Integrand size = 24, antiderivative size = 221 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \text {arcsinh}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))+\frac {d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+d^3 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b d^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]
-7/72*b*c*d^3*x*(c^2*x^2+1)^(3/2)-1/36*b*c*d^3*x*(c^2*x^2+1)^(5/2)-19/48*b *d^3*arcsinh(c*x)+1/2*d^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))+1/4*d^3*(c^2*x^2+ 1)^2*(a+b*arcsinh(c*x))+1/6*d^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))+1/2*d^3*( a+b*arcsinh(c*x))^2/b+d^3*(a+b*arcsinh(c*x))*ln(1-1/(c*x+(c^2*x^2+1)^(1/2) )^2)-1/2*b*d^3*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-19/48*b*c*d^3*x*(c^2 *x^2+1)^(1/2)
Time = 0.17 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {d^3 \left (-72 a^2+132 a b+216 a b c^2 x^2+108 a b c^4 x^4+24 a b c^6 x^6-75 b^2 c x \sqrt {1+c^2 x^2}-22 b^2 c^3 x^3 \sqrt {1+c^2 x^2}-4 b^2 c^5 x^5 \sqrt {1+c^2 x^2}-72 b^2 \text {arcsinh}(c x)^2+144 a b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3 b \text {arcsinh}(c x) \left (-48 a+b \left (25+72 c^2 x^2+36 c^4 x^4+8 c^6 x^6\right )+48 b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )+72 b^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{144 b} \]
(d^3*(-72*a^2 + 132*a*b + 216*a*b*c^2*x^2 + 108*a*b*c^4*x^4 + 24*a*b*c^6*x ^6 - 75*b^2*c*x*Sqrt[1 + c^2*x^2] - 22*b^2*c^3*x^3*Sqrt[1 + c^2*x^2] - 4*b ^2*c^5*x^5*Sqrt[1 + c^2*x^2] - 72*b^2*ArcSinh[c*x]^2 + 144*a*b*Log[1 - E^( 2*ArcSinh[c*x])] + 3*b*ArcSinh[c*x]*(-48*a + b*(25 + 72*c^2*x^2 + 36*c^4*x ^4 + 8*c^6*x^6) + 48*b*Log[1 - E^(2*ArcSinh[c*x])]) + 72*b^2*PolyLog[2, E^ (2*ArcSinh[c*x])]))/(144*b)
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.57, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6216, 27, 211, 211, 211, 222, 6216, 211, 211, 222, 6216, 211, 222, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx\) |
\(\Big \downarrow \) 6216 |
\(\displaystyle d \int \frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{6} b c d^3 \int \left (c^2 x^2+1\right )^{5/2}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{6} b c d^3 \int \left (c^2 x^2+1\right )^{5/2}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{6} b c d^3 \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 222 |
\(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 6216 |
\(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{4} b c \int \left (c^2 x^2+1\right )^{3/2}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{4} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 6216 |
\(\displaystyle d^3 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d^3 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle d^3 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 6190 |
\(\displaystyle d^3 \left (\frac {\int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle d^3 \left (-\frac {\int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^3 \left (-\frac {\int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle d^3 \left (\frac {i \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle d^3 \left (\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle d^3 \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle d^3 \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle d^3 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\) |
(d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/6 - (b*c*d^3*((x*(1 + c^2*x^2)^ (5/2))/6 + (5*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 + c^2*x^2])/2 + A rcSinh[c*x]/(2*c)))/4))/6))/6 + d^3*(((1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/ 2 + ((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/4 - (b*c*((x*Sqrt[1 + c^2*x^2]) /2 + ArcSinh[c*x]/(2*c)))/2 - (b*c*((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))/4 + (I*((-1/2*I)*(a + b*ArcSi nh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c*x])*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c*x]))/b)]) + (b^2*PolyLog[2, -a - b*ArcSinh[c*x]])/4 )))/b)
3.1.24.3.1 Defintions of rubi rules used
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[((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[(((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[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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSinh[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcSinh[c*x])/x), x], x] - Simp[b*c*(d^p /(2*p)) Int[(1 + c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.23
method | result | size |
parts | \(d^{3} a \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (x \right )\right )-\frac {d^{3} b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}+\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {25 b \,d^{3} \operatorname {arcsinh}\left (c x \right )}{48}+d^{3} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )\) | \(271\) |
derivativedivides | \(d^{3} a \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (c x \right )\right )+\frac {25 b \,d^{3} \operatorname {arcsinh}\left (c x \right )}{48}+d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}+d^{3} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}\) | \(273\) |
default | \(d^{3} a \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (c x \right )\right )+\frac {25 b \,d^{3} \operatorname {arcsinh}\left (c x \right )}{48}+d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}+d^{3} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}\) | \(273\) |
d^3*a*(1/6*c^6*x^6+3/4*c^4*x^4+3/2*c^2*x^2+ln(x))-1/36*d^3*b*c^5*x^5*(c^2* x^2+1)^(1/2)-11/72*d^3*b*c^3*x^3*(c^2*x^2+1)^(1/2)-25/48*b*c*d^3*x*(c^2*x^ 2+1)^(1/2)+1/6*d^3*b*arcsinh(c*x)*c^6*x^6+3/4*d^3*b*arcsinh(c*x)*c^4*x^4+3 /2*d^3*b*arcsinh(c*x)*c^2*x^2+25/48*b*d^3*arcsinh(c*x)+d^3*b*polylog(2,-c* x-(c^2*x^2+1)^(1/2))+d^3*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))-1/2*d^3*b*arcs inh(c*x)^2+d^3*b*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+d^3*b*arcsinh(c* x)*ln(1-c*x-(c^2*x^2+1)^(1/2))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
integral((a*c^6*d^3*x^6 + 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 + a*d^3 + (b*c ^6*d^3*x^6 + 3*b*c^4*d^3*x^4 + 3*b*c^2*d^3*x^2 + b*d^3)*arcsinh(c*x))/x, x )
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=d^{3} \left (\int \frac {a}{x}\, dx + \int 3 a c^{2} x\, dx + \int 3 a c^{4} x^{3}\, dx + \int a c^{6} x^{5}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 3 b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{5} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(a/x, x) + Integral(3*a*c**2*x, x) + Integral(3*a*c**4*x**3, x) + Integral(a*c**6*x**5, x) + Integral(b*asinh(c*x)/x, x) + Integral(3* b*c**2*x*asinh(c*x), x) + Integral(3*b*c**4*x**3*asinh(c*x), x) + Integral (b*c**6*x**5*asinh(c*x), x))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
1/6*a*c^6*d^3*x^6 + 3/4*a*c^4*d^3*x^4 + 3/2*a*c^2*d^3*x^2 + a*d^3*log(x) + integrate(b*c^6*d^3*x^5*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b*c^4*d^3*x^3*lo g(c*x + sqrt(c^2*x^2 + 1)) + 3*b*c^2*d^3*x*log(c*x + sqrt(c^2*x^2 + 1)) + b*d^3*log(c*x + sqrt(c^2*x^2 + 1))/x, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x} \,d x \]