Integrand size = 15, antiderivative size = 62 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=-\frac {\cosh (a+b x)}{2 b}+\frac {\cosh (a-2 c+(b-2 d) x)}{4 (b-2 d)}+\frac {\cosh (a+2 c+(b+2 d) x)}{4 (b+2 d)} \] Output:
-1/2*cosh(b*x+a)/b+cosh(a-2*c+(b-2*d)*x)/(4*b-8*d)+cosh(a+2*c+(b+2*d)*x)/( 4*b+8*d)
Time = 0.51 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\frac {1}{4} \left (-\frac {2 \cosh (a) \cosh (b x)}{b}+\frac {\cosh (a-2 c+b x-2 d x)}{b-2 d}+\frac {\cosh (a+2 c+b x+2 d x)}{b+2 d}-\frac {2 \sinh (a) \sinh (b x)}{b}\right ) \] Input:
Integrate[Sinh[a + b*x]*Sinh[c + d*x]^2,x]
Output:
((-2*Cosh[a]*Cosh[b*x])/b + Cosh[a - 2*c + b*x - 2*d*x]/(b - 2*d) + Cosh[a + 2*c + b*x + 2*d*x]/(b + 2*d) - (2*Sinh[a]*Sinh[b*x])/b)/4
Time = 0.28 (sec) , antiderivative size = 62, 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 = {6147, 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.
\(\displaystyle \int \sinh (a+b x) \sinh ^2(c+d x) \, dx\) |
\(\Big \downarrow \) 6147 |
\(\displaystyle \int \left (\frac {1}{4} \sinh (a+x (b-2 d)-2 c)+\frac {1}{4} \sinh (a+x (b+2 d)+2 c)-\frac {1}{2} \sinh (a+b x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\cosh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac {\cosh (a+x (b+2 d)+2 c)}{4 (b+2 d)}-\frac {\cosh (a+b x)}{2 b}\) |
Input:
Int[Sinh[a + b*x]*Sinh[c + d*x]^2,x]
Output:
-1/2*Cosh[a + b*x]/b + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))
Int[Sinh[v_]^(p_.)*Sinh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v ]^p*Sinh[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 = 1.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\cosh \left (b x +a \right )}{2 b}+\frac {\cosh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\cosh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d}\) | \(57\) |
parallelrisch | \(\frac {b \left (b +2 d \right ) \cosh \left (a -2 c +\left (b -2 d \right ) x \right )+b \left (b -2 d \right ) \cosh \left (a +2 c +\left (b +2 d \right ) x \right )+\left (-2 b^{2}+8 d^{2}\right ) \cosh \left (b x +a \right )-8 d^{2}}{4 b^{3}-16 b \,d^{2}}\) | \(80\) |
risch | \(-\frac {{\mathrm e}^{b x +a}}{4 b}-\frac {{\mathrm e}^{-b x -a}}{4 b}+\frac {\left (b \,{\mathrm e}^{2 b x +2 a}-2 d \,{\mathrm e}^{2 b x +2 a}+b +2 d \right ) {\mathrm e}^{-b x +2 d x -a +2 c}}{8 \left (b +2 d \right ) \left (b -2 d \right )}+\frac {\left (b \,{\mathrm e}^{2 b x +2 a}+2 d \,{\mathrm e}^{2 b x +2 a}+b -2 d \right ) {\mathrm e}^{-b x -2 d x -a -2 c}}{8 \left (b +2 d \right ) \left (b -2 d \right )}\) | \(147\) |
orering | \(\frac {\left (3 b^{4}+16 d^{4}\right ) \left (b \cosh \left (b x +a \right ) \sinh \left (d x +c \right )^{2}+2 \sinh \left (b x +a \right ) \sinh \left (d x +c \right ) d \cosh \left (d x +c \right )\right )}{b^{2} \left (b^{4}-8 b^{2} d^{2}+16 d^{4}\right )}-\frac {\left (3 b^{2}+8 d^{2}\right ) \left (b^{3} \cosh \left (b x +a \right ) \sinh \left (d x +c \right )^{2}+6 b^{2} \sinh \left (b x +a \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) d +6 \cosh \left (b x +a \right ) d^{2} \sinh \left (d x +c \right )^{2} b +6 \cosh \left (b x +a \right ) \cosh \left (d x +c \right )^{2} d^{2} b +8 \sinh \left (b x +a \right ) \sinh \left (d x +c \right ) d^{3} \cosh \left (d x +c \right )\right )}{b^{2} \left (b^{4}-8 b^{2} d^{2}+16 d^{4}\right )}+\frac {b^{5} \cosh \left (b x +a \right ) \sinh \left (d x +c \right )^{2}+10 b^{4} \sinh \left (b x +a \right ) \sinh \left (d x +c \right ) d \cosh \left (d x +c \right )+20 b^{3} \cosh \left (b x +a \right ) d^{2} \sinh \left (d x +c \right )^{2}+20 b^{3} \cosh \left (b x +a \right ) \cosh \left (d x +c \right )^{2} d^{2}+80 b^{2} \sinh \left (b x +a \right ) d^{3} \sinh \left (d x +c \right ) \cosh \left (d x +c \right )+40 b \cosh \left (b x +a \right ) d^{4} \cosh \left (d x +c \right )^{2}+40 b \cosh \left (b x +a \right ) \sinh \left (d x +c \right )^{2} d^{4}+32 \sinh \left (b x +a \right ) d^{5} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{b^{2} \left (b^{4}-8 b^{2} d^{2}+16 d^{4}\right )}\) | \(414\) |
Input:
int(sinh(b*x+a)*sinh(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*cosh(b*x+a)/b+1/4/(b-2*d)*cosh(a-2*c+(b-2*d)*x)+1/4/(b+2*d)*cosh(a+2* c+(b+2*d)*x)
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (56) = 112\).
Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.94 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\frac {b^{2} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - 4 \, b d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right ) + b^{2} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - {\left (b^{2} - 4 \, d^{2}\right )} \cosh \left (b x + a\right )}{2 \, {\left ({\left (b^{3} - 4 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{3} - 4 \, b d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )}} \] Input:
integrate(sinh(b*x+a)*sinh(d*x+c)^2,x, algorithm="fricas")
Output:
1/2*(b^2*cosh(b*x + a)*cosh(d*x + c)^2 - 4*b*d*cosh(d*x + c)*sinh(b*x + a) *sinh(d*x + c) + b^2*cosh(b*x + a)*sinh(d*x + c)^2 - (b^2 - 4*d^2)*cosh(b* x + a))/((b^3 - 4*b*d^2)*cosh(b*x + a)^2 - (b^3 - 4*b*d^2)*sinh(b*x + a)^2 )
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (49) = 98\).
Time = 0.69 (sec) , antiderivative size = 405, normalized size of antiderivative = 6.53 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\begin {cases} x \sinh {\left (a \right )} \sinh ^{2}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\left (\frac {x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d}\right ) \sinh {\left (a \right )} & \text {for}\: b = 0 \\\frac {x \sinh {\left (a - 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac {x \sinh {\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {x \sinh {\left (c + d x \right )} \cosh {\left (a - 2 d x \right )} \cosh {\left (c + d x \right )}}{2} - \frac {\sinh {\left (a - 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} - \frac {\sinh ^{2}{\left (c + d x \right )} \cosh {\left (a - 2 d x \right )}}{2 d} & \text {for}\: b = - 2 d \\\frac {x \sinh {\left (a + 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac {x \sinh {\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {x \sinh {\left (c + d x \right )} \cosh {\left (a + 2 d x \right )} \cosh {\left (c + d x \right )}}{2} - \frac {\sinh {\left (a + 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} + \frac {\sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + 2 d x \right )}}{2 d} & \text {for}\: b = 2 d \\\frac {b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 b d \sinh {\left (a + b x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 d^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{3} - 4 b d^{2}} + \frac {2 d^{2} \cosh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(sinh(b*x+a)*sinh(d*x+c)**2,x)
Output:
Piecewise((x*sinh(a)*sinh(c)**2, Eq(b, 0) & Eq(d, 0)), ((x*sinh(c + d*x)** 2/2 - x*cosh(c + d*x)**2/2 + sinh(c + d*x)*cosh(c + d*x)/(2*d))*sinh(a), E q(b, 0)), (x*sinh(a - 2*d*x)*sinh(c + d*x)**2/4 + x*sinh(a - 2*d*x)*cosh(c + d*x)**2/4 + x*sinh(c + d*x)*cosh(a - 2*d*x)*cosh(c + d*x)/2 - sinh(a - 2*d*x)*sinh(c + d*x)*cosh(c + d*x)/(4*d) - sinh(c + d*x)**2*cosh(a - 2*d*x )/(2*d), Eq(b, -2*d)), (x*sinh(a + 2*d*x)*sinh(c + d*x)**2/4 + x*sinh(a + 2*d*x)*cosh(c + d*x)**2/4 - x*sinh(c + d*x)*cosh(a + 2*d*x)*cosh(c + d*x)/ 2 - sinh(a + 2*d*x)*sinh(c + d*x)*cosh(c + d*x)/(4*d) + sinh(c + d*x)**2*c osh(a + 2*d*x)/(2*d), Eq(b, 2*d)), (b**2*sinh(c + d*x)**2*cosh(a + b*x)/(b **3 - 4*b*d**2) - 2*b*d*sinh(a + b*x)*sinh(c + d*x)*cosh(c + d*x)/(b**3 - 4*b*d**2) - 2*d**2*sinh(c + d*x)**2*cosh(a + b*x)/(b**3 - 4*b*d**2) + 2*d* *2*cosh(a + b*x)*cosh(c + d*x)**2/(b**3 - 4*b*d**2), True))
Exception generated. \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sinh(b*x+a)*sinh(d*x+c)^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(-(2*d)/b>0)', see `assume?` for more detai
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.94 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\frac {e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} + \frac {e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} - \frac {e^{\left (b x + a\right )}}{4 \, b} + \frac {e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} + \frac {e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \] Input:
integrate(sinh(b*x+a)*sinh(d*x+c)^2,x, algorithm="giac")
Output:
1/8*e^(b*x + 2*d*x + a + 2*c)/(b + 2*d) + 1/8*e^(b*x - 2*d*x + a - 2*c)/(b - 2*d) - 1/4*e^(b*x + a)/b + 1/8*e^(-b*x + 2*d*x - a + 2*c)/(b - 2*d) + 1 /8*e^(-b*x - 2*d*x - a - 2*c)/(b + 2*d) - 1/4*e^(-b*x - a)/b
Time = 1.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\frac {b^2\,\left (\mathrm {cosh}\left (a+b\,x\right )-\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\right )-2\,d^2\,\mathrm {cosh}\left (a+b\,x\right )+2\,b\,d\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^2-b^3} \] Input:
int(sinh(a + b*x)*sinh(c + d*x)^2,x)
Output:
(b^2*(cosh(a + b*x) - cosh(a + b*x)*cosh(c + d*x)^2) - 2*d^2*cosh(a + b*x) + 2*b*d*cosh(c + d*x)*sinh(a + b*x)*sinh(c + d*x))/(4*b*d^2 - b^3)
Time = 0.23 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.42 \[ \int \sinh (a+b x) \sinh ^2(c+d x) \, dx=\frac {e^{2 b x +4 d x +2 a +4 c} b^{2}-2 e^{2 b x +4 d x +2 a +4 c} b d -2 e^{2 b x +2 d x +2 a +2 c} b^{2}+8 e^{2 b x +2 d x +2 a +2 c} d^{2}+e^{2 b x +2 a} b^{2}+2 e^{2 b x +2 a} b d +e^{4 d x +4 c} b^{2}+2 e^{4 d x +4 c} b d -2 e^{2 d x +2 c} b^{2}+8 e^{2 d x +2 c} d^{2}+b^{2}-2 b d}{8 e^{b x +2 d x +a +2 c} b \left (b^{2}-4 d^{2}\right )} \] Input:
int(sinh(b*x+a)*sinh(d*x+c)^2,x)
Output:
(e**(2*a + 2*b*x + 4*c + 4*d*x)*b**2 - 2*e**(2*a + 2*b*x + 4*c + 4*d*x)*b* d - 2*e**(2*a + 2*b*x + 2*c + 2*d*x)*b**2 + 8*e**(2*a + 2*b*x + 2*c + 2*d* x)*d**2 + e**(2*a + 2*b*x)*b**2 + 2*e**(2*a + 2*b*x)*b*d + e**(4*c + 4*d*x )*b**2 + 2*e**(4*c + 4*d*x)*b*d - 2*e**(2*c + 2*d*x)*b**2 + 8*e**(2*c + 2* d*x)*d**2 + b**2 - 2*b*d)/(8*e**(a + b*x + 2*c + 2*d*x)*b*(b**2 - 4*d**2))