Integrand size = 37, antiderivative size = 548 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac {245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (1+c^2 x^2\right )^2}+\frac {65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac {115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \text {arcsinh}(c x)}{1152 c \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac {5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))}{48 c \sqrt {1+c^2 x^2}}-\frac {b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{18 c}+\frac {1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{16 \left (1+c^2 x^2\right )^2}+\frac {5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2}{24 \left (1+c^2 x^2\right )}+\frac {5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^3}{48 b c \left (1+c^2 x^2\right )^{5/2}} \]
1/108*b^2*x*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)+245/1152*b^2*x*(d+I*c*d*x) ^(5/2)*(f-I*c*f*x)^(5/2)/(c^2*x^2+1)^2+65/1728*b^2*x*(d+I*c*d*x)^(5/2)*(f- I*c*f*x)^(5/2)/(c^2*x^2+1)-115/1152*b^2*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2 )*arcsinh(c*x)/c/(c^2*x^2+1)^(5/2)-5/16*b*c*x^2*(d+I*c*d*x)^(5/2)*(f-I*c*f *x)^(5/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(5/2)+1/6*x*(d+I*c*d*x)^(5/2)*(f- I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2+5/16*x*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^( 5/2)*(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^2+5/24*x*(d+I*c*d*x)^(5/2)*(f-I*c*f* x)^(5/2)*(a+b*arcsinh(c*x))^2/(c^2*x^2+1)+5/48*(d+I*c*d*x)^(5/2)*(f-I*c*f* x)^(5/2)*(a+b*arcsinh(c*x))^3/b/c/(c^2*x^2+1)^(5/2)-5/48*b*(d+I*c*d*x)^(5/ 2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))/c/(c^2*x^2+1)^(1/2)-1/18*b*(d+I*c* d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 3.95 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.34 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9504 a^2 c d^2 f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+7488 a^2 c^3 d^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2304 a^2 c^5 d^2 f^2 x^5 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1440 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-3240 a b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-324 a b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))-24 a b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (6 \text {arcsinh}(c x))+4320 a^2 d^{5/2} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+1620 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+81 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (4 \text {arcsinh}(c x))+4 b^2 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (6 \text {arcsinh}(c x))-12 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) (270 b \cosh (2 \text {arcsinh}(c x))+27 b \cosh (4 \text {arcsinh}(c x))+2 b \cosh (6 \text {arcsinh}(c x))-540 a \sinh (2 \text {arcsinh}(c x))-108 a \sinh (4 \text {arcsinh}(c x))-12 a \sinh (6 \text {arcsinh}(c x)))+72 b d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 (60 a+45 b \sinh (2 \text {arcsinh}(c x))+9 b \sinh (4 \text {arcsinh}(c x))+b \sinh (6 \text {arcsinh}(c x)))}{13824 c \sqrt {1+c^2 x^2}} \]
(9504*a^2*c*d^2*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2 ] + 7488*a^2*c^3*d^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2304*a^2*c^5*d^2*f^2*x^5*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sq rt[1 + c^2*x^2] + 1440*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Arc Sinh[c*x]^3 - 3240*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2* ArcSinh[c*x]] - 324*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4 *ArcSinh[c*x]] - 24*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[6 *ArcSinh[c*x]] + 4320*a^2*d^(5/2)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 1620*b^2*d^2*f^2*Sq rt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 81*b^2*d^2*f^2*Sq rt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 4*b^2*d^2*f^2*Sqr t[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[6*ArcSinh[c*x]] - 12*b*d^2*f^2*Sqrt[ d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh[c*x]] + 27*b*Cosh[4*ArcSinh[c*x]] + 2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*ArcSin h[c*x]] - 108*a*Sinh[4*ArcSinh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]) + 72*b*d ^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + 45*b*Sin h[2*ArcSinh[c*x]] + 9*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]))/(1 3824*c*Sqrt[1 + c^2*x^2])
Time = 1.71 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.78, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {6211, 6201, 6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 211, 222}
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.
| \(\displaystyle \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \int \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2dx}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx-b c \int x (a+b \text {arcsinh}(c x))dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \left (-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \left (-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )+\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \int x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{5/2}dx}{6 c}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{3/2}dx}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c \left (\frac {\left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )}{6 c}\right )+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )\right )}{\left (c^2 x^2+1\right )^{5/2}}\) |
((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*((x*(1 + c^2*x^2)^(5/2)*(a + b*Ar cSinh[c*x])^2)/6 - (b*c*(((1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(6*c^2) - (b*((x*(1 + c^2*x^2)^(5/2))/6 + (5*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqr t[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/6))/(6*c)))/3 + (5*((x*(1 + c ^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*((x*Sqrt[1 + c^2*x^2]*(a + b* ArcSinh[c*x])^2)/2 + (a + b*ArcSinh[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*Arc Sinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3) ))/2)))/4 - (b*c*(((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(4*c^2) - (b*((x* (1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)) )/4))/(4*c)))/2))/6))/(1 + c^2*x^2)^(5/2)
3.6.82.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ ) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 + c^2*x ^2)^q) Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^ 2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
\[\int \left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}d x\]
\[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
integral((b^2*c^4*d^2*f^2*x^4 + 2*b^2*c^2*d^2*f^2*x^2 + b^2*d^2*f^2)*sqrt( I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*(a*b*c^ 4*d^2*f^2*x^4 + 2*a*b*c^2*d^2*f^2*x^2 + a*b*d^2*f^2)*sqrt(I*c*d*x + d)*sqr t(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (a^2*c^4*d^2*f^2*x^4 + 2*a^ 2*c^2*d^2*f^2*x^2 + a^2*d^2*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f), x)
Timed out. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \]
Exception generated. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2} \,d x \]