\(\int \frac {a+b \text {arctanh}(c x^2)}{(d x)^{7/2}} \, dx\) [88]

Optimal result

Integrand size = 18, antiderivative size = 257 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=-\frac {8 b c}{5 d^3 \sqrt {d x}}-\frac {2 b c^{5/4} \arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {\sqrt {2} b c^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {\sqrt {2} b c^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}+\frac {2 b c^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {\sqrt {2} b c^{5/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d} \left (1+\sqrt {c} x\right )}\right )}{5 d^{7/2}} \] Output:

-8/5*b*c/d^3/(d*x)^(1/2)-2/5*b*c^(5/4)*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2)) 
/d^(7/2)-1/5*2^(1/2)*b*c^(5/4)*arctan(-1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/ 
2))/d^(7/2)-1/5*2^(1/2)*b*c^(5/4)*arctan(1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^( 
1/2))/d^(7/2)-2/5*(a+b*arctanh(c*x^2))/d/(d*x)^(5/2)+2/5*b*c^(5/4)*arctanh 
(c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(7/2)+1/5*2^(1/2)*b*c^(5/4)*arctanh(2^(1/2 
)*c^(1/4)*(d*x)^(1/2)/d^(1/2)/(1+c^(1/2)*x))/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=\frac {x \left (-4 a-16 b c x^2+2 \sqrt {2} b c^{5/4} x^{5/2} \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-2 \sqrt {2} b c^{5/4} x^{5/2} \arctan \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-4 b c^{5/4} x^{5/2} \arctan \left (\sqrt [4]{c} \sqrt {x}\right )-4 b \text {arctanh}\left (c x^2\right )-2 b c^{5/4} x^{5/2} \log \left (1-\sqrt [4]{c} \sqrt {x}\right )+2 b c^{5/4} x^{5/2} \log \left (1+\sqrt [4]{c} \sqrt {x}\right )-\sqrt {2} b c^{5/4} x^{5/2} \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )+\sqrt {2} b c^{5/4} x^{5/2} \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{10 (d x)^{7/2}} \] Input:

Integrate[(a + b*ArcTanh[c*x^2])/(d*x)^(7/2),x]
 

Output:

(x*(-4*a - 16*b*c*x^2 + 2*Sqrt[2]*b*c^(5/4)*x^(5/2)*ArcTan[1 - Sqrt[2]*c^( 
1/4)*Sqrt[x]] - 2*Sqrt[2]*b*c^(5/4)*x^(5/2)*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqr 
t[x]] - 4*b*c^(5/4)*x^(5/2)*ArcTan[c^(1/4)*Sqrt[x]] - 4*b*ArcTanh[c*x^2] - 
 2*b*c^(5/4)*x^(5/2)*Log[1 - c^(1/4)*Sqrt[x]] + 2*b*c^(5/4)*x^(5/2)*Log[1 
+ c^(1/4)*Sqrt[x]] - Sqrt[2]*b*c^(5/4)*x^(5/2)*Log[1 - Sqrt[2]*c^(1/4)*Sqr 
t[x] + Sqrt[c]*x] + Sqrt[2]*b*c^(5/4)*x^(5/2)*Log[1 + Sqrt[2]*c^(1/4)*Sqrt 
[x] + Sqrt[c]*x]))/(10*(d*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.37, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {6464, 847, 851, 27, 830, 826, 827, 218, 221, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6464

\(\displaystyle \frac {4 b c \int \frac {1}{(d x)^{3/2} \left (1-c^2 x^4\right )}dx}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {4 b c \left (\frac {c^2 \int \frac {(d x)^{5/2}}{1-c^2 x^4}dx}{d^4}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \int \frac {d^7 x^3}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{d^5}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \int \frac {d^3 x^3}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 830

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 c}-\frac {\int \frac {d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {c} x d+d}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {4 b c \left (\frac {2 c^2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {c} d x+\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\log \left (\sqrt {c} d x-\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )}{d}-\frac {2}{d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{5 d (d x)^{5/2}}\)

Input:

Int[(a + b*ArcTanh[c*x^2])/(d*x)^(7/2),x]
 

Output:

(-2*(a + b*ArcTanh[c*x^2]))/(5*d*(d*x)^(5/2)) + (4*b*c*(-2/(d*Sqrt[d*x]) + 
 (2*c^2*((-1/2*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(c^(3/4)*Sqrt[d]) + Arc 
Tanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2*c^(3/4)*Sqrt[d]))/(2*c) - ((-(ArcTan[ 
1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d])) + ArcT 
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d]))/(2* 
Sqrt[c]) - (-1/2*Log[d + Sqrt[c]*d*x - Sqrt[2]*c^(1/4)*Sqrt[d]*Sqrt[d*x]]/ 
(Sqrt[2]*c^(1/4)*Sqrt[d]) + Log[d + Sqrt[c]*d*x + Sqrt[2]*c^(1/4)*Sqrt[d]* 
Sqrt[d*x]]/(2*Sqrt[2]*c^(1/4)*Sqrt[d]))/(2*Sqrt[c]))/(2*c)))/d))/(5*d^2)
 

Defintions of rubi rules used

rule 25

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

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 830

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt 
[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[x^(m - n/2)/( 
r + s*x^(n/2)), x], x] - Simp[s/(2*b)   Int[x^(m - n/2)/(r - s*x^(n/2)), x] 
, x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L 
tQ[m, n] &&  !GtQ[a/b, 0]
 

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 6464

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] : 
> Simp[(d*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(d*(m + 1))), x] - Simp[b*c*(n 
/(d^n*(m + 1)))   Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{a, 
b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{5 \left (d x \right )^{\frac {5}{2}}}+2 b \left (-\frac {\operatorname {arctanh}\left (c \,x^{2}\right )}{5 \left (d x \right )^{\frac {5}{2}}}+\frac {4 c \,d^{2} \left (-\frac {1}{d^{4} \sqrt {d x}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 d^{4} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 d^{4} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{5}\right )}{d}\)	\(250\)
default	\(\frac {-\frac {2 a}{5 \left (d x \right )^{\frac {5}{2}}}+2 b \left (-\frac {\operatorname {arctanh}\left (c \,x^{2}\right )}{5 \left (d x \right )^{\frac {5}{2}}}+\frac {4 c \,d^{2} \left (-\frac {1}{d^{4} \sqrt {d x}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 d^{4} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 d^{4} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{5}\right )}{d}\)	\(250\)
parts	\(-\frac {2 a}{5 \left (d x \right )^{\frac {5}{2}} d}+\frac {2 b \left (-\frac {\operatorname {arctanh}\left (c \,x^{2}\right )}{5 \left (d x \right )^{\frac {5}{2}}}+\frac {4 c \,d^{2} \left (-\frac {1}{d^{4} \sqrt {d x}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 d^{4} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 d^{4} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{5}\right )}{d}\)	\(252\)

Input:

int((a+b*arctanh(c*x^2))/(d*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

2/d*(-1/5*a/(d*x)^(5/2)+b*(-1/5/(d*x)^(5/2)*arctanh(c*x^2)+4/5*c*d^2*(-1/d 
^4/(d*x)^(1/2)-1/16/d^4/(d^2/c)^(1/4)*2^(1/2)*(ln((d*x-(d^2/c)^(1/4)*(d*x) 
^(1/2)*2^(1/2)+(d^2/c)^(1/2))/(d*x+(d^2/c)^(1/4)*(d*x)^(1/2)*2^(1/2)+(d^2/ 
c)^(1/2)))+2*arctan(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/ 
(d^2/c)^(1/4)*(d*x)^(1/2)-1))-1/8/d^4/(d^2/c)^(1/4)*(2*arctan((d*x)^(1/2)/ 
(d^2/c)^(1/4))-ln(((d*x)^(1/2)+(d^2/c)^(1/4))/((d*x)^(1/2)-(d^2/c)^(1/4))) 
))))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.79 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=\frac {d^{4} x^{3} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (d^{11} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) - i \, d^{4} x^{3} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (i \, d^{11} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) + i \, d^{4} x^{3} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (-i \, d^{11} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) - d^{4} x^{3} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (-d^{11} \left (\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) - d^{4} x^{3} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (d^{11} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) + i \, d^{4} x^{3} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (i \, d^{11} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) - i \, d^{4} x^{3} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (-i \, d^{11} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) + d^{4} x^{3} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {1}{4}} \log \left (-d^{11} \left (-\frac {b^{4} c^{5}}{d^{14}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c^{4}\right ) - {\left (8 \, b c x^{2} + b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )} \sqrt {d x}}{5 \, d^{4} x^{3}} \] Input:

integrate((a+b*arctanh(c*x^2))/(d*x)^(7/2),x, algorithm="fricas")
 

Output:

1/5*(d^4*x^3*(b^4*c^5/d^14)^(1/4)*log(d^11*(b^4*c^5/d^14)^(3/4) + sqrt(d*x 
)*b^3*c^4) - I*d^4*x^3*(b^4*c^5/d^14)^(1/4)*log(I*d^11*(b^4*c^5/d^14)^(3/4 
) + sqrt(d*x)*b^3*c^4) + I*d^4*x^3*(b^4*c^5/d^14)^(1/4)*log(-I*d^11*(b^4*c 
^5/d^14)^(3/4) + sqrt(d*x)*b^3*c^4) - d^4*x^3*(b^4*c^5/d^14)^(1/4)*log(-d^ 
11*(b^4*c^5/d^14)^(3/4) + sqrt(d*x)*b^3*c^4) - d^4*x^3*(-b^4*c^5/d^14)^(1/ 
4)*log(d^11*(-b^4*c^5/d^14)^(3/4) + sqrt(d*x)*b^3*c^4) + I*d^4*x^3*(-b^4*c 
^5/d^14)^(1/4)*log(I*d^11*(-b^4*c^5/d^14)^(3/4) + sqrt(d*x)*b^3*c^4) - I*d 
^4*x^3*(-b^4*c^5/d^14)^(1/4)*log(-I*d^11*(-b^4*c^5/d^14)^(3/4) + sqrt(d*x) 
*b^3*c^4) + d^4*x^3*(-b^4*c^5/d^14)^(1/4)*log(-d^11*(-b^4*c^5/d^14)^(3/4) 
+ sqrt(d*x)*b^3*c^4) - (8*b*c*x^2 + b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a) 
*sqrt(d*x))/(d^4*x^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*atanh(c*x**2))/(d*x)**(7/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=-\frac {b {\left (\frac {{\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )} + 2 \, c {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}}\right )} + \frac {16}{\sqrt {d x}}\right )} c}{d^{2}} + \frac {4 \, \operatorname {artanh}\left (c x^{2}\right )}{\left (d x\right )^{\frac {5}{2}}}\right )} + \frac {4 \, a}{\left (d x\right )^{\frac {5}{2}}}}{10 \, d} \] Input:

integrate((a+b*arctanh(c*x^2))/(d*x)^(7/2),x, algorithm="maxima")
 

Output:

-1/10*(b*((c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*sqrt(d) + 2*sq 
rt(d*x)*sqrt(c))/sqrt(sqrt(c)*d))/(sqrt(sqrt(c)*d)*sqrt(c)) + 2*sqrt(2)*ar 
ctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*sqrt(d) - 2*sqrt(d*x)*sqrt(c))/sqrt(sqr 
t(c)*d))/(sqrt(sqrt(c)*d)*sqrt(c)) - sqrt(2)*log(sqrt(c)*d*x + sqrt(2)*sqr 
t(d*x)*c^(1/4)*sqrt(d) + d)/(c^(3/4)*sqrt(d)) + sqrt(2)*log(sqrt(c)*d*x - 
sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/(c^(3/4)*sqrt(d))) + 2*c*(2*arctan( 
sqrt(d*x)*sqrt(c)/sqrt(sqrt(c)*d))/(sqrt(sqrt(c)*d)*sqrt(c)) + log((sqrt(d 
*x)*sqrt(c) - sqrt(sqrt(c)*d))/(sqrt(d*x)*sqrt(c) + sqrt(sqrt(c)*d)))/(sqr 
t(sqrt(c)*d)*sqrt(c))) + 16/sqrt(d*x))*c/d^2 + 4*arctanh(c*x^2)/(d*x)^(5/2 
)) + 4*a/(d*x)^(5/2))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (174) = 348\).

Time = 1.59 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.07 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=-\frac {\frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} + \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c d^{4}} - \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x + \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c d^{4}} + \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x - \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c d^{4}} + \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x + \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c d^{4}} - \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x - \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c d^{4}} + \frac {2 \, b \log \left (-\frac {c d^{2} x^{2} + d^{2}}{c d^{2} x^{2} - d^{2}}\right )}{\sqrt {d x} d^{2} x^{2}} + \frac {4 \, {\left (4 \, b c d^{2} x^{2} + a d^{2}\right )}}{\sqrt {d x} d^{4} x^{2}}}{10 \, d} \] Input:

integrate((a+b*arctanh(c*x^2))/(d*x)^(7/2),x, algorithm="giac")
 

Output:

-1/10*(2*sqrt(2)*(c^3*d^2)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/ 
4) + 2*sqrt(d*x))/(d^2/c)^(1/4))/(c*d^4) + 2*sqrt(2)*(c^3*d^2)^(3/4)*b*arc 
tan(-1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4) - 2*sqrt(d*x))/(d^2/c)^(1/4))/(c*d 
^4) - 2*sqrt(2)*(-c^3*d^2)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1 
/4) + 2*sqrt(d*x))/(-d^2/c)^(1/4))/(c*d^4) - 2*sqrt(2)*(-c^3*d^2)^(3/4)*b* 
arctan(-1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1/4) - 2*sqrt(d*x))/(-d^2/c)^(1/4)) 
/(c*d^4) - sqrt(2)*(c^3*d^2)^(3/4)*b*log(d*x + sqrt(2)*sqrt(d*x)*(d^2/c)^( 
1/4) + sqrt(d^2/c))/(c*d^4) + sqrt(2)*(c^3*d^2)^(3/4)*b*log(d*x - sqrt(2)* 
sqrt(d*x)*(d^2/c)^(1/4) + sqrt(d^2/c))/(c*d^4) + sqrt(2)*(-c^3*d^2)^(3/4)* 
b*log(d*x + sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt(-d^2/c))/(c*d^4) - sqr 
t(2)*(-c^3*d^2)^(3/4)*b*log(d*x - sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt( 
-d^2/c))/(c*d^4) + 2*b*log(-(c*d^2*x^2 + d^2)/(c*d^2*x^2 - d^2))/(sqrt(d*x 
)*d^2*x^2) + 4*(4*b*c*d^2*x^2 + a*d^2)/(sqrt(d*x)*d^4*x^2))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{{\left (d\,x\right )}^{7/2}} \,d x \] Input:

int((a + b*atanh(c*x^2))/(d*x)^(7/2),x)
 

Output:

int((a + b*atanh(c*x^2))/(d*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{7/2}} \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b \,x^{2}-2 \sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b \,x^{2}-4 \sqrt {x}\, c^{\frac {5}{4}} \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}}}\right ) b \,x^{2}+2 \sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathit {atanh} \left (c \,x^{2}\right ) b \,x^{2}-4 \mathit {atanh} \left (c \,x^{2}\right ) b +\sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b \,x^{2}+\sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b \,x^{2}-2 \sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x +1\right ) b \,x^{2}+\sqrt {x}\, c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {c}\, x +1\right ) b \,x^{2}+2 \sqrt {x}\, c^{\frac {5}{4}} \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b \,x^{2}-2 \sqrt {x}\, c^{\frac {5}{4}} \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b \,x^{2}-4 a -16 b c \,x^{2}\right )}{10 \sqrt {x}\, d^{4} x^{2}} \] Input:

int((a+b*atanh(c*x^2))/(d*x)^(7/2),x)
 

Output:

(sqrt(d)*(2*sqrt(x)*c**(1/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) - 2*sqrt(x)*sq 
rt(c))/(c**(1/4)*sqrt(2)))*b*c*x**2 - 2*sqrt(x)*c**(1/4)*sqrt(2)*atan((c** 
(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*sqrt(2)))*b*c*x**2 - 4*sqrt(x 
)*c**(1/4)*atan((sqrt(x)*sqrt(c))/c**(1/4))*b*c*x**2 + 2*sqrt(x)*c**(1/4)* 
sqrt(2)*atanh(c*x**2)*b*c*x**2 - 4*atanh(c*x**2)*b + sqrt(x)*c**(1/4)*sqrt 
(2)*log(c**(1/4) + sqrt(x)*sqrt(c))*b*c*x**2 + sqrt(x)*c**(1/4)*sqrt(2)*lo 
g( - c**(1/4) + sqrt(x)*sqrt(c))*b*c*x**2 - 2*sqrt(x)*c**(1/4)*sqrt(2)*log 
( - sqrt(x)*c**(1/4)*sqrt(2) + sqrt(c)*x + 1)*b*c*x**2 + sqrt(x)*c**(1/4)* 
sqrt(2)*log(sqrt(c)*x + 1)*b*c*x**2 + 2*sqrt(x)*c**(1/4)*log(c**(1/4) + sq 
rt(x)*sqrt(c))*b*c*x**2 - 2*sqrt(x)*c**(1/4)*log( - c**(1/4) + sqrt(x)*sqr 
t(c))*b*c*x**2 - 4*a - 16*b*c*x**2))/(10*sqrt(x)*d**4*x**2)