Integrand size = 15, antiderivative size = 91 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\frac {\sinh (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac {3 \sinh (a-c+(b-d) x)}{8 (b-d)}+\frac {3 \sinh (a+c+(b+d) x)}{8 (b+d)}+\frac {\sinh (a+3 c+(b+3 d) x)}{8 (b+3 d)} \] Output:
sinh(a-3*c+(b-3*d)*x)/(8*b-24*d)+3*sinh(a-c+(b-d)*x)/(8*b-8*d)+3*sinh(a+c+ (b+d)*x)/(8*b+8*d)+sinh(a+3*c+(b+3*d)*x)/(8*b+24*d)
Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\frac {1}{8} \left (\frac {\sinh (a-3 c+b x-3 d x)}{b-3 d}+\frac {3 \sinh (a-c+b x-d x)}{b-d}+\frac {\sinh (a+3 c+b x+3 d x)}{b+3 d}+\frac {3 \sinh (a+c+(b+d) x)}{b+d}\right ) \] Input:
Integrate[Cosh[a + b*x]*Cosh[c + d*x]^3,x]
Output:
(Sinh[a - 3*c + b*x - 3*d*x]/(b - 3*d) + (3*Sinh[a - c + b*x - d*x])/(b - d) + Sinh[a + 3*c + b*x + 3*d*x]/(b + 3*d) + (3*Sinh[a + c + (b + d)*x])/( b + d))/8
Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6148, 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.
\(\displaystyle \int \cosh (a+b x) \cosh ^3(c+d x) \, dx\) |
\(\Big \downarrow \) 6148 |
\(\displaystyle \int \left (\frac {1}{8} \cosh (a+x (b-3 d)-3 c)+\frac {3}{8} \cosh (a+x (b-d)-c)+\frac {3}{8} \cosh (a+x (b+d)+c)+\frac {1}{8} \cosh (a+x (b+3 d)+3 c)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sinh (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \sinh (a+x (b-d)-c)}{8 (b-d)}+\frac {3 \sinh (a+x (b+d)+c)}{8 (b+d)}+\frac {\sinh (a+x (b+3 d)+3 c)}{8 (b+3 d)}\) |
Input:
Int[Cosh[a + b*x]*Cosh[c + d*x]^3,x]
Output:
Sinh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sinh[a - c + (b - d)*x])/(8 *(b - d)) + (3*Sinh[a + c + (b + d)*x])/(8*(b + d)) + Sinh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))
Int[Cosh[v_]^(p_.)*Cosh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Cosh[v ]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x ]))
Time = 2.88 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\sinh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 b -24 d}+\frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}\) | \(84\) |
risch | \(\frac {\left (b \,{\mathrm e}^{2 b x +2 a}-3 d \,{\mathrm e}^{2 b x +2 a}-b -3 d \right ) {\mathrm e}^{-b x +3 d x -a +3 c}}{16 \left (b +3 d \right ) \left (b -3 d \right )}+\frac {3 \left (b \,{\mathrm e}^{2 b x +2 a}-d \,{\mathrm e}^{2 b x +2 a}-b -d \right ) {\mathrm e}^{-b x +d x -a +c}}{16 \left (b +d \right ) \left (b -d \right )}+\frac {3 \left (b \,{\mathrm e}^{2 b x +2 a}+d \,{\mathrm e}^{2 b x +2 a}-b +d \right ) {\mathrm e}^{-b x -d x -a -c}}{16 \left (b +d \right ) \left (b -d \right )}+\frac {\left (b \,{\mathrm e}^{2 b x +2 a}+3 d \,{\mathrm e}^{2 b x +2 a}-b +3 d \right ) {\mathrm e}^{-b x -3 d x -a -3 c}}{16 \left (b +3 d \right ) \left (b -3 d \right )}\) | \(240\) |
parallelrisch | \(\frac {2 b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (b^{2}-7 d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-6 d \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \left (b^{2}-3 d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+6 b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (b^{2}+d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-12 d \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \left (b^{2}+d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+6 b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (b^{2}+d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-6 d \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \left (b^{2}-3 d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (b^{2}-7 d^{2}\right )}{\left (b -d \right ) \left (b +d \right ) \left (b +3 d \right ) \left (b -3 d \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (-1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\) | \(298\) |
orering | \(\text {Expression too large to display}\) | \(948\) |
Input:
int(cosh(b*x+a)*cosh(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
1/8*sinh(a-3*c+(b-3*d)*x)/(b-3*d)+3/8*sinh(a-c+(b-d)*x)/(b-d)+3/8/(b+d)*si nh(a+c+(b+d)*x)+1/8/(b+3*d)*sinh(a+3*c+(b+3*d)*x)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (83) = 166\).
Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.38 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\frac {3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - 3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} + {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \, {\left (3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} + {\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )}} \] Input:
integrate(cosh(b*x+a)*cosh(d*x+c)^3,x, algorithm="fricas")
Output:
1/4*(3*(b^3 - b*d^2)*cosh(d*x + c)*sinh(b*x + a)*sinh(d*x + c)^2 - 3*(b^2* d - d^3)*cosh(b*x + a)*sinh(d*x + c)^3 + ((b^3 - b*d^2)*cosh(d*x + c)^3 + 3*(b^3 - 9*b*d^2)*cosh(d*x + c))*sinh(b*x + a) - 3*(3*(b^2*d - d^3)*cosh(b *x + a)*cosh(d*x + c)^2 + (b^2*d - 9*d^3)*cosh(b*x + a))*sinh(d*x + c))/(( b^4 - 10*b^2*d^2 + 9*d^4)*cosh(b*x + a)^2 - (b^4 - 10*b^2*d^2 + 9*d^4)*sin h(b*x + a)^2)
Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (76) = 152\).
Time = 1.91 (sec) , antiderivative size = 921, normalized size of antiderivative = 10.12 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(b*x+a)*cosh(d*x+c)**3,x)
Output:
Piecewise((x*cosh(a)*cosh(c)**3, Eq(b, 0) & Eq(d, 0)), (x*sinh(a - 3*d*x)* sinh(c + d*x)**3/8 + 3*x*sinh(a - 3*d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 + 3*x*sinh(c + d*x)**2*cosh(a - 3*d*x)*cosh(c + d*x)/8 + x*cosh(a - 3*d*x) *cosh(c + d*x)**3/8 + sinh(a - 3*d*x)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) - 7*sinh(a - 3*d*x)*cosh(c + d*x)**3/(24*d) + sinh(c + d*x)**3*cosh(a - 3 *d*x)/(8*d), Eq(b, -3*d)), (-3*x*sinh(a - d*x)*sinh(c + d*x)**3/8 + 3*x*si nh(a - d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 - 3*x*sinh(c + d*x)**2*cosh(a - d*x)*cosh(c + d*x)/8 + 3*x*cosh(a - d*x)*cosh(c + d*x)**3/8 + 3*sinh(a - d*x)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) - 5*sinh(a - d*x)*cosh(c + d*x )**3/(8*d) + 3*sinh(c + d*x)**3*cosh(a - d*x)/(8*d), Eq(b, -d)), (3*x*sinh (a + d*x)*sinh(c + d*x)**3/8 - 3*x*sinh(a + d*x)*sinh(c + d*x)*cosh(c + d* x)**2/8 - 3*x*sinh(c + d*x)**2*cosh(a + d*x)*cosh(c + d*x)/8 + 3*x*cosh(a + d*x)*cosh(c + d*x)**3/8 - 3*sinh(a + d*x)*sinh(c + d*x)**2*cosh(c + d*x) /(4*d) + 5*sinh(a + d*x)*cosh(c + d*x)**3/(8*d) + 3*sinh(c + d*x)**3*cosh( a + d*x)/(8*d), Eq(b, d)), (-x*sinh(a + 3*d*x)*sinh(c + d*x)**3/8 - 3*x*si nh(a + 3*d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 + 3*x*sinh(c + d*x)**2*cosh (a + 3*d*x)*cosh(c + d*x)/8 + x*cosh(a + 3*d*x)*cosh(c + d*x)**3/8 - sinh( a + 3*d*x)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) + 7*sinh(a + 3*d*x)*cosh(c + d*x)**3/(24*d) + sinh(c + d*x)**3*cosh(a + 3*d*x)/(8*d), Eq(b, 3*d)), ( b**3*sinh(a + b*x)*cosh(c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) - 3*...
Exception generated. \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(b*x+a)*cosh(d*x+c)^3,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(-(3*d)/b>0)', see `assume?` for more detai
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (83) = 166\).
Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.97 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\frac {e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{16 \, {\left (b + 3 \, d\right )}} + \frac {3 \, e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} + \frac {3 \, e^{\left (b x - d x + a - c\right )}}{16 \, {\left (b - d\right )}} + \frac {e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{16 \, {\left (b - 3 \, d\right )}} - \frac {e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{16 \, {\left (b - 3 \, d\right )}} - \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x - d x - a - c\right )}}{16 \, {\left (b + d\right )}} - \frac {e^{\left (-b x - 3 \, d x - a - 3 \, c\right )}}{16 \, {\left (b + 3 \, d\right )}} \] Input:
integrate(cosh(b*x+a)*cosh(d*x+c)^3,x, algorithm="giac")
Output:
1/16*e^(b*x + 3*d*x + a + 3*c)/(b + 3*d) + 3/16*e^(b*x + d*x + a + c)/(b + d) + 3/16*e^(b*x - d*x + a - c)/(b - d) + 1/16*e^(b*x - 3*d*x + a - 3*c)/ (b - 3*d) - 1/16*e^(-b*x + 3*d*x - a + 3*c)/(b - 3*d) - 3/16*e^(-b*x + d*x - a + c)/(b - d) - 3/16*e^(-b*x - d*x - a - c)/(b + d) - 1/16*e^(-b*x - 3 *d*x - a - 3*c)/(b + 3*d)
Time = 0.54 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.98 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\frac {b\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2-7\,d^2\right )}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (b^2\,d-3\,d^3\right )}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {6\,d^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{b^4-10\,b^2\,d^2+9\,d^4}+\frac {6\,b\,d^2\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{b^4-10\,b^2\,d^2+9\,d^4} \] Input:
int(cosh(a + b*x)*cosh(c + d*x)^3,x)
Output:
(b*cosh(c + d*x)^3*sinh(a + b*x)*(b^2 - 7*d^2))/(b^4 + 9*d^4 - 10*b^2*d^2) - (3*cosh(a + b*x)*cosh(c + d*x)^2*sinh(c + d*x)*(b^2*d - 3*d^3))/(b^4 + 9*d^4 - 10*b^2*d^2) - (6*d^3*cosh(a + b*x)*sinh(c + d*x)^3)/(b^4 + 9*d^4 - 10*b^2*d^2) + (6*b*d^2*cosh(c + d*x)*sinh(a + b*x)*sinh(c + d*x)^2)/(b^4 + 9*d^4 - 10*b^2*d^2)
Time = 0.25 (sec) , antiderivative size = 574, normalized size of antiderivative = 6.31 \[ \int \cosh (a+b x) \cosh ^3(c+d x) \, dx=\frac {-b^{3}-3 d^{3}+b \,d^{2}+e^{6 d x +6 c} b \,d^{2}-e^{2 b x +2 a} b \,d^{2}-e^{2 b x +6 d x +2 a +6 c} b \,d^{2}+27 e^{2 b x +4 d x +2 a +4 c} d^{3}+3 e^{2 b x +2 d x +2 a +2 c} b^{3}-27 e^{2 b x +2 d x +2 a +2 c} d^{3}-3 e^{2 b x +2 a} d^{3}+3 e^{6 d x +6 c} d^{3}+27 e^{4 d x +4 c} d^{3}-3 e^{2 d x +2 c} b^{3}-27 e^{2 d x +2 c} d^{3}+3 b^{2} d -3 e^{2 b x +6 d x +2 a +6 c} b^{2} d -3 e^{2 b x +4 d x +2 a +4 c} b^{2} d -27 e^{2 b x +4 d x +2 a +4 c} b \,d^{2}+3 e^{2 b x +2 d x +2 a +2 c} b^{2} d -27 e^{2 b x +2 d x +2 a +2 c} b \,d^{2}+3 e^{2 b x +2 a} b^{2} d -3 e^{6 d x +6 c} b^{2} d -3 e^{4 d x +4 c} b^{2} d +27 e^{4 d x +4 c} b \,d^{2}+3 e^{2 d x +2 c} b^{2} d +27 e^{2 d x +2 c} b \,d^{2}-3 e^{4 d x +4 c} b^{3}+e^{2 b x +6 d x +2 a +6 c} b^{3}+e^{2 b x +2 a} b^{3}-e^{6 d x +6 c} b^{3}+3 e^{2 b x +6 d x +2 a +6 c} d^{3}+3 e^{2 b x +4 d x +2 a +4 c} b^{3}}{16 e^{b x +3 d x +a +3 c} \left (b^{4}-10 b^{2} d^{2}+9 d^{4}\right )} \] Input:
int(cosh(b*x+a)*cosh(d*x+c)^3,x)
Output:
(e**(2*a + 2*b*x + 6*c + 6*d*x)*b**3 - 3*e**(2*a + 2*b*x + 6*c + 6*d*x)*b* *2*d - e**(2*a + 2*b*x + 6*c + 6*d*x)*b*d**2 + 3*e**(2*a + 2*b*x + 6*c + 6 *d*x)*d**3 + 3*e**(2*a + 2*b*x + 4*c + 4*d*x)*b**3 - 3*e**(2*a + 2*b*x + 4 *c + 4*d*x)*b**2*d - 27*e**(2*a + 2*b*x + 4*c + 4*d*x)*b*d**2 + 27*e**(2*a + 2*b*x + 4*c + 4*d*x)*d**3 + 3*e**(2*a + 2*b*x + 2*c + 2*d*x)*b**3 + 3*e **(2*a + 2*b*x + 2*c + 2*d*x)*b**2*d - 27*e**(2*a + 2*b*x + 2*c + 2*d*x)*b *d**2 - 27*e**(2*a + 2*b*x + 2*c + 2*d*x)*d**3 + e**(2*a + 2*b*x)*b**3 + 3 *e**(2*a + 2*b*x)*b**2*d - e**(2*a + 2*b*x)*b*d**2 - 3*e**(2*a + 2*b*x)*d* *3 - e**(6*c + 6*d*x)*b**3 - 3*e**(6*c + 6*d*x)*b**2*d + e**(6*c + 6*d*x)* b*d**2 + 3*e**(6*c + 6*d*x)*d**3 - 3*e**(4*c + 4*d*x)*b**3 - 3*e**(4*c + 4 *d*x)*b**2*d + 27*e**(4*c + 4*d*x)*b*d**2 + 27*e**(4*c + 4*d*x)*d**3 - 3*e **(2*c + 2*d*x)*b**3 + 3*e**(2*c + 2*d*x)*b**2*d + 27*e**(2*c + 2*d*x)*b*d **2 - 27*e**(2*c + 2*d*x)*d**3 - b**3 + 3*b**2*d + b*d**2 - 3*d**3)/(16*e* *(a + b*x + 3*c + 3*d*x)*(b**4 - 10*b**2*d**2 + 9*d**4))