Integrand size = 10, antiderivative size = 69 \[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=-\frac {10 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{21 b}+\frac {10 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{21 b}+\frac {2 \cosh ^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{7 b} \] Output:
-10/21*I*InverseJacobiAM(1/2*I*(b*x+a),2^(1/2))/b+10/21*cosh(b*x+a)^(1/2)* sinh(b*x+a)/b+2/7*cosh(b*x+a)^(5/2)*sinh(b*x+a)/b
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\frac {-20 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sqrt {\cosh (a+b x)} (23 \sinh (a+b x)+3 \sinh (3 (a+b x)))}{42 b} \] Input:
Integrate[Cosh[a + b*x]^(7/2),x]
Output:
((-20*I)*EllipticF[(I/2)*(a + b*x), 2] + Sqrt[Cosh[a + b*x]]*(23*Sinh[a + b*x] + 3*Sinh[3*(a + b*x)]))/(42*b)
Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3115, 3042, 3115, 3042, 3120}
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.
\(\displaystyle \int \cosh ^{\frac {7}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^{7/2}dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{7} \int \cosh ^{\frac {3}{2}}(a+b x)dx+\frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{7} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^{3/2}dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cosh (a+b x)}}dx+\frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}\right )+\frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{7} \left (\frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}}dx\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \sinh (a+b x) \cosh ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{7} \left (\frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b}\right )\) |
Input:
Int[Cosh[a + b*x]^(7/2),x]
Output:
(2*Cosh[a + b*x]^(5/2)*Sinh[a + b*x])/(7*b) + (5*((((-2*I)/3)*EllipticF[(I /2)*(a + b*x), 2])/b + (2*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(3*b)))/7
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(200\) vs. \(2(57)=114\).
Time = 7.47 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.91
method | result | size |
default | \(\frac {2 \sqrt {\left (2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (48 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}-120 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}+128 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}-72 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+5 \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+16 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{21 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\) | \(201\) |
Input:
int(cosh(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
2/21*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(48*cosh(1/ 2*b*x+1/2*a)^9-120*cosh(1/2*b*x+1/2*a)^7+128*cosh(1/2*b*x+1/2*a)^5-72*cosh (1/2*b*x+1/2*a)^3+5*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a) ^2+1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+16*cosh(1/2*b*x+1/2*a)) /(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/sinh(1/2*b*x+1/2*a) /(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (56) = 112\).
Time = 0.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 4.72 \[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\frac {40 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{6} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + {\left (45 \, \cosh \left (b x + a\right )^{2} + 23\right )} \sinh \left (b x + a\right )^{4} + 23 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{3} + 23 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (45 \, \cosh \left (b x + a\right )^{4} + 138 \, \cosh \left (b x + a\right )^{2} - 23\right )} \sinh \left (b x + a\right )^{2} - 23 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (9 \, \cosh \left (b x + a\right )^{5} + 46 \, \cosh \left (b x + a\right )^{3} - 23 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 3\right )} \sqrt {\cosh \left (b x + a\right )}}{84 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \] Input:
integrate(cosh(b*x+a)^(7/2),x, algorithm="fricas")
Output:
1/84*(40*(sqrt(2)*cosh(b*x + a)^3 + 3*sqrt(2)*cosh(b*x + a)^2*sinh(b*x + a ) + 3*sqrt(2)*cosh(b*x + a)*sinh(b*x + a)^2 + sqrt(2)*sinh(b*x + a)^3)*wei erstrassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a)) + (3*cosh(b*x + a)^ 6 + 18*cosh(b*x + a)*sinh(b*x + a)^5 + 3*sinh(b*x + a)^6 + (45*cosh(b*x + a)^2 + 23)*sinh(b*x + a)^4 + 23*cosh(b*x + a)^4 + 4*(15*cosh(b*x + a)^3 + 23*cosh(b*x + a))*sinh(b*x + a)^3 + (45*cosh(b*x + a)^4 + 138*cosh(b*x + a )^2 - 23)*sinh(b*x + a)^2 - 23*cosh(b*x + a)^2 + 2*(9*cosh(b*x + a)^5 + 46 *cosh(b*x + a)^3 - 23*cosh(b*x + a))*sinh(b*x + a) - 3)*sqrt(cosh(b*x + a) ))/(b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a)^2*sinh(b*x + a) + 3*b*cosh(b*x + a)*sinh(b*x + a)^2 + b*sinh(b*x + a)^3)
Timed out. \[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\text {Timed out} \] Input:
integrate(cosh(b*x+a)**(7/2),x)
Output:
Timed out
\[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\int { \cosh \left (b x + a\right )^{\frac {7}{2}} \,d x } \] Input:
integrate(cosh(b*x+a)^(7/2),x, algorithm="maxima")
Output:
integrate(cosh(b*x + a)^(7/2), x)
\[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\int { \cosh \left (b x + a\right )^{\frac {7}{2}} \,d x } \] Input:
integrate(cosh(b*x+a)^(7/2),x, algorithm="giac")
Output:
integrate(cosh(b*x + a)^(7/2), x)
Timed out. \[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\int {\mathrm {cosh}\left (a+b\,x\right )}^{7/2} \,d x \] Input:
int(cosh(a + b*x)^(7/2),x)
Output:
int(cosh(a + b*x)^(7/2), x)
\[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx=\int \sqrt {\cosh \left (b x +a \right )}\, \cosh \left (b x +a \right )^{3}d x \] Input:
int(cosh(b*x+a)^(7/2),x)
Output:
int(sqrt(cosh(a + b*x))*cosh(a + b*x)**3,x)