Integrand size = 16, antiderivative size = 117 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {b d \arctan \left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b d \text {arctanh}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {(d+e x)^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{2 e}+\frac {b \left (c d^2+e^2\right ) \log \left (1-c x^2\right )}{4 c e}-\frac {b \left (c d^2-e^2\right ) \log \left (1+c x^2\right )}{4 c e} \] Output:
b*d*arctan(c^(1/2)*x)/c^(1/2)-b*d*arctanh(c^(1/2)*x)/c^(1/2)+1/2*(e*x+d)^2 *(a+b*arctanh(c*x^2))/e+1/4*b*(c*d^2+e^2)*ln(-c*x^2+1)/c/e-1/4*b*(c*d^2-e^ 2)*ln(c*x^2+1)/c/e
Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.89 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=a d x+\frac {1}{2} a e x^2+b d x \text {arctanh}\left (c x^2\right )+\frac {1}{2} b e x^2 \text {arctanh}\left (c x^2\right )+\frac {b d \left (2 \arctan \left (\sqrt {c} x\right )+\log \left (1-\sqrt {c} x\right )-\log \left (1+\sqrt {c} x\right )\right )}{2 \sqrt {c}}+\frac {b e \log \left (1-c^2 x^4\right )}{4 c} \] Input:
Integrate[(d + e*x)*(a + b*ArcTanh[c*x^2]),x]
Output:
a*d*x + (a*e*x^2)/2 + b*d*x*ArcTanh[c*x^2] + (b*e*x^2*ArcTanh[c*x^2])/2 + (b*d*(2*ArcTan[Sqrt[c]*x] + Log[1 - Sqrt[c]*x] - Log[1 + Sqrt[c]*x]))/(2*S qrt[c]) + (b*e*Log[1 - c^2*x^4])/(4*c)
Time = 0.37 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6486, 2370, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6486 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{2 e}-\frac {b c \int \frac {x (d+e x)^2}{1-c^2 x^4}dx}{e}\) |
| \(\Big \downarrow \) 2370 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{2 e}-\frac {b c \int \left (\frac {2 d e x^2}{1-c^2 x^4}+\frac {\left (d^2+e^2 x^2\right ) x}{1-c^2 x^4}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{2 e}-\frac {b c \left (-\frac {d e \arctan \left (\sqrt {c} x\right )}{c^{3/2}}+\frac {d e \text {arctanh}\left (\sqrt {c} x\right )}{c^{3/2}}+\frac {d^2 \text {arctanh}\left (c x^2\right )}{2 c}-\frac {e^2 \log \left (1-c^2 x^4\right )}{4 c^2}\right )}{e}\) |
Input:
Int[(d + e*x)*(a + b*ArcTanh[c*x^2]),x]
Output:
((d + e*x)^2*(a + b*ArcTanh[c*x^2]))/(2*e) - (b*c*(-((d*e*ArcTan[Sqrt[c]*x ])/c^(3/2)) + (d*e*ArcTanh[Sqrt[c]*x])/c^(3/2) + (d^2*ArcTanh[c*x^2])/(2*c ) - (e^2*Log[1 - c^2*x^4])/(4*c^2)))/e
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 - c^2*x^(2 *n))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \,\operatorname {arctanh}\left (c \,x^{2}\right ) x^{2} e}{2}+b \,\operatorname {arctanh}\left (c \,x^{2}\right ) d x +\frac {b e \ln \left (c \,x^{2}-1\right )}{4 c}-\frac {b d \,\operatorname {arctanh}\left (\sqrt {c}\, x \right )}{\sqrt {c}}+\frac {b e \ln \left (c \,x^{2}+1\right )}{4 c}+\frac {b d \arctan \left (\sqrt {c}\, x \right )}{\sqrt {c}}\) | \(92\) |
| parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \,\operatorname {arctanh}\left (c \,x^{2}\right ) x^{2} e}{2}+b \,\operatorname {arctanh}\left (c \,x^{2}\right ) d x +\frac {b e \ln \left (c \,x^{2}-1\right )}{4 c}-\frac {b d \,\operatorname {arctanh}\left (\sqrt {c}\, x \right )}{\sqrt {c}}+\frac {b e \ln \left (c \,x^{2}+1\right )}{4 c}+\frac {b d \arctan \left (\sqrt {c}\, x \right )}{\sqrt {c}}\) | \(92\) |
| risch | \(\frac {b x \left (e x +2 d \right ) \ln \left (c \,x^{2}+1\right )}{4}-\frac {b e \,x^{2} \ln \left (-c \,x^{2}+1\right )}{4}-\frac {b d x \ln \left (-c \,x^{2}+1\right )}{2}+\frac {a e \,x^{2}}{2}-\frac {b \ln \left (1+\sqrt {c}\, x \right ) d}{2 \sqrt {c}}+\frac {b \ln \left (1-\sqrt {-c}\, x \right ) \sqrt {-c}\, d}{2 c}-\frac {b \ln \left (1+\sqrt {-c}\, x \right ) \sqrt {-c}\, d}{2 c}+\frac {b \ln \left (1-\sqrt {c}\, x \right ) d}{2 \sqrt {c}}+a d x +\frac {b \ln \left (1+\sqrt {c}\, x \right ) e}{4 c}+\frac {b \ln \left (1-\sqrt {-c}\, x \right ) e}{4 c}+\frac {b \ln \left (1+\sqrt {-c}\, x \right ) e}{4 c}+\frac {b \ln \left (1-\sqrt {c}\, x \right ) e}{4 c}\) | \(204\) |
Input:
int((e*x+d)*(a+b*arctanh(c*x^2)),x,method=_RETURNVERBOSE)
Output:
a*(1/2*e*x^2+d*x)+1/2*b*arctanh(c*x^2)*x^2*e+b*arctanh(c*x^2)*d*x+1/4*b*e/ c*ln(c*x^2-1)-b*d*arctanh(c^(1/2)*x)/c^(1/2)+1/4*b*e/c*ln(c*x^2+1)+b*d*arc tan(c^(1/2)*x)/c^(1/2)
Time = 0.10 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.13 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\left [\frac {2 \, a c e x^{2} + 4 \, a c d x + 4 \, b \sqrt {c} d \arctan \left (\sqrt {c} x\right ) + 2 \, b \sqrt {c} d \log \left (\frac {c x^{2} - 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right ) + b e \log \left (c x^{2} + 1\right ) + b e \log \left (c x^{2} - 1\right ) + {\left (b c e x^{2} + 2 \, b c d x\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{4 \, c}, \frac {2 \, a c e x^{2} + 4 \, a c d x + 4 \, b \sqrt {-c} d \arctan \left (\sqrt {-c} x\right ) - 2 \, b \sqrt {-c} d \log \left (\frac {c x^{2} - 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right ) + b e \log \left (c x^{2} + 1\right ) + b e \log \left (c x^{2} - 1\right ) + {\left (b c e x^{2} + 2 \, b c d x\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{4 \, c}\right ] \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x^2)),x, algorithm="fricas")
Output:
[1/4*(2*a*c*e*x^2 + 4*a*c*d*x + 4*b*sqrt(c)*d*arctan(sqrt(c)*x) + 2*b*sqrt (c)*d*log((c*x^2 - 2*sqrt(c)*x + 1)/(c*x^2 - 1)) + b*e*log(c*x^2 + 1) + b* e*log(c*x^2 - 1) + (b*c*e*x^2 + 2*b*c*d*x)*log(-(c*x^2 + 1)/(c*x^2 - 1)))/ c, 1/4*(2*a*c*e*x^2 + 4*a*c*d*x + 4*b*sqrt(-c)*d*arctan(sqrt(-c)*x) - 2*b* sqrt(-c)*d*log((c*x^2 - 2*sqrt(-c)*x - 1)/(c*x^2 + 1)) + b*e*log(c*x^2 + 1 ) + b*e*log(c*x^2 - 1) + (b*c*e*x^2 + 2*b*c*d*x)*log(-(c*x^2 + 1)/(c*x^2 - 1)))/c]
Time = 4.89 (sec) , antiderivative size = 753, normalized size of antiderivative = 6.44 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)*(a+b*atanh(c*x**2)),x)
Output:
a*d*x + a*e*x**2/2 + b*d*Piecewise((4*c*x*sqrt(-1/c)*sqrt(1/c)*atanh(c*x** 2)/(c**2*(-1/c)**(3/2)*sqrt(1/c) - c**2*sqrt(-1/c)*(1/c)**(3/2) + 6*c*sqrt (-1/c)*sqrt(1/c)) - c*(-1/c)**(3/2)*log(x + sqrt(-1/c))/(c**2*(-1/c)**(3/2 )*sqrt(1/c) - c**2*sqrt(-1/c)*(1/c)**(3/2) + 6*c*sqrt(-1/c)*sqrt(1/c)) + c *(1/c)**(3/2)*log(x + sqrt(-1/c))/(c**2*(-1/c)**(3/2)*sqrt(1/c) - c**2*sqr t(-1/c)*(1/c)**(3/2) + 6*c*sqrt(-1/c)*sqrt(1/c)) - 2*sqrt(-1/c)*log(x - sq rt(-1/c))/(c**2*(-1/c)**(3/2)*sqrt(1/c) - c**2*sqrt(-1/c)*(1/c)**(3/2) + 6 *c*sqrt(-1/c)*sqrt(1/c)) - 3*sqrt(-1/c)*log(x + sqrt(-1/c))/(c**2*(-1/c)** (3/2)*sqrt(1/c) - c**2*sqrt(-1/c)*(1/c)**(3/2) + 6*c*sqrt(-1/c)*sqrt(1/c)) + 4*sqrt(-1/c)*log(x - sqrt(1/c))/(c**2*(-1/c)**(3/2)*sqrt(1/c) - c**2*sq rt(-1/c)*(1/c)**(3/2) + 6*c*sqrt(-1/c)*sqrt(1/c)) + 4*sqrt(-1/c)*atanh(c*x **2)/(c**2*(-1/c)**(3/2)*sqrt(1/c) - c**2*sqrt(-1/c)*(1/c)**(3/2) + 6*c*sq rt(-1/c)*sqrt(1/c)) + 2*sqrt(1/c)*log(x - sqrt(-1/c))/(c**2*(-1/c)**(3/2)* sqrt(1/c) - c**2*sqrt(-1/c)*(1/c)**(3/2) + 6*c*sqrt(-1/c)*sqrt(1/c)) - 3*s qrt(1/c)*log(x + sqrt(-1/c))/(c**2*(-1/c)**(3/2)*sqrt(1/c) - c**2*sqrt(-1/ c)*(1/c)**(3/2) + 6*c*sqrt(-1/c)*sqrt(1/c)), Ne(c, 0)), (0, True)) + b*e*P iecewise((x**2*atanh(c*x**2)/2 + log(c*x**2 + 1)/(2*c) - atanh(c*x**2)/(2* c), Ne(c, 0)), (0, True))
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\sqrt {c} x\right )}{c^{\frac {3}{2}}} + \frac {\log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} + 2 \, x \operatorname {artanh}\left (c x^{2}\right )\right )} b d + a d x + \frac {{\left (2 \, c x^{2} \operatorname {artanh}\left (c x^{2}\right ) + \log \left (-c^{2} x^{4} + 1\right )\right )} b e}{4 \, c} \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x^2)),x, algorithm="maxima")
Output:
1/2*a*e*x^2 + 1/2*(c*(2*arctan(sqrt(c)*x)/c^(3/2) + log((c*x - sqrt(c))/(c *x + sqrt(c)))/c^(3/2)) + 2*x*arctanh(c*x^2))*b*d + a*d*x + 1/4*(2*c*x^2*a rctanh(c*x^2) + log(-c^2*x^4 + 1))*b*e/c
Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {1}{2} \, a e x^{2} + a d x + \frac {b c d \arctan \left (x \sqrt {{\left | c \right |}}\right )}{{\left | c \right |}^{\frac {3}{2}}} + \frac {b e \log \left (x^{2} + \frac {1}{{\left | c \right |}}\right )}{4 \, c} + \frac {1}{4} \, {\left (b e x^{2} + 2 \, b d x\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) - \frac {{\left (2 \, b c d \sqrt {{\left | c \right |}} - b c e\right )} \log \left ({\left | x + \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{4 \, c^{2}} + \frac {{\left (2 \, b c d \sqrt {{\left | c \right |}} + b c e\right )} \log \left ({\left | x - \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{4 \, c^{2}} \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x^2)),x, algorithm="giac")
Output:
1/2*a*e*x^2 + a*d*x + b*c*d*arctan(x*sqrt(abs(c)))/abs(c)^(3/2) + 1/4*b*e* log(x^2 + 1/abs(c))/c + 1/4*(b*e*x^2 + 2*b*d*x)*log(-(c*x^2 + 1)/(c*x^2 - 1)) - 1/4*(2*b*c*d*sqrt(abs(c)) - b*c*e)*log(abs(x + 1/sqrt(abs(c))))/c^2 + 1/4*(2*b*c*d*sqrt(abs(c)) + b*c*e)*log(abs(x - 1/sqrt(abs(c))))/c^2
Time = 3.84 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.07 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=a\,d\,x+\frac {a\,e\,x^2}{2}+\frac {b\,d\,x\,\ln \left (c\,x^2+1\right )}{2}-\frac {b\,d\,x\,\ln \left (1-c\,x^2\right )}{2}+\frac {b\,e\,\ln \left (c+x\,\sqrt {c^3}\right )}{4\,c}+\frac {b\,e\,\ln \left (c-x\,\sqrt {c^3}\right )}{4\,c}+\frac {b\,e\,x^2\,\ln \left (c\,x^2+1\right )}{4}-\frac {b\,e\,x^2\,\ln \left (1-c\,x^2\right )}{4}+\frac {b\,e\,\ln \left (c+x\,\sqrt {-c^3}\right )}{4\,c}+\frac {b\,e\,\ln \left (c-x\,\sqrt {-c^3}\right )}{4\,c}-\frac {b\,d\,\ln \left (c+x\,\sqrt {c^3}\right )\,\sqrt {c^3}}{2\,c^2}+\frac {b\,d\,\ln \left (c-x\,\sqrt {c^3}\right )\,\sqrt {c^3}}{2\,c^2}-\frac {b\,d\,\ln \left (c+x\,\sqrt {-c^3}\right )\,\sqrt {-c^3}}{2\,c^2}+\frac {b\,d\,\ln \left (c-x\,\sqrt {-c^3}\right )\,\sqrt {-c^3}}{2\,c^2} \] Input:
int((a + b*atanh(c*x^2))*(d + e*x),x)
Output:
a*d*x + (a*e*x^2)/2 + (b*d*x*log(c*x^2 + 1))/2 - (b*d*x*log(1 - c*x^2))/2 + (b*e*log(c + x*(c^3)^(1/2)))/(4*c) + (b*e*log(c - x*(c^3)^(1/2)))/(4*c) + (b*e*x^2*log(c*x^2 + 1))/4 - (b*e*x^2*log(1 - c*x^2))/4 + (b*e*log(c + x *(-c^3)^(1/2)))/(4*c) + (b*e*log(c - x*(-c^3)^(1/2)))/(4*c) - (b*d*log(c + x*(c^3)^(1/2))*(c^3)^(1/2))/(2*c^2) + (b*d*log(c - x*(c^3)^(1/2))*(c^3)^( 1/2))/(2*c^2) - (b*d*log(c + x*(-c^3)^(1/2))*(-c^3)^(1/2))/(2*c^2) + (b*d* log(c - x*(-c^3)^(1/2))*(-c^3)^(1/2))/(2*c^2)
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {2 \sqrt {c}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}}\right ) b d +2 \sqrt {c}\, \mathit {atanh} \left (c \,x^{2}\right ) b d +2 \mathit {atanh} \left (c \,x^{2}\right ) b c d x +\mathit {atanh} \left (c \,x^{2}\right ) b c e \,x^{2}-\mathit {atanh} \left (c \,x^{2}\right ) b e +2 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}\, x -1\right ) b d -\sqrt {c}\, \mathrm {log}\left (c \,x^{2}+1\right ) b d +\mathrm {log}\left (c \,x^{2}+1\right ) b e +2 a c d x +a c e \,x^{2}}{2 c} \] Input:
int((e*x+d)*(a+b*atanh(c*x^2)),x)
Output:
(2*sqrt(c)*atan((c*x)/sqrt(c))*b*d + 2*sqrt(c)*atanh(c*x**2)*b*d + 2*atanh (c*x**2)*b*c*d*x + atanh(c*x**2)*b*c*e*x**2 - atanh(c*x**2)*b*e + 2*sqrt(c )*log(sqrt(c)*x - 1)*b*d - sqrt(c)*log(c*x**2 + 1)*b*d + log(c*x**2 + 1)*b *e + 2*a*c*d*x + a*c*e*x**2)/(2*c)