B*a*sinh(x)^5 + 5*(2*A + 3*B)*a*cosh(x )^4 + 30*(3*A + 2*B)*a*cosh(x)^3 + 5*(3*B*a*cosh(x) + (2*A + 3*B)*a)*sinh( x)^4 - 30*(3*A + 2*B)*a*cosh(x)^2 + 10*(3*B*a*cosh(x)^2 + 2*(2*A + 3*B)*a* cosh(x) + 3*(3*A + 2*B)*a)*sinh(x)^3 - 5*(2*A + 3*B)*a*cosh(x) + 30*(B*a*c osh(x)^3 + (2*A + 3*B)*a*cosh(x)^2 + 3*(3*A + 2*B)*a*cosh(x) - (3*A + 2*B) *a)*sinh(x)^2 - 3*B*a + 5*(3*B*a*cosh(x)^4 + 4*(2*A + 3*B)*a*cosh(x)^3 + 1 8*(3*A + 2*B)*a*cosh(x)^2 - 12*(3*A + 2*B)*a*cosh(x) - (2*A + 3*B)*a)*sinh (x))*sqrt(a/(cosh(x) + sinh(x)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^ 2)
\[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int \left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \cosh {\left (x \right )}\right )\, dx \] Input:
integrate((a+a*cosh(x))**(3/2)*(A+B*cosh(x)),x)
Output:
Integral((a*(cosh(x) + 1))**(3/2)*(A + B*cosh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (56) = 112\).
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.40 \[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {1}{6} \, {\left (\sqrt {2} a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} + 9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} - 9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} - \sqrt {2} a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}\right )} A + \frac {1}{20} \, {\left ({\left (\sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} + 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}\right )} e^{\left (\frac {7}{2} \, x\right )} + {\left (5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} - 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}\right )} e^{\left (\frac {3}{2} \, x\right )}\right )} B \] Input:
integrate((a+a*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="maxima")
Output:
1/6*(sqrt(2)*a^(3/2)*e^(3/2*x) + 9*sqrt(2)*a^(3/2)*e^(1/2*x) - 9*sqrt(2)*a ^(3/2)*e^(-1/2*x) - sqrt(2)*a^(3/2)*e^(-3/2*x))*A + 1/20*((sqrt(2)*a^(3/2) *e^(-x) + 5*sqrt(2)*a^(3/2)*e^(-2*x) + 15*sqrt(2)*a^(3/2)*e^(-3*x) - 5*sqr t(2)*a^(3/2)*e^(-4*x))*e^(7/2*x) + (5*sqrt(2)*a^(3/2)*e^(-x) - 15*sqrt(2)* a^(3/2)*e^(-2*x) - 5*sqrt(2)*a^(3/2)*e^(-3*x) - sqrt(2)*a^(3/2)*e^(-4*x))* e^(3/2*x))*B
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.66 \[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {1}{60} \, \sqrt {2} {\left (\frac {{\left (90 \, A a^{4} e^{\left (2 \, x\right )} + 60 \, B a^{4} e^{\left (2 \, x\right )} + 10 \, A a^{4} e^{x} + 15 \, B a^{4} e^{x} + 3 \, B a^{4}\right )} e^{\left (-\frac {5}{2} \, x\right )}}{a^{\frac {5}{2}}} - \frac {3 \, B a^{\frac {13}{2}} e^{\left (\frac {5}{2} \, x\right )} + 10 \, A a^{\frac {13}{2}} e^{\left (\frac {3}{2} \, x\right )} + 15 \, B a^{\frac {13}{2}} e^{\left (\frac {3}{2} \, x\right )} + 90 \, A a^{\frac {13}{2}} e^{\left (\frac {1}{2} \, x\right )} + 60 \, B a^{\frac {13}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{5}}\right )} \] Input:
integrate((a+a*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="giac")
Output:
-1/60*sqrt(2)*((90*A*a^4*e^(2*x) + 60*B*a^4*e^(2*x) + 10*A*a^4*e^x + 15*B* a^4*e^x + 3*B*a^4)*e^(-5/2*x)/a^(5/2) - (3*B*a^(13/2)*e^(5/2*x) + 10*A*a^( 13/2)*e^(3/2*x) + 15*B*a^(13/2)*e^(3/2*x) + 90*A*a^(13/2)*e^(1/2*x) + 60*B *a^(13/2)*e^(1/2*x))/a^5)
Timed out. \[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \] Input:
int((A + B*cosh(x))*(a + a*cosh(x))^(3/2),x)
Output:
int((A + B*cosh(x))*(a + a*cosh(x))^(3/2), x)
\[ \int (a+a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {\cosh \left (x \right )+1}d x \right ) a +\left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) a +\left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) b +\left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )^{2}d x \right ) b \right ) \] Input:
int((a+a*cosh(x))^(3/2)*(A+B*cosh(x)),x)
Output:
sqrt(a)*a*(int(sqrt(cosh(x) + 1),x)*a + int(sqrt(cosh(x) + 1)*cosh(x),x)*a + int(sqrt(cosh(x) + 1)*cosh(x),x)*b + int(sqrt(cosh(x) + 1)*cosh(x)**2,x )*b)