\(\int \frac {(d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x))}{x} \, dx\) [24]

Optimal result

Integrand size = 24, antiderivative size = 221 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \text {arcsinh}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{2 b}+d^3 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\frac {1}{2} b d^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) \] Output:

-19/48*b*c*d^3*x*(c^2*x^2+1)^(1/2)-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-(c*x+(c^2*x^2+1)^(1/2))^2)+1/2*b*d^3*polylog(2,(c*x+(c^2*x^2+1)^ 
(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {d^3 \left (-72 a^2+132 a b+216 a b c^2 x^2+108 a b c^4 x^4+24 a b c^6 x^6-75 b^2 c x \sqrt {1+c^2 x^2}-22 b^2 c^3 x^3 \sqrt {1+c^2 x^2}-4 b^2 c^5 x^5 \sqrt {1+c^2 x^2}-72 b^2 \text {arcsinh}(c x)^2+144 a b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3 b \text {arcsinh}(c x) \left (-48 a+b \left (25+72 c^2 x^2+36 c^4 x^4+8 c^6 x^6\right )+48 b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )+72 b^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{144 b} \] Input:

Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x,x]
 

Output:

(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)
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.25 (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

\(\displaystyle d^3 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {1}{4} b c \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2}\right )\)

Input:

Int[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x,x]
 

Output:

(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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 211

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])
 

rule 222

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]
 

rule 2620

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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]
 

rule 6190

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]
 

rule 6216

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]
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.23

method	result	size
parts	\(d^{3} a \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (x \right )\right )+d^{3} b \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{2}+\frac {25 b \,d^{3} \operatorname {arcsinh}\left (x c \right )}{48}-\frac {d^{3} b \operatorname {arcsinh}\left (x c \right )^{2}}{2}+d^{3} b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,x^{5} c^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,x^{3} c^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}\)	\(271\)
derivativedivides	\(d^{3} a \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (x c \right )\right )+d^{3} b \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {25 b \,d^{3} \operatorname {arcsinh}\left (x c \right )}{48}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{2}-\frac {d^{3} b \operatorname {arcsinh}\left (x c \right )^{2}}{2}+d^{3} b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,x^{5} c^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,x^{3} c^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}+\frac {d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}\)	\(273\)
default	\(d^{3} a \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (x c \right )\right )+d^{3} b \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\frac {25 b \,d^{3} \operatorname {arcsinh}\left (x c \right )}{48}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{4} c^{4}}{4}+\frac {3 d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}}{2}-\frac {d^{3} b \operatorname {arcsinh}\left (x c \right )^{2}}{2}+d^{3} b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,x^{5} c^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,x^{3} c^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}+\frac {d^{3} b \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{6}\)	\(273\)

Input:

int((c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))/x,x,method=_RETURNVERBOSE)
 

Output:

d^3*a*(1/6*c^6*x^6+3/4*c^4*x^4+3/2*c^2*x^2+ln(x))+d^3*b*polylog(2,x*c+(c^2 
*x^2+1)^(1/2))+d^3*b*polylog(2,-x*c-(c^2*x^2+1)^(1/2))+1/6*d^3*b*arcsinh(x 
*c)*x^6*c^6+3/4*d^3*b*arcsinh(x*c)*x^4*c^4+3/2*d^3*b*arcsinh(x*c)*x^2*c^2+ 
25/48*b*d^3*arcsinh(x*c)-1/2*d^3*b*arcsinh(x*c)^2+d^3*b*arcsinh(x*c)*ln(1- 
x*c-(c^2*x^2+1)^(1/2))+d^3*b*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))-1/36 
*d^3*b*x^5*c^5*(c^2*x^2+1)^(1/2)-11/72*d^3*b*x^3*c^3*(c^2*x^2+1)^(1/2)-25/ 
48*b*c*d^3*x*(c^2*x^2+1)^(1/2)
 

Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))/x,x, algorithm="fricas")
 

Output:

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 
)
 

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=d^{3} \left (\int \frac {a}{x}\, dx + \int 3 a c^{2} x\, dx + \int 3 a c^{4} x^{3}\, dx + \int a c^{6} x^{5}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 3 b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{5} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:

integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))/x,x)
 

Output:

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))
 

Maxima [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))/x,x, algorithm="maxima")
 

Output:

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)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))/x,x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x} \,d x \] Input:

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^3)/x,x)
 

Output:

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^3)/x, x)
 

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {d^{3} \left (24 \mathit {asinh} \left (c x \right ) b \,c^{6} x^{6}+108 \mathit {asinh} \left (c x \right ) b \,c^{4} x^{4}+216 \mathit {asinh} \left (c x \right ) b \,c^{2} x^{2}-4 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}-22 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}-75 \sqrt {c^{2} x^{2}+1}\, b c x +144 \left (\int \frac {\mathit {asinh} \left (c x \right )}{x}d x \right ) b +75 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b +144 \,\mathrm {log}\left (x \right ) a +24 a \,c^{6} x^{6}+108 a \,c^{4} x^{4}+216 a \,c^{2} x^{2}\right )}{144} \] Input:

int((c^2*d*x^2+d)^3*(a+b*asinh(c*x))/x,x)
                                                                                    
                                                                                    
 

Output:

(d**3*(24*asinh(c*x)*b*c**6*x**6 + 108*asinh(c*x)*b*c**4*x**4 + 216*asinh( 
c*x)*b*c**2*x**2 - 4*sqrt(c**2*x**2 + 1)*b*c**5*x**5 - 22*sqrt(c**2*x**2 + 
 1)*b*c**3*x**3 - 75*sqrt(c**2*x**2 + 1)*b*c*x + 144*int(asinh(c*x)/x,x)*b 
 + 75*log(sqrt(c**2*x**2 + 1) + c*x)*b + 144*log(x)*a + 24*a*c**6*x**6 + 1 
08*a*c**4*x**4 + 216*a*c**2*x**2))/144