Integrand size = 15, antiderivative size = 97 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=-\frac {3 \cosh (a-c+(b-d) x)}{8 (b-d)}+\frac {\cosh (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac {3 \cosh (a+c+(b+d) x)}{8 (b+d)}+\frac {\cosh (3 a+c+(3 b+d) x)}{8 (3 b+d)} \] Output:
-3*cosh(a-c+(b-d)*x)/(8*b-8*d)+cosh(3*a-c+(3*b-d)*x)/(24*b-8*d)-3*cosh(a+c +(b+d)*x)/(8*b+8*d)+cosh(3*a+c+(3*b+d)*x)/(24*b+8*d)
Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\frac {1}{8} \left (-\frac {3 \cosh (a-c+b x-d x)}{b-d}+\frac {\cosh (3 a-c+3 b x-d x)}{3 b-d}+\frac {\cosh (3 a+c+3 b x+d x)}{3 b+d}-\frac {3 \cosh (a+c+(b+d) x)}{b+d}\right ) \] Input:
Integrate[Cosh[c + d*x]*Sinh[a + b*x]^3,x]
Output:
((-3*Cosh[a - c + b*x - d*x])/(b - d) + Cosh[3*a - c + 3*b*x - d*x]/(3*b - d) + Cosh[3*a + c + 3*b*x + d*x]/(3*b + d) - (3*Cosh[a + c + (b + d)*x])/ (b + d))/8
Time = 0.30 (sec) , antiderivative size = 97, 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 = {6152, 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.
\(\displaystyle \int \sinh ^3(a+b x) \cosh (c+d x) \, dx\) |
\(\Big \downarrow \) 6152 |
\(\displaystyle \int \left (-\frac {3}{8} \sinh (a+x (b-d)-c)+\frac {1}{8} \sinh (3 a+x (3 b-d)-c)-\frac {3}{8} \sinh (a+x (b+d)+c)+\frac {1}{8} \sinh (3 a+x (3 b+d)+c)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac {\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac {\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)}\) |
Input:
Int[Cosh[c + d*x]*Sinh[a + b*x]^3,x]
Output:
(-3*Cosh[a - c + (b - d)*x])/(8*(b - d)) + Cosh[3*a - c + (3*b - d)*x]/(8* (3*b - d)) - (3*Cosh[a + c + (b + d)*x])/(8*(b + d)) + Cosh[3*a + c + (3*b + d)*x]/(8*(3*b + d))
Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[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 = 2.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}\) | \(90\) |
parallelrisch | \(\frac {-12 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{6} b^{3}+24 b^{2} d \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-12 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b \,d^{2}+36 b^{3}-12 b \,d^{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\left (-64 b^{2} d +16 d^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\left (\left (36 b^{3}-12 b \,d^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-12 b \,d^{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+24 \tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d -12 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{3}}{9 \left (-1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (b +\frac {d}{3}\right ) \left (b +d \right ) \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (b -d \right ) \left (b -\frac {d}{3}\right )}\) | \(281\) |
risch | \(\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}+d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}+27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}-3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}+3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}+b^{2} d -3 b \,d^{2}-d^{3}\right ) {\mathrm e}^{-3 b x +d x -3 a +c}}{16 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}+\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}+b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}-d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}+3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}+27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}-3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}-b^{2} d -3 b \,d^{2}+d^{3}\right ) {\mathrm e}^{-3 b x -d x -3 a -c}}{16 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}\) | \(480\) |
orering | \(\text {Expression too large to display}\) | \(948\) |
Input:
int(cosh(d*x+c)*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-3/8*cosh(a-c+(b-d)*x)/(b-d)-3/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(3*a-c+(3 *b-d)*x)/(3*b-d)+1/8*cosh(3*a+c+(3*b+d)*x)/(3*b+d)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (89) = 178\).
Time = 0.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.51 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\frac {9 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) - {\left ({\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d - d^{3} - {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)^3,x, algorithm="fricas")
Output:
1/4*(9*(b^3 - b*d^2)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^2 + 3*((b^3 - b*d^2)*cosh(b*x + a)^3 - (9*b^3 - b*d^2)*cosh(b*x + a))*cosh(d*x + c) - ((b^2*d - d^3)*sinh(b*x + a)^3 - 3*(9*b^2*d - d^3 - (b^2*d - d^3)*cosh(b* x + a)^2)*sinh(b*x + a))*sinh(d*x + c))/((9*b^4 - 10*b^2*d^2 + d^4)*cosh(b *x + a)^4 - 2*(9*b^4 - 10*b^2*d^2 + d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (9*b^4 - 10*b^2*d^2 + d^4)*sinh(b*x + a)^4)
Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (76) = 152\).
Time = 1.97 (sec) , antiderivative size = 937, normalized size of antiderivative = 9.66 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)**3,x)
Output:
Piecewise((x*sinh(a)**3*cosh(c), Eq(b, 0) & Eq(d, 0)), (3*x*sinh(a - d*x)* *3*cosh(c + d*x)/8 + 3*x*sinh(a - d*x)**2*sinh(c + d*x)*cosh(a - d*x)/8 - 3*x*sinh(a - d*x)*cosh(a - d*x)**2*cosh(c + d*x)/8 - 3*x*sinh(c + d*x)*cos h(a - d*x)**3/8 + 5*sinh(a - d*x)**3*sinh(c + d*x)/(8*d) - 3*sinh(a - d*x) *sinh(c + d*x)*cosh(a - d*x)**2/(4*d) - 3*cosh(a - d*x)**3*cosh(c + d*x)/( 8*d), Eq(b, -d)), (x*sinh(a - d*x/3)**3*cosh(c + d*x)/8 + 3*x*sinh(a - d*x /3)**2*sinh(c + d*x)*cosh(a - d*x/3)/8 + 3*x*sinh(a - d*x/3)*cosh(a - d*x/ 3)**2*cosh(c + d*x)/8 + x*sinh(c + d*x)*cosh(a - d*x/3)**3/8 + 7*sinh(a - d*x/3)**3*sinh(c + d*x)/(8*d) - 3*sinh(a - d*x/3)*sinh(c + d*x)*cosh(a - d *x/3)**2/(4*d) - 3*cosh(a - d*x/3)**3*cosh(c + d*x)/(8*d), Eq(b, -d/3)), ( x*sinh(a + d*x/3)**3*cosh(c + d*x)/8 - 3*x*sinh(a + d*x/3)**2*sinh(c + d*x )*cosh(a + d*x/3)/8 + 3*x*sinh(a + d*x/3)*cosh(a + d*x/3)**2*cosh(c + d*x) /8 - x*sinh(c + d*x)*cosh(a + d*x/3)**3/8 + 7*sinh(a + d*x/3)**3*sinh(c + d*x)/(8*d) - 3*sinh(a + d*x/3)*sinh(c + d*x)*cosh(a + d*x/3)**2/(4*d) + 3* cosh(a + d*x/3)**3*cosh(c + d*x)/(8*d), Eq(b, d/3)), (3*x*sinh(a + d*x)**3 *cosh(c + d*x)/8 - 3*x*sinh(a + d*x)**2*sinh(c + d*x)*cosh(a + d*x)/8 - 3* x*sinh(a + d*x)*cosh(a + d*x)**2*cosh(c + d*x)/8 + 3*x*sinh(c + d*x)*cosh( a + d*x)**3/8 - sinh(a + d*x)**3*sinh(c + d*x)/(8*d) + 3*sinh(a + d*x)**2* cosh(a + d*x)*cosh(c + d*x)/(4*d) - 3*cosh(a + d*x)**3*cosh(c + d*x)/(8*d) , Eq(b, d)), (9*b**3*sinh(a + b*x)**2*cosh(a + b*x)*cosh(c + d*x)/(9*b*...
Exception generated. \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)^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(-d/b>0)', see `assume?` for more details)I
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (89) = 178\).
Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.89 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{16 \, {\left (3 \, b + d\right )}} + \frac {e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{16 \, {\left (3 \, b - d\right )}} - \frac {3 \, e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (b x - d x + a - c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-b x - d x - a - c\right )}}{16 \, {\left (b + d\right )}} + \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{16 \, {\left (3 \, b - d\right )}} + \frac {e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{16 \, {\left (3 \, b + d\right )}} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)^3,x, algorithm="giac")
Output:
1/16*e^(3*b*x + d*x + 3*a + c)/(3*b + d) + 1/16*e^(3*b*x - d*x + 3*a - c)/ (3*b - d) - 3/16*e^(b*x + d*x + a + c)/(b + d) - 3/16*e^(b*x - d*x + a - c )/(b - 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^(-3*b*x + d*x - 3*a + c)/(3*b - d) + 1/16*e^(-3*b*x - d*x - 3*a - c)/(3*b + d)
Time = 0.55 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.89 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\frac {6\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {d\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (7\,b^2-d^2\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (b\,d^2-3\,b^3\right )}{9\,b^4-10\,b^2\,d^2+d^4}-\frac {6\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {cosh}\left (c+d\,x\right )}{9\,b^4-10\,b^2\,d^2+d^4} \] Input:
int(cosh(c + d*x)*sinh(a + b*x)^3,x)
Output:
(6*b^2*d*cosh(a + b*x)^2*sinh(a + b*x)*sinh(c + d*x))/(9*b^4 + d^4 - 10*b^ 2*d^2) - (d*sinh(a + b*x)^3*sinh(c + d*x)*(7*b^2 - d^2))/(9*b^4 + d^4 - 10 *b^2*d^2) - (3*cosh(a + b*x)*cosh(c + d*x)*sinh(a + b*x)^2*(b*d^2 - 3*b^3) )/(9*b^4 + d^4 - 10*b^2*d^2) - (6*b^3*cosh(a + b*x)^3*cosh(c + d*x))/(9*b^ 4 + d^4 - 10*b^2*d^2)
Time = 0.24 (sec) , antiderivative size = 573, normalized size of antiderivative = 5.91 \[ \int \cosh (c+d x) \sinh ^3(a+b x) \, dx=\frac {27 e^{4 b x +2 d x +4 a +2 c} b^{2} d +3 b^{3}+d^{3}-3 e^{6 b x +2 d x +6 a +2 c} b \,d^{2}-3 e^{6 b x +6 a} b \,d^{2}+3 e^{4 b x +2 d x +4 a +2 c} b \,d^{2}-3 e^{4 b x +2 d x +4 a +2 c} d^{3}-3 b \,d^{2}+3 e^{4 b x +4 a} b \,d^{2}-27 e^{4 b x +4 a} b^{2} d +3 e^{2 b x +2 a} b \,d^{2}-27 e^{2 b x +2 d x +2 a +2 c} b^{3}+3 e^{2 b x +2 d x +2 a +2 c} d^{3}-3 e^{2 b x +2 a} d^{3}+3 e^{2 d x +2 c} b^{3}-e^{2 d x +2 c} d^{3}-b^{2} d -27 e^{2 b x +2 d x +2 a +2 c} b^{2} d +3 e^{2 b x +2 d x +2 a +2 c} b \,d^{2}+27 e^{2 b x +2 a} b^{2} d +e^{2 d x +2 c} b^{2} d -3 e^{2 d x +2 c} b \,d^{2}-e^{6 b x +2 d x +6 a +2 c} b^{2} d +e^{6 b x +2 d x +6 a +2 c} d^{3}-e^{6 b x +6 a} d^{3}+e^{6 b x +6 a} b^{2} d +3 e^{6 b x +2 d x +6 a +2 c} b^{3}+3 e^{6 b x +6 a} b^{3}-27 e^{4 b x +2 d x +4 a +2 c} b^{3}-27 e^{4 b x +4 a} b^{3}+3 e^{4 b x +4 a} d^{3}-27 e^{2 b x +2 a} b^{3}}{16 e^{3 b x +d x +3 a +c} \left (9 b^{4}-10 b^{2} d^{2}+d^{4}\right )} \] Input:
int(cosh(d*x+c)*sinh(b*x+a)^3,x)
Output:
(3*e**(6*a + 6*b*x + 2*c + 2*d*x)*b**3 - e**(6*a + 6*b*x + 2*c + 2*d*x)*b* *2*d - 3*e**(6*a + 6*b*x + 2*c + 2*d*x)*b*d**2 + e**(6*a + 6*b*x + 2*c + 2 *d*x)*d**3 + 3*e**(6*a + 6*b*x)*b**3 + e**(6*a + 6*b*x)*b**2*d - 3*e**(6*a + 6*b*x)*b*d**2 - e**(6*a + 6*b*x)*d**3 - 27*e**(4*a + 4*b*x + 2*c + 2*d* x)*b**3 + 27*e**(4*a + 4*b*x + 2*c + 2*d*x)*b**2*d + 3*e**(4*a + 4*b*x + 2 *c + 2*d*x)*b*d**2 - 3*e**(4*a + 4*b*x + 2*c + 2*d*x)*d**3 - 27*e**(4*a + 4*b*x)*b**3 - 27*e**(4*a + 4*b*x)*b**2*d + 3*e**(4*a + 4*b*x)*b*d**2 + 3*e **(4*a + 4*b*x)*d**3 - 27*e**(2*a + 2*b*x + 2*c + 2*d*x)*b**3 - 27*e**(2*a + 2*b*x + 2*c + 2*d*x)*b**2*d + 3*e**(2*a + 2*b*x + 2*c + 2*d*x)*b*d**2 + 3*e**(2*a + 2*b*x + 2*c + 2*d*x)*d**3 - 27*e**(2*a + 2*b*x)*b**3 + 27*e** (2*a + 2*b*x)*b**2*d + 3*e**(2*a + 2*b*x)*b*d**2 - 3*e**(2*a + 2*b*x)*d**3 + 3*e**(2*c + 2*d*x)*b**3 + e**(2*c + 2*d*x)*b**2*d - 3*e**(2*c + 2*d*x)* b*d**2 - e**(2*c + 2*d*x)*d**3 + 3*b**3 - b**2*d - 3*b*d**2 + d**3)/(16*e* *(3*a + 3*b*x + c + d*x)*(9*b**4 - 10*b**2*d**2 + d**4))