Integrand size = 18, antiderivative size = 544 \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}-\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}+\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2}+\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))} \] Output:
3/2*a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(5/2)/d-1 /2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d-3/2*a^ 2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(5/2)/d+1/2*(f* x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d+3/2*a*f*ln(a +b*sinh(d*x+c))/(a^2+b^2)^2/d^2+3/2*a^2*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+ b^2)^(1/2)))/(a^2+b^2)^(5/2)/d^2-1/2*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2 )^(1/2)))/(a^2+b^2)^(3/2)/d^2-3/2*a^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^ 2)^(1/2)))/(a^2+b^2)^(5/2)/d^2+1/2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^ (1/2)))/(a^2+b^2)^(3/2)/d^2-1/2*b*(f*x+e)*cosh(d*x+c)/(a^2+b^2)/d/(a+b*sin h(d*x+c))^2-1/2*f/(a^2+b^2)/d^2/(a+b*sinh(d*x+c))-3/2*a*b*(f*x+e)*cosh(d*x +c)/(a^2+b^2)^2/d/(a+b*sinh(d*x+c))
Time = 4.73 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.42 \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=-\frac {-\frac {-3 a \sqrt {-\left (a^2+b^2\right )^2} f (c+d x)+6 a^2 \sqrt {a^2+b^2} f \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )-4 a^2 \sqrt {-a^2-b^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 b^2 \sqrt {-a^2-b^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+6 a^2 \sqrt {-a^2-b^2} f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+4 a^2 \sqrt {-a^2-b^2} c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 b^2 \sqrt {-a^2-b^2} c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 a^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-b^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 a^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+b^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 a \sqrt {-\left (a^2+b^2\right )^2} f \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+\sqrt {-a^2-b^2} \left (2 a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+\sqrt {-a^2-b^2} \left (-2 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}+\frac {b \left (a^2+b^2\right ) d (e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^2}+\frac {\left (a^2+b^2\right ) f+3 a b d (e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}}{2 \left (a^2+b^2\right )^2 d^2} \] Input:
Integrate[(e + f*x)/(a + b*Sinh[c + d*x])^3,x]
Output:
-1/2*(-((-3*a*Sqrt[-(a^2 + b^2)^2]*f*(c + d*x) + 6*a^2*Sqrt[a^2 + b^2]*f*A rcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] - 4*a^2*Sqrt[-a^2 - b^2]*d*e*A rcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*b^2*Sqrt[-a^2 - b^2]*d*e*A rcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 6*a^2*Sqrt[-a^2 - b^2]*f*Arc Tanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 4*a^2*Sqrt[-a^2 - b^2]*c*f*Arc Tanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*b^2*Sqrt[-a^2 - b^2]*c*f*Arc Tanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*a^2*Sqrt[-a^2 - b^2]*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - b^2*Sqrt[-a^2 - b^2] *f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*a^2*Sqrt[- a^2 - b^2]*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + b^ 2*Sqrt[-a^2 - b^2]*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2 ])] + 3*a*Sqrt[-(a^2 + b^2)^2]*f*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d *x)))] + Sqrt[-a^2 - b^2]*(2*a^2 - b^2)*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + Sqrt[-a^2 - b^2]*(-2*a^2 + b^2)*f*PolyLog[2, -((b*E^( c + d*x))/(a + Sqrt[a^2 + b^2]))])/Sqrt[-(a^2 + b^2)^2]) + (b*(a^2 + b^2)* d*(e + f*x)*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^2 + ((a^2 + b^2)*f + 3*a* b*d*(e + f*x)*Cosh[c + d*x])/(a + b*Sinh[c + d*x]))/((a^2 + b^2)^2*d^2)
Time = 3.79 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.53, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {3042, 3806, 26, 3042, 3147, 17, 3805, 3042, 3147, 16, 3803, 25, 2694, 27, 2620, 2715, 2838, 7293, 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.
| \(\displaystyle \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {e+f x}{(a-i b \sin (i c+i d x))^3}dx\) |
| \(\Big \downarrow \) 3806 |
| \(\displaystyle \frac {a \int \frac {e+f x}{(a+b \sinh (c+d x))^2}dx}{a^2+b^2}+\frac {i b \int \frac {i (e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}+\frac {b f \int \frac {\cosh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {a \int \frac {e+f x}{(a+b \sinh (c+d x))^2}dx}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}+\frac {b f \int \frac {\cosh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {e+f x}{(a-i b \sin (i c+i d x))^2}dx}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}+\frac {b f \int \frac {\cos (i c+i d x)}{(a-i b \sin (i c+i d x))^2}dx}{2 d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {f \int \frac {1}{(a+b \sinh (c+d x))^2}d(b \sinh (c+d x))}{2 d^2 \left (a^2+b^2\right )}+\frac {a \int \frac {e+f x}{(a-i b \sin (i c+i d x))^2}dx}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {a \int \frac {e+f x}{(a-i b \sin (i c+i d x))^2}dx}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 3805 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {a \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {b f \int \frac {\cosh (c+d x)}{a+b \sinh (c+d x)}dx}{d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}+\frac {a \left (\frac {a \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {b f \int \frac {\cos (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{d \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {a \left (\frac {f \int \frac {1}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}+\frac {a \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {a \left (\frac {a \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {a \left (\frac {2 a \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (-\frac {2 a \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {a \left (-\frac {2 a \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {2 a \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {a \left (-\frac {2 a \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {a \left (-\frac {2 a \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2}dx}{2 \left (a^2+b^2\right )}+\frac {a \left (-\frac {2 a \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b \int \left (\frac {e+f x}{b (a+b \sinh (c+d x))}-\frac {a (e+f x)}{b (a+b \sinh (c+d x))^2}\right )dx}{2 \left (a^2+b^2\right )}+\frac {a \left (-\frac {2 a \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {f \log (a+b \sinh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac {b (e+f x) \cosh (c+d x)}{d \left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )}{a^2+b^2}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {b \left (-\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^2}{b \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^2}{b \left (a^2+b^2\right )^{3/2} d}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {f \log (a+b \sinh (c+d x)) a}{b \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \cosh (c+d x) a}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))}+\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{b \sqrt {a^2+b^2} d}-\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{b \sqrt {a^2+b^2} d}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2} d^2}\right )}{2 \left (a^2+b^2\right )}+\frac {a \left (-\frac {b (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right ) d (a+b \sinh (c+d x))}+\frac {f \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d^2}-\frac {2 a \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a^2+b^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}\) |
Input:
Int[(e + f*x)/(a + b*Sinh[c + d*x])^3,x]
Output:
-1/2*(b*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d*(a + b*Sinh[c + d*x])^2) - f/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x])) - (b*(-((a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d)) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d ) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2] )])/(b*Sqrt[a^2 + b^2]*d) - (a*f*Log[a + b*Sinh[c + d*x]])/(b*(a^2 + b^2)* d^2) - (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^ 2 + b^2)^(3/2)*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2] ))])/(b*Sqrt[a^2 + b^2]*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sq rt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) - (f*PolyLog[2, -((b*E^(c + d* x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2) + (a*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d*(a + b*Sinh[c + d*x]))))/(2*(a^2 + b^2)) + (a*((f* Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)*d^2) - (2*a*(-1/2*(b*(((e + f*x)*Lo g[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E ^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2 , -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)))/(2*Sqrt[a^2 + b^2]) ))/(a^2 + b^2) - (b*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d*(a + b*Sinh[c + d*x]))))/(a^2 + b^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f *x]))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] , x] - Simp[b*d*(m/(f*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*(c + d*x)^m*Cos[e + f*x]*((a + b*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(a^2 - b^2))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m*(a + b*Sin[e + f*x])^(n + 1), x], x] - Simp[b*((n + 2)/((n + 1)*(a^2 - b^2))) Int[(c + d*x)^m*Sin[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x], x] + Simp [b*d*(m/(f*(n + 1)*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1231\) vs. \(2(480)=960\).
Time = 0.59 (sec) , antiderivative size = 1232, normalized size of antiderivative = 2.26
Input:
int((f*x+e)/(a+b*sinh(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
(2*a^2*b*d*f*x*exp(3*d*x+3*c)-b^3*d*f*x*exp(3*d*x+3*c)+6*a^3*d*f*x*exp(2*d *x+2*c)+2*a^2*b*d*e*exp(3*d*x+3*c)-3*a*b^2*d*f*x*exp(2*d*x+2*c)-b^3*d*e*ex p(3*d*x+3*c)+6*a^3*d*e*exp(2*d*x+2*c)-10*a^2*b*d*f*x*exp(d*x+c)-a^2*b*f*ex p(3*d*x+3*c)-3*a*b^2*d*e*exp(2*d*x+2*c)-b^3*d*f*x*exp(d*x+c)-b^3*f*exp(3*d *x+3*c)-2*a^3*f*exp(2*d*x+2*c)-10*a^2*b*d*e*exp(d*x+c)+3*a*b^2*d*f*x-2*a*b ^2*f*exp(2*d*x+2*c)-b^3*d*e*exp(d*x+c)+a^2*b*f*exp(d*x+c)+3*a*b^2*d*e+b^3* f*exp(d*x+c))/d^2/(a^2+b^2)^2/(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)^2-1/(a^2 +b^2)^(5/2)/d^2*b^2*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))- 3/(a^2+b^2)^2/d^2*a*f*ln(exp(d*x+c))+3/2/(a^2+b^2)^2/d^2*a*f*ln(b*exp(2*d* x+2*c)+2*a*exp(d*x+c)-b)-2/(a^2+b^2)^(5/2)/d*a^2*e*arctanh(1/2*(2*b*exp(d* x+c)+2*a)/(a^2+b^2)^(1/2))+2/(a^2+b^2)^(5/2)/d^2*a^2*f*c*arctanh(1/2*(2*b* exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(5/2)/d*a^2*f*ln((-b*exp(d*x+ c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(5/2)/d*a^2*f*ln ((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2) /d^2*a^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/ (a^2+b^2)^(5/2)/d^2*a^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2) ^(1/2)))*c+1/(a^2+b^2)^(5/2)/d^2*a^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2 )-a)/(-a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)^(5/2)/d^2*a^2*f*dilog((b*exp(d*x+c) +(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/(a^2+b^2)^(5/2)/d*b^2*e*arctanh (1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/2/(a^2+b^2)^(5/2)/d*b^2*f*...
Leaf count of result is larger than twice the leaf count of optimal. 6396 vs. \(2 (476) = 952\).
Time = 0.21 (sec) , antiderivative size = 6396, normalized size of antiderivative = 11.76 \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)/(a+b*sinh(d*x+c))**3,x)
Output:
Timed out
\[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {f x + e}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*(4*a^2*d*integrate(x*e^(d*x + c)/(a^4*b*d*e^(2*d*x + 2*c) + 2*a^2*b^3* d*e^(2*d*x + 2*c) + b^5*d*e^(2*d*x + 2*c) + 2*a^5*d*e^(d*x + c) + 4*a^3*b^ 2*d*e^(d*x + c) + 2*a*b^4*d*e^(d*x + c) - a^4*b*d - 2*a^2*b^3*d - b^5*d), x) - 2*b^2*d*integrate(x*e^(d*x + c)/(a^4*b*d*e^(2*d*x + 2*c) + 2*a^2*b^3* d*e^(2*d*x + 2*c) + b^5*d*e^(2*d*x + 2*c) + 2*a^5*d*e^(d*x + c) + 4*a^3*b^ 2*d*e^(d*x + c) + 2*a*b^4*d*e^(d*x + c) - a^4*b*d - 2*a^2*b^3*d - b^5*d), x) + 3*a*b*(a*log((b*e^(d*x + c) + a - sqrt(a^2 + b^2))/(b*e^(d*x + c) + a + sqrt(a^2 + b^2)))/((a^4*b + 2*a^2*b^3 + b^5)*sqrt(a^2 + b^2)*d^2) - 2*( d*x + c)/((a^4*b + 2*a^2*b^3 + b^5)*d^2) + log(b*e^(2*d*x + 2*c) + 2*a*e^( d*x + c) - b)/((a^4*b + 2*a^2*b^3 + b^5)*d^2)) + 2*(3*a*b^2*d*x - (a^2*b*e ^(3*c) + b^3*e^(3*c) - (2*a^2*b*d*e^(3*c) - b^3*d*e^(3*c))*x)*e^(3*d*x) - (2*a^3*e^(2*c) + 2*a*b^2*e^(2*c) - 3*(2*a^3*d*e^(2*c) - a*b^2*d*e^(2*c))*x )*e^(2*d*x) + (a^2*b*e^c + b^3*e^c - (10*a^2*b*d*e^c + b^3*d*e^c)*x)*e^(d* x))/(a^4*b^2*d^2 + 2*a^2*b^4*d^2 + b^6*d^2 + (a^4*b^2*d^2*e^(4*c) + 2*a^2* b^4*d^2*e^(4*c) + b^6*d^2*e^(4*c))*e^(4*d*x) + 4*(a^5*b*d^2*e^(3*c) + 2*a^ 3*b^3*d^2*e^(3*c) + a*b^5*d^2*e^(3*c))*e^(3*d*x) + 2*(2*a^6*d^2*e^(2*c) + 3*a^4*b^2*d^2*e^(2*c) - b^6*d^2*e^(2*c))*e^(2*d*x) - 4*(a^5*b*d^2*e^c + 2* a^3*b^3*d^2*e^c + a*b^5*d^2*e^c)*e^(d*x)) - 3*a^2*log((b*e^(d*x + c) + a - sqrt(a^2 + b^2))/(b*e^(d*x + c) + a + sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)*d^2))*f + 1/2*e*((2*a^2 - b^2)*log((b*e^(-d*x -...
\[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {f x + e}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="giac")
Output:
integrate((f*x + e)/(b*sinh(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\int \frac {e+f\,x}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \] Input:
int((e + f*x)/(a + b*sinh(c + d*x))^3,x)
Output:
int((e + f*x)/(a + b*sinh(c + d*x))^3, x)
\[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\text {too large to display} \] Input:
int((f*x+e)/(a+b*sinh(d*x+c))^3,x)
Output:
(16*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt( a**2 + b**2))*a**7*b**2*f*i + 40*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan(( e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**5*b**4*f*i + 24*e**(4*c + 4* d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a* *3*b**6*d*e*i + 8*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b* i + a*i)/sqrt(a**2 + b**2))*a**3*b**6*f*i - 12*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a*b**8*d*e*i - 16 *e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a** 2 + b**2))*a*b**8*f*i + 64*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**8*b*f*i + 160*e**(3*c + 3*d*x)*sqr t(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**6*b**3* f*i + 96*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/ sqrt(a**2 + b**2))*a**4*b**5*d*e*i + 32*e**(3*c + 3*d*x)*sqrt(a**2 + b**2) *atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**4*b**5*f*i - 48*e**(3 *c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b* *2))*a**2*b**7*d*e*i - 64*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**7*f*i + 64*e**(2*c + 2*d*x)*sq rt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**9*f*i + 128*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqr t(a**2 + b**2))*a**7*b**2*f*i + 96*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*a...