Integrand size = 34, antiderivative size = 565 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^2}{4 b^2 d}+\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d} \] Output:
-1/4*a*(f*x+e)^2/b^2/d+1/3*a^3*(f*x+e)^3/b^4/f-2*a^2*f*(f*x+e)*cosh(d*x+c) /b^3/d^2+4/9*f*(f*x+e)*cosh(d*x+c)/b/d^2-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/( a-(a^2+b^2)^(1/2)))/b^4/d-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/ 2)))/b^4/d-2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^ 4/d^2-2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^2 +2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^3+2*a^3*f^2* polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^3+2*a^2*f^2*sinh(d*x+c) /b^3/d^3-4/9*f^2*sinh(d*x+c)/b/d^3+a^2*(f*x+e)^2*sinh(d*x+c)/b^3/d+1/2*a*f *(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2-1/4*a*f^2*sinh(d*x+c)^2/b^2/d^3-1 /2*a*(f*x+e)^2*sinh(d*x+c)^2/b^2/d-2/9*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)^2 /b/d^2+2/27*f^2*sinh(d*x+c)^3/b/d^3+1/3*(f*x+e)^2*sinh(d*x+c)^3/b/d
Leaf count is larger than twice the leaf count of optimal. \(1961\) vs. \(2(565)=1130\).
Time = 7.88 (sec) , antiderivative size = 1961, normalized size of antiderivative = 3.47 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x] ),x]
Output:
-1/12*(f^2*((4*a*x^3)/(-1 + E^(2*c)) - 2*a*x^3*Coth[c] - (6*a*b^2*(d^2*x^2 *Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[2, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*PolyLog[3, ((-a + Sqrt[a^2 + b^2])* E^(-c - d*x))/b]))/(Sqrt[a^2 + b^2]*(-a + Sqrt[a^2 + b^2])*d^3) - (6*a*b^2 *(d^2*x^2*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[ 2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] - 2*PolyLog[3, -(((a + Sqrt[ a^2 + b^2])*E^(-c - d*x))/b)]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^3 ) + (6*a^2*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*d*x *PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (6*a^2*(d^2*x^2 *Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((b*E^ (c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((b*E^(c + d*x))/(a + S qrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*b*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (6*b*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[ c])*Sinh[d*x])/d^3))/b^2 + (e^2*((a*Log[a + b*Sinh[c + d*x]])/b^2 - Sinh[c + d*x]/b))/(2*d) - (e*f*(-2*b*Cosh[c + d*x] - a*(2*c*(c + d*x) - (c + d*x )^2 + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) + 2...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^3(c+d x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 f \int (e+f x) \sinh ^3(c+d x)dx}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 f \int i (e+f x) \sin (i c+i d x)^3dx}{3 d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \int (e+f x) \sin (i c+i d x)^3dx}{3 d}}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \int i (e+f x) \sinh (c+d x)dx+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} i \int (e+f x) \sinh (c+d x)dx+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} i \int -i (e+f x) \sin (i c+i d x)dx+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \int (e+f x) \sin (i c+i d x)dx+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle -\frac {a \left (\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}-\frac {f \int (e+f x) \sinh ^2(c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}-\frac {f \int -\left ((e+f x) \sin (i c+i d x)^2\right )dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}+\frac {f \int (e+f x) \sin (i c+i d x)^2dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 6113 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^3}{3 b f}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 d}-\frac {2 i f \left (\frac {2}{3} \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )+\frac {i f \sinh ^3(c+d x)}{9 d^2}-\frac {i (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{3 d}}{b}-\frac {a \left (\frac {\frac {f \left (\frac {f \sinh ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{d}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{b}+\frac {\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 4263 vs. \(2 (525) = 1050\).
Time = 0.15 (sec) , antiderivative size = 4263, normalized size of antiderivative = 7.55 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*cosh(d*x+c)*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
-1/24*e^2*(24*(d*x + c)*a^3/(b^4*d) + 24*a^3*log(-2*a*e^(-d*x - c) + b*e^( -2*d*x - 2*c) - b)/(b^4*d) + (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 - b^2)*e ^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e ^(-3*d*x - 3*c) + 3*(4*a^2 - b^2)*e^(-d*x - c))/(b^3*d)) - 1/432*(144*a^3* d^3*f^2*x^3*e^(3*c) + 432*a^3*d^3*e*f*x^2*e^(3*c) - 2*(9*b^3*d^2*f^2*x^2*e ^(6*c) + 6*(3*d^2*e*f - d*f^2)*b^3*x*e^(6*c) - 2*(3*d*e*f - f^2)*b^3*e^(6* c))*e^(3*d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)*a* b^2*x*e^(5*c) - (2*d*e*f - f^2)*a*b^2*e^(5*c))*e^(2*d*x) + 54*(8*(d*e*f - f^2)*a^2*b*e^(4*c) - 2*(d*e*f - f^2)*b^3*e^(4*c) - (4*a^2*b*d^2*f^2*e^(4*c ) - b^3*d^2*f^2*e^(4*c))*x^2 - 2*(4*(d^2*e*f - d*f^2)*a^2*b*e^(4*c) - (d^2 *e*f - d*f^2)*b^3*e^(4*c))*x)*e^(d*x) + 54*(8*(d*e*f + f^2)*a^2*b*e^(2*c) - 2*(d*e*f + f^2)*b^3*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) - b^3*d^2*f^2*e^( 2*c))*x^2 + 2*(4*(d^2*e*f + d*f^2)*a^2*b*e^(2*c) - (d^2*e*f + d*f^2)*b^3*e ^(2*c))*x)*e^(-d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^c + 2*(2*d^2*e*f + d*f^2)* a*b^2*x*e^c + (2*d*e*f + f^2)*a*b^2*e^c)*e^(-2*d*x) + 2*(9*b^3*d^2*f^2*x^2 + 6*(3*d^2*e*f + d*f^2)*b^3*x + 2*(3*d*e*f + f^2)*b^3)*e^(-3*d*x))*e^(-3* c)/(b^4*d^3) + integrate(-2*(a^3*b*f^2*x^2 + 2*a^3*b*e*f*x - (a^4*f^2*x^2* e^c + 2*a^4*e*f*x*e^c)*e^(d*x))/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x)
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
integrate((f*x + e)^2*cosh(d*x + c)*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(18*e**(6*c + 6*d*x)*b**6*d**2*e**2 + 36*e**(6*c + 6*d*x)*b**6*d**2*e*f*x + 18*e**(6*c + 6*d*x)*b**6*d**2*f**2*x**2 - 12*e**(6*c + 6*d*x)*b**6*d*e*f - 12*e**(6*c + 6*d*x)*b**6*d*f**2*x + 4*e**(6*c + 6*d*x)*b**6*f**2 - 54*e **(5*c + 5*d*x)*a*b**5*d**2*e**2 - 108*e**(5*c + 5*d*x)*a*b**5*d**2*e*f*x - 54*e**(5*c + 5*d*x)*a*b**5*d**2*f**2*x**2 + 54*e**(5*c + 5*d*x)*a*b**5*d *e*f + 54*e**(5*c + 5*d*x)*a*b**5*d*f**2*x - 27*e**(5*c + 5*d*x)*a*b**5*f* *2 + 216*e**(4*c + 4*d*x)*a**2*b**4*d**2*e**2 + 432*e**(4*c + 4*d*x)*a**2* b**4*d**2*e*f*x + 216*e**(4*c + 4*d*x)*a**2*b**4*d**2*f**2*x**2 - 432*e**( 4*c + 4*d*x)*a**2*b**4*d*e*f - 432*e**(4*c + 4*d*x)*a**2*b**4*d*f**2*x + 4 32*e**(4*c + 4*d*x)*a**2*b**4*f**2 - 54*e**(4*c + 4*d*x)*b**6*d**2*e**2 - 108*e**(4*c + 4*d*x)*b**6*d**2*e*f*x - 54*e**(4*c + 4*d*x)*b**6*d**2*f**2* x**2 + 108*e**(4*c + 4*d*x)*b**6*d*e*f + 108*e**(4*c + 4*d*x)*b**6*d*f**2* x - 108*e**(4*c + 4*d*x)*b**6*f**2 + 3456*e**(3*c + 3*d*x)*int(x**2/(e**(5 *c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**6*b*d**3* f**2 + 2592*e**(3*c + 3*d*x)*int(x**2/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4* d*x)*a - e**(3*c + 3*d*x)*b),x)*a**4*b**3*d**3*f**2 + 6912*e**(3*c + 3*d*x )*int(x/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x )*a**6*b*d**3*e*f + 5184*e**(3*c + 3*d*x)*int(x/(e**(5*c + 5*d*x)*b + 2*e* *(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**4*b**3*d**3*e*f - 432*e**(3*c + 3*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**3*b**3*d**2...