Integrand size = 15, antiderivative size = 68 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\frac {\sinh (2 a-c+(2 b-d) x)}{4 (2 b-d)}-\frac {\sinh (c+d x)}{2 d}+\frac {\sinh (2 a+c+(2 b+d) x)}{4 (2 b+d)} \] Output:
sinh(2*a-c+(2*b-d)*x)/(8*b-4*d)-1/2*sinh(d*x+c)/d+sinh(2*a+c+(2*b+d)*x)/(8 *b+4*d)
Time = 0.54 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\frac {1}{4} \left (-\frac {2 \cosh (d x) \sinh (c)}{d}-\frac {2 \cosh (c) \sinh (d x)}{d}+\frac {\sinh (2 a-c+2 b x-d x)}{2 b-d}+\frac {\sinh (2 a+c+2 b x+d x)}{2 b+d}\right ) \] Input:
Integrate[Cosh[c + d*x]*Sinh[a + b*x]^2,x]
Output:
((-2*Cosh[d*x]*Sinh[c])/d - (2*Cosh[c]*Sinh[d*x])/d + Sinh[2*a - c + 2*b*x - d*x]/(2*b - d) + Sinh[2*a + c + 2*b*x + d*x]/(2*b + d))/4
Time = 0.25 (sec) , antiderivative size = 68, 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.
\(\displaystyle \int \sinh ^2(a+b x) \cosh (c+d x) \, dx\) |
\(\Big \downarrow \) 6152 |
\(\displaystyle \int \left (\frac {1}{4} \cosh (2 a+x (2 b-d)-c)+\frac {1}{4} \cosh (2 a+x (2 b+d)+c)-\frac {1}{2} \cosh (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac {\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac {\sinh (c+d x)}{2 d}\) |
Input:
Int[Cosh[c + d*x]*Sinh[a + b*x]^2,x]
Output:
Sinh[2*a - c + (2*b - d)*x]/(4*(2*b - d)) - Sinh[c + d*x]/(2*d) + Sinh[2*a + c + (2*b + d)*x]/(4*(2*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 = 1.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{8 b -4 d}-\frac {\sinh \left (d x +c \right )}{2 d}+\frac {\sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 b +4 d}\) | \(63\) |
parallelrisch | \(\frac {\left (2 b d +d^{2}\right ) \sinh \left (2 a -c +\left (2 b -d \right ) x \right )+\left (2 b d -d^{2}\right ) \sinh \left (2 a +c +\left (2 b +d \right ) x \right )+\left (-8 b^{2}+2 d^{2}\right ) \sinh \left (d x +c \right )}{16 b^{2} d -4 d^{3}}\) | \(85\) |
risch | \(-\frac {\left (-2 d \,{\mathrm e}^{4 b x +4 a} b +d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}+2 b d +d^{2}\right ) {\mathrm e}^{-2 b x +d x -2 a +c}}{8 \left (2 b +d \right ) \left (2 b -d \right ) d}+\frac {\left (2 d \,{\mathrm e}^{4 b x +4 a} b +d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}-2 b d +d^{2}\right ) {\mathrm e}^{-2 b x -d x -2 a -c}}{8 \left (2 b +d \right ) \left (2 b -d \right ) d}\) | \(197\) |
orering | \(\frac {\left (16 b^{4}+3 d^{4}\right ) \left (d \sinh \left (d x +c \right ) \sinh \left (b x +a \right )^{2}+2 \cosh \left (d x +c \right ) \sinh \left (b x +a \right ) b \cosh \left (b x +a \right )\right )}{d^{2} \left (16 b^{4}-8 b^{2} d^{2}+d^{4}\right )}-\frac {\left (8 b^{2}+3 d^{2}\right ) \left (d^{3} \sinh \left (d x +c \right ) \sinh \left (b x +a \right )^{2}+6 d^{2} \cosh \left (d x +c \right ) \sinh \left (b x +a \right ) b \cosh \left (b x +a \right )+6 d \sinh \left (d x +c \right ) b^{2} \cosh \left (b x +a \right )^{2}+6 d \sinh \left (d x +c \right ) \sinh \left (b x +a \right )^{2} b^{2}+8 \cosh \left (d x +c \right ) b^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )\right )}{d^{2} \left (16 b^{4}-8 b^{2} d^{2}+d^{4}\right )}+\frac {d^{5} \sinh \left (d x +c \right ) \sinh \left (b x +a \right )^{2}+10 d^{4} \cosh \left (d x +c \right ) \sinh \left (b x +a \right ) b \cosh \left (b x +a \right )+20 d^{3} \sinh \left (d x +c \right ) b^{2} \cosh \left (b x +a \right )^{2}+20 d^{3} \sinh \left (d x +c \right ) \sinh \left (b x +a \right )^{2} b^{2}+80 d^{2} \cosh \left (d x +c \right ) b^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )+40 d \sinh \left (d x +c \right ) b^{4} \sinh \left (b x +a \right )^{2}+40 d \sinh \left (d x +c \right ) b^{4} \cosh \left (b x +a \right )^{2}+32 \cosh \left (d x +c \right ) b^{5} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )}{d^{2} \left (16 b^{4}-8 b^{2} d^{2}+d^{4}\right )}\) | \(414\) |
Input:
int(cosh(d*x+c)*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*sinh(d*x+c)/d+1/4*sinh(2*a-c+(2*b-d)*x)/(2*b-d)+1/4*sinh(2*a+c+(2*b+d )*x)/(2*b+d)
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.68 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\frac {4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left (d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} - d^{2}\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} - {\left (4 \, b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)^2,x, algorithm="fricas")
Output:
1/2*(4*b*d*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a) - (d^2*cosh(b*x + a)^ 2 + d^2*sinh(b*x + a)^2 + 4*b^2 - d^2)*sinh(d*x + c))/((4*b^2*d - d^3)*cos h(b*x + a)^2 - (4*b^2*d - d^3)*sinh(b*x + a)^2)
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (49) = 98\).
Time = 0.73 (sec) , antiderivative size = 408, normalized size of antiderivative = 6.00 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\begin {cases} x \sinh ^{2}{\left (a \right )} \cosh {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh ^{2}{\left (a - \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} + \frac {x \sinh {\left (a - \frac {d x}{2} \right )} \sinh {\left (c + d x \right )} \cosh {\left (a - \frac {d x}{2} \right )}}{2} + \frac {x \cosh ^{2}{\left (a - \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} + \frac {\sinh ^{2}{\left (a - \frac {d x}{2} \right )} \sinh {\left (c + d x \right )}}{d} + \frac {\sinh {\left (a - \frac {d x}{2} \right )} \cosh {\left (a - \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{2 d} & \text {for}\: b = - \frac {d}{2} \\\frac {x \sinh ^{2}{\left (a + \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} - \frac {x \sinh {\left (a + \frac {d x}{2} \right )} \sinh {\left (c + d x \right )} \cosh {\left (a + \frac {d x}{2} \right )}}{2} + \frac {x \cosh ^{2}{\left (a + \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{4} + \frac {\sinh ^{2}{\left (a + \frac {d x}{2} \right )} \sinh {\left (c + d x \right )}}{d} - \frac {\sinh {\left (a + \frac {d x}{2} \right )} \cosh {\left (a + \frac {d x}{2} \right )} \cosh {\left (c + d x \right )}}{2 d} & \text {for}\: b = \frac {d}{2} \\\left (\frac {x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b}\right ) \cosh {\left (c \right )} & \text {for}\: d = 0 \\\frac {2 b^{2} \sinh ^{2}{\left (a + b x \right )} \sinh {\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac {2 b^{2} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} + \frac {2 b d \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac {d^{2} \sinh ^{2}{\left (a + b x \right )} \sinh {\left (c + d x \right )}}{4 b^{2} d - d^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)**2,x)
Output:
Piecewise((x*sinh(a)**2*cosh(c), Eq(b, 0) & Eq(d, 0)), (x*sinh(a - d*x/2)* *2*cosh(c + d*x)/4 + x*sinh(a - d*x/2)*sinh(c + d*x)*cosh(a - d*x/2)/2 + x *cosh(a - d*x/2)**2*cosh(c + d*x)/4 + sinh(a - d*x/2)**2*sinh(c + d*x)/d + sinh(a - d*x/2)*cosh(a - d*x/2)*cosh(c + d*x)/(2*d), Eq(b, -d/2)), (x*sin h(a + d*x/2)**2*cosh(c + d*x)/4 - x*sinh(a + d*x/2)*sinh(c + d*x)*cosh(a + d*x/2)/2 + x*cosh(a + d*x/2)**2*cosh(c + d*x)/4 + sinh(a + d*x/2)**2*sinh (c + d*x)/d - sinh(a + d*x/2)*cosh(a + d*x/2)*cosh(c + d*x)/(2*d), Eq(b, d /2)), ((x*sinh(a + b*x)**2/2 - x*cosh(a + b*x)**2/2 + sinh(a + b*x)*cosh(a + b*x)/(2*b))*cosh(c), Eq(d, 0)), (2*b**2*sinh(a + b*x)**2*sinh(c + d*x)/ (4*b**2*d - d**3) - 2*b**2*sinh(c + d*x)*cosh(a + b*x)**2/(4*b**2*d - d**3 ) + 2*b*d*sinh(a + b*x)*cosh(a + b*x)*cosh(c + d*x)/(4*b**2*d - d**3) - d* *2*sinh(a + b*x)**2*sinh(c + d*x)/(4*b**2*d - d**3), True))
Exception generated. \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)^2,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(1-d/b>0)', see `assume?` for mor e details)
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.82 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\frac {e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \, {\left (2 \, b + d\right )}} + \frac {e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{8 \, {\left (2 \, b - d\right )}} - \frac {e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \, {\left (2 \, b - d\right )}} - \frac {e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{8 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (d x + c\right )}}{4 \, d} + \frac {e^{\left (-d x - c\right )}}{4 \, d} \] Input:
integrate(cosh(d*x+c)*sinh(b*x+a)^2,x, algorithm="giac")
Output:
1/8*e^(2*b*x + d*x + 2*a + c)/(2*b + d) + 1/8*e^(2*b*x - d*x + 2*a - c)/(2 *b - d) - 1/8*e^(-2*b*x + d*x - 2*a + c)/(2*b - d) - 1/8*e^(-2*b*x - d*x - 2*a - c)/(2*b + d) - 1/4*e^(d*x + c)/d + 1/4*e^(-d*x - c)/d
Time = 1.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\frac {d^2\,\left (\mathrm {sinh}\left (c+d\,x\right )-{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\right )-2\,b^2\,\mathrm {sinh}\left (c+d\,x\right )+2\,b\,d\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{4\,b^2\,d-d^3} \] Input:
int(cosh(c + d*x)*sinh(a + b*x)^2,x)
Output:
(d^2*(sinh(c + d*x) - cosh(a + b*x)^2*sinh(c + d*x)) - 2*b^2*sinh(c + d*x) + 2*b*d*cosh(a + b*x)*cosh(c + d*x)*sinh(a + b*x))/(4*b^2*d - d^3)
Time = 0.22 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.18 \[ \int \cosh (c+d x) \sinh ^2(a+b x) \, dx=\frac {2 e^{4 b x +2 d x +4 a +2 c} b d -e^{4 b x +2 d x +4 a +2 c} d^{2}+2 e^{4 b x +4 a} b d +e^{4 b x +4 a} d^{2}-8 e^{2 b x +2 d x +2 a +2 c} b^{2}+2 e^{2 b x +2 d x +2 a +2 c} d^{2}+8 e^{2 b x +2 a} b^{2}-2 e^{2 b x +2 a} d^{2}-2 e^{2 d x +2 c} b d -e^{2 d x +2 c} d^{2}-2 b d +d^{2}}{8 e^{2 b x +d x +2 a +c} d \left (4 b^{2}-d^{2}\right )} \] Input:
int(cosh(d*x+c)*sinh(b*x+a)^2,x)
Output:
(2*e**(4*a + 4*b*x + 2*c + 2*d*x)*b*d - e**(4*a + 4*b*x + 2*c + 2*d*x)*d** 2 + 2*e**(4*a + 4*b*x)*b*d + e**(4*a + 4*b*x)*d**2 - 8*e**(2*a + 2*b*x + 2 *c + 2*d*x)*b**2 + 2*e**(2*a + 2*b*x + 2*c + 2*d*x)*d**2 + 8*e**(2*a + 2*b *x)*b**2 - 2*e**(2*a + 2*b*x)*d**2 - 2*e**(2*c + 2*d*x)*b*d - e**(2*c + 2* d*x)*d**2 - 2*b*d + d**2)/(8*e**(2*a + 2*b*x + c + d*x)*d*(4*b**2 - d**2))