3.4.19 \(\int \tanh ^{-1}(c+d \tan (a+b x)) \, dx\) [319]

Optimal. Leaf size=194 \[ x \tanh ^{-1}(c+d \tan (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right )-\frac {i \text {PolyLog}\left (2,-\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right )}{4 b}+\frac {i \text {PolyLog}\left (2,-\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right )}{4 b} \]

[Out]

x*arctanh(c+d*tan(b*x+a))+1/2*x*ln(1+(1-c+I*d)*exp(2*I*a+2*I*b*x)/(1-c-I*d))-1/2*x*ln(1+(1+c-I*d)*exp(2*I*a+2*
I*b*x)/(1+c+I*d))-1/4*I*polylog(2,-(1-c+I*d)*exp(2*I*a+2*I*b*x)/(1-c-I*d))/b+1/4*I*polylog(2,-(1+c-I*d)*exp(2*
I*a+2*I*b*x)/(1+c+I*d))/b

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Rubi [A]
time = 0.17, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6394, 2221, 2317, 2438} \begin {gather*} -\frac {i \text {Li}_2\left (-\frac {(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right )}{4 b}+\frac {i \text {Li}_2\left (-\frac {(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right )}{4 b}+\frac {1}{2} x \log \left (1+\frac {(-c+i d+1) e^{2 i a+2 i b x}}{-c-i d+1}\right )-\frac {1}{2} x \log \left (1+\frac {(c-i d+1) e^{2 i a+2 i b x}}{c+i d+1}\right )+x \tanh ^{-1}(d \tan (a+b x)+c) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[c + d*Tan[a + b*x]],x]

[Out]

x*ArcTanh[c + d*Tan[a + b*x]] + (x*Log[1 + ((1 - c + I*d)*E^((2*I)*a + (2*I)*b*x))/(1 - c - I*d)])/2 - (x*Log[
1 + ((1 + c - I*d)*E^((2*I)*a + (2*I)*b*x))/(1 + c + I*d)])/2 - ((I/4)*PolyLog[2, -(((1 - c + I*d)*E^((2*I)*a
+ (2*I)*b*x))/(1 - c - I*d))])/b + ((I/4)*PolyLog[2, -(((1 + c - I*d)*E^((2*I)*a + (2*I)*b*x))/(1 + c + I*d))]
)/b

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 6394

Int[ArcTanh[(c_.) + (d_.)*Tan[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*ArcTanh[c + d*Tan[a + b*x]], x] + (-Di
st[I*b*(1 + c - I*d), Int[x*(E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x))), x], x] +
Dist[I*b*(1 - c + I*d), Int[x*(E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x))), x], x])
/; FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]

Rubi steps

\begin {align*} \int \tanh ^{-1}(c+d \tan (a+b x)) \, dx &=x \tanh ^{-1}(c+d \tan (a+b x))+(b (i (1-c)-d)) \int \frac {e^{2 i a+2 i b x} x}{1-c-i d+(1-c+i d) e^{2 i a+2 i b x}} \, dx-(b (i+i c+d)) \int \frac {e^{2 i a+2 i b x} x}{1+c+i d+(1+c-i d) e^{2 i a+2 i b x}} \, dx\\ &=x \tanh ^{-1}(c+d \tan (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right )-\frac {1}{2} \int \log \left (1+\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right ) \, dx+\frac {1}{2} \int \log \left (1+\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right ) \, dx\\ &=x \tanh ^{-1}(c+d \tan (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right )+\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {(1-c+i d) x}{1-c-i d}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {(1+c-i d) x}{1+c+i d}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b}\\ &=x \tanh ^{-1}(c+d \tan (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right )-\frac {i \text {Li}_2\left (-\frac {(1-c+i d) e^{2 i a+2 i b x}}{1-c-i d}\right )}{4 b}+\frac {i \text {Li}_2\left (-\frac {(1+c-i d) e^{2 i a+2 i b x}}{1+c+i d}\right )}{4 b}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4654\) vs. \(2(194)=388\).
time = 30.41, size = 4654, normalized size = 23.99 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[c + d*Tan[a + b*x]],x]

[Out]

x*ArcTanh[c + d*Tan[a + b*x]] + (d*(-(a*Log[-(Sec[(a + b*x)/2]^2*((-1 + c)*Cos[a + b*x] + d*Sin[a + b*x]))]) +
a*Log[Sec[(a + b*x)/2]^2*(Cos[a + b*x] + c*Cos[a + b*x] + d*Sin[a + b*x])] + (a + b*x)*Log[(-d + Sqrt[1 - 2*c
+ c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]] + I*Log[((-1 + c)*(1 + I*Tan[(a + b*x)/2]))/(-1 + c + I*d - I*Sqrt
[1 - 2*c + c^2 + d^2])]*Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]] - I*Log[-(((-1 + c)*
(I + Tan[(a + b*x)/2]))/(I - I*c - d + Sqrt[1 - 2*c + c^2 + d^2]))]*Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 +
c) + Tan[(a + b*x)/2]] + (a + b*x)*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] + I*Log[((
-1 + c)*(-I + Tan[(a + b*x)/2]))/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Log[(d + Sqrt[1 - 2*c + c^2 + d^2]
)/(1 - c) + Tan[(a + b*x)/2]] - I*Log[((-1 + c)*(I + Tan[(a + b*x)/2]))/(-I + I*c + d + Sqrt[1 - 2*c + c^2 + d
^2])]*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] - (a + b*x)*Log[-((d + Sqrt[1 + 2*c + c^
2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] - I*Log[((1 + c)*(-I + Tan[(a + b*x)/2]))/(-I - I*c + d + Sqrt[1 + 2*c
+ c^2 + d^2])]*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] + I*Log[((1 + c)*(I + Tan[(a
+ b*x)/2]))/(I + I*c + d + Sqrt[1 + 2*c + c^2 + d^2])]*Log[-((d + Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(
a + b*x)/2]] - (a + b*x)*Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] + I*Log[((1
+ c)*(1 - I*Tan[(a + b*x)/2]))/(1 + c - I*d + I*Sqrt[1 + 2*c + c^2 + d^2])]*Log[(-d + Sqrt[1 + 2*c + c^2 + d^2
] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] - I*Log[((1 + c)*(1 + I*Tan[(a + b*x)/2]))/(1 + c + I*d - I*Sqrt[1 + 2*
c + c^2 + d^2])]*Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)] + I*PolyLog[2, (d +
Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2])/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])] - I*PolyLog[
2, (d + Sqrt[1 - 2*c + c^2 + d^2] - (-1 + c)*Tan[(a + b*x)/2])/(-I + I*c + d + Sqrt[1 - 2*c + c^2 + d^2])] - I
*PolyLog[2, (-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2])/(I - I*c - d + Sqrt[1 - 2*c + c^2 + d
^2])] + I*PolyLog[2, (-d + Sqrt[1 - 2*c + c^2 + d^2] + (-1 + c)*Tan[(a + b*x)/2])/(-I + I*c - d + Sqrt[1 - 2*c
+ c^2 + d^2])] - I*PolyLog[2, (d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2])/(-I - I*c + d + Sqrt
[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (d + Sqrt[1 + 2*c + c^2 + d^2] - (1 + c)*Tan[(a + b*x)/2])/(I + I*c + d
+ Sqrt[1 + 2*c + c^2 + d^2])] + I*PolyLog[2, (-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(-I
- I*c - d + Sqrt[1 + 2*c + c^2 + d^2])] - I*PolyLog[2, (-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)
/2])/(I + I*c - d + Sqrt[1 + 2*c + c^2 + d^2])])*((-2*a)/(b*(-1 + c^2 + d^2 - Cos[2*(a + b*x)] + c^2*Cos[2*(a
+ b*x)] - d^2*Cos[2*(a + b*x)] + 2*c*d*Sin[2*(a + b*x)])) + (2*(a + b*x))/(b*(-1 + c^2 + d^2 - Cos[2*(a + b*x)
] + c^2*Cos[2*(a + b*x)] - d^2*Cos[2*(a + b*x)] + 2*c*d*Sin[2*(a + b*x)]))))/(Log[(-d + Sqrt[1 - 2*c + c^2 + d
^2])/(-1 + c) + Tan[(a + b*x)/2]] + Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]] - Log[-((d
+ Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]] - Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan
[(a + b*x)/2])/(1 + c)] + (Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/2])/(1 + c)]*Sec[(a + b
*x)/2]^2)/(2*(1 - I*Tan[(a + b*x)/2])) - (Log[(-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]]*Se
c[(a + b*x)/2]^2)/(2*(1 + I*Tan[(a + b*x)/2])) + (Log[(-d + Sqrt[1 + 2*c + c^2 + d^2] + (1 + c)*Tan[(a + b*x)/
2])/(1 + c)]*Sec[(a + b*x)/2]^2)/(2*(1 + I*Tan[(a + b*x)/2])) + ((I/2)*Log[(d + Sqrt[1 - 2*c + c^2 + d^2])/(1
- c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) - ((I/2)*Log[-((d + Sqrt[1 + 2*c + c^2 +
d^2])/(1 + c)) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(-I + Tan[(a + b*x)/2]) - ((I/2)*Log[(-d + Sqrt[1 - 2*c
+ c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) - ((I/2)*Log[(d + Sqrt[
1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) + ((I/2)*Log[-((d
+ Sqrt[1 + 2*c + c^2 + d^2])/(1 + c)) + Tan[(a + b*x)/2]]*Sec[(a + b*x)/2]^2)/(I + Tan[(a + b*x)/2]) + ((a +
b*x)*Sec[(a + b*x)/2]^2)/(2*((-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2])) + ((I/2)*Log[((-1
+ c)*(1 + I*Tan[(a + b*x)/2]))/(-1 + c + I*d - I*Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/((-d + Sqrt[1
- 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x)/2]) - ((I/2)*Log[-(((-1 + c)*(I + Tan[(a + b*x)/2]))/(I - I*c -
d + Sqrt[1 - 2*c + c^2 + d^2]))]*Sec[(a + b*x)/2]^2)/((-d + Sqrt[1 - 2*c + c^2 + d^2])/(-1 + c) + Tan[(a + b*x
)/2]) + ((a + b*x)*Sec[(a + b*x)/2]^2)/(2*((d + Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2])) + ((I/
2)*Log[((-1 + c)*(-I + Tan[(a + b*x)/2]))/(I - I*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b*x)/2]^2)/((d +
Sqrt[1 - 2*c + c^2 + d^2])/(1 - c) + Tan[(a + b*x)/2]) - ((I/2)*Log[((-1 + c)*(I + Tan[(a + b*x)/2]))/(-I + I
*c + d + Sqrt[1 - 2*c + c^2 + d^2])]*Sec[(a + b...

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (164 ) = 328\).
time = 1.28, size = 563, normalized size = 2.90

method	result	size
derivativedivides	\(\frac {d \arctan \left (\tan \left (b x +a \right )\right ) \arctanh \left (c +d \tan \left (b x +a \right )\right )-d^{2} \left (-\frac {\arctan \left (-\frac {c +d \tan \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right )}{2 d}+\frac {\arctan \left (-\frac {c +d \tan \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right )}{2 d}+\frac {i \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}+\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )}{4 d}-\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )}{4 d}-\frac {i \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}-\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )}{4 d}+\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )}{4 d}\right )}{b d}\)	\(563\)
default	\(\frac {d \arctan \left (\tan \left (b x +a \right )\right ) \arctanh \left (c +d \tan \left (b x +a \right )\right )-d^{2} \left (-\frac {\arctan \left (-\frac {c +d \tan \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right )}{2 d}+\frac {\arctan \left (-\frac {c +d \tan \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right )}{2 d}+\frac {i \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}+\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )}{4 d}-\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )}{4 d}-\frac {i \ln \left (d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}-\frac {i \dilog \left (\frac {i d -d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )}{4 d}+\frac {i \dilog \left (\frac {i d +d \left (\frac {c +d \tan \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )}{4 d}\right )}{b d}\)	\(563\)
risch	\(\text {Expression too large to display}\)	\(3967\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(c+d*tan(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/d*(d*arctan(tan(b*x+a))*arctanh(c+d*tan(b*x+a))-d^2*(-1/2*arctan(-(c+d*tan(b*x+a))/d+c/d)/d*ln(d*((c+d*tan
(b*x+a))/d-c/d)+c+1)+1/2*arctan(-(c+d*tan(b*x+a))/d+c/d)/d*ln(d*((c+d*tan(b*x+a))/d-c/d)+c-1)+1/4*I*ln(d*((c+d
*tan(b*x+a))/d-c/d)+c+1)*(ln((I*d-d*((c+d*tan(b*x+a))/d-c/d))/(1+c+I*d))-ln((I*d+d*((c+d*tan(b*x+a))/d-c/d))/(
I*d-c-1)))/d+1/4*I/d*dilog((I*d-d*((c+d*tan(b*x+a))/d-c/d))/(1+c+I*d))-1/4*I/d*dilog((I*d+d*((c+d*tan(b*x+a))/
d-c/d))/(I*d-c-1))-1/4*I*ln(d*((c+d*tan(b*x+a))/d-c/d)+c-1)*(ln((I*d-d*((c+d*tan(b*x+a))/d-c/d))/(I*d+c-1))-ln
((I*d+d*((c+d*tan(b*x+a))/d-c/d))/(1-c+I*d)))/d-1/4*I/d*dilog((I*d-d*((c+d*tan(b*x+a))/d-c/d))/(I*d+c-1))+1/4*
I/d*dilog((I*d+d*((c+d*tan(b*x+a))/d-c/d))/(1-c+I*d))))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (136) = 272\).
time = 0.54, size = 372, normalized size = 1.92 \begin {gather*} \frac {4 \, {\left (b x + a\right )} \operatorname {artanh}\left (d \tan \left (b x + a\right ) + c\right ) + {\left (\arctan \left (\frac {d^{2} \tan \left (b x + a\right ) + {\left (c + 1\right )} d}{c^{2} + d^{2} + 2 \, c + 1}, \frac {{\left (c + 1\right )} d \tan \left (b x + a\right ) + c^{2} + 2 \, c + 1}{c^{2} + d^{2} + 2 \, c + 1}\right ) - \arctan \left (\frac {d^{2} \tan \left (b x + a\right ) + {\left (c - 1\right )} d}{c^{2} + d^{2} - 2 \, c + 1}, \frac {{\left (c - 1\right )} d \tan \left (b x + a\right ) + c^{2} - 2 \, c + 1}{c^{2} + d^{2} - 2 \, c + 1}\right )\right )} \log \left (\tan \left (b x + a\right )^{2} + 1\right ) - {\left (b x + a\right )} \log \left (\frac {d^{2} \tan \left (b x + a\right )^{2} + 2 \, {\left (c + 1\right )} d \tan \left (b x + a\right ) + c^{2} + 2 \, c + 1}{c^{2} + d^{2} + 2 \, c + 1}\right ) + {\left (b x + a\right )} \log \left (\frac {d^{2} \tan \left (b x + a\right )^{2} + 2 \, {\left (c - 1\right )} d \tan \left (b x + a\right ) + c^{2} - 2 \, c + 1}{c^{2} + d^{2} - 2 \, c + 1}\right ) - i \, {\rm Li}_2\left (-\frac {i \, d \tan \left (b x + a\right ) - d}{i \, c + d + i}\right ) + i \, {\rm Li}_2\left (-\frac {i \, d \tan \left (b x + a\right ) - d}{i \, c + d - i}\right ) - i \, {\rm Li}_2\left (\frac {i \, d \tan \left (b x + a\right ) + d}{-i \, c + d + i}\right ) + i \, {\rm Li}_2\left (\frac {i \, d \tan \left (b x + a\right ) + d}{-i \, c + d - i}\right )}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(c+d*tan(b*x+a)),x, algorithm="maxima")

[Out]

1/4*(4*(b*x + a)*arctanh(d*tan(b*x + a) + c) + (arctan2((d^2*tan(b*x + a) + (c + 1)*d)/(c^2 + d^2 + 2*c + 1),
((c + 1)*d*tan(b*x + a) + c^2 + 2*c + 1)/(c^2 + d^2 + 2*c + 1)) - arctan2((d^2*tan(b*x + a) + (c - 1)*d)/(c^2
+ d^2 - 2*c + 1), ((c - 1)*d*tan(b*x + a) + c^2 - 2*c + 1)/(c^2 + d^2 - 2*c + 1)))*log(tan(b*x + a)^2 + 1) - (
b*x + a)*log((d^2*tan(b*x + a)^2 + 2*(c + 1)*d*tan(b*x + a) + c^2 + 2*c + 1)/(c^2 + d^2 + 2*c + 1)) + (b*x + a
)*log((d^2*tan(b*x + a)^2 + 2*(c - 1)*d*tan(b*x + a) + c^2 - 2*c + 1)/(c^2 + d^2 - 2*c + 1)) - I*dilog(-(I*d*t
an(b*x + a) - d)/(I*c + d + I)) + I*dilog(-(I*d*tan(b*x + a) - d)/(I*c + d - I)) - I*dilog((I*d*tan(b*x + a) +
d)/(-I*c + d + I)) + I*dilog((I*d*tan(b*x + a) + d)/(-I*c + d - I)))/b

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1185 vs. \(2 (136) = 272\).
time = 0.53, size = 1185, normalized size = 6.11 \begin {gather*} \frac {4 \, b x \log \left (-\frac {d \tan \left (b x + a\right ) + c + 1}{d \tan \left (b x + a\right ) + c - 1}\right ) - 2 \, {\left (b x + a\right )} \log \left (-\frac {2 \, {\left ({\left (i \, {\left (c + 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} - i \, {\left (c + 1\right )} d + {\left (i \, c^{2} - 2 \, {\left (c + 1\right )} d - i \, d^{2} + 2 i \, c + i\right )} \tan \left (b x + a\right ) - 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} + 2 \, c + 1}\right ) - 2 \, {\left (b x + a\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, {\left (c + 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} + i \, {\left (c + 1\right )} d + {\left (-i \, c^{2} - 2 \, {\left (c + 1\right )} d + i \, d^{2} - 2 i \, c - i\right )} \tan \left (b x + a\right ) - 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} + 2 \, c + 1}\right ) + 2 \, {\left (b x + a\right )} \log \left (-\frac {2 \, {\left ({\left (i \, {\left (c - 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} - i \, {\left (c - 1\right )} d + {\left (i \, c^{2} - 2 \, {\left (c - 1\right )} d - i \, d^{2} - 2 i \, c + i\right )} \tan \left (b x + a\right ) + 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} - 2 \, c + 1}\right ) + 2 \, {\left (b x + a\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, {\left (c - 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} + i \, {\left (c - 1\right )} d + {\left (-i \, c^{2} - 2 \, {\left (c - 1\right )} d + i \, d^{2} + 2 i \, c - i\right )} \tan \left (b x + a\right ) + 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} - 2 \, c + 1}\right ) + 2 \, a \log \left (\frac {{\left (i \, {\left (c + 1\right )} d + d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} + i \, {\left (c + 1\right )} d + {\left (i \, c^{2} + i \, d^{2} + 2 i \, c + i\right )} \tan \left (b x + a\right ) - 2 \, c - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) + 2 \, a \log \left (\frac {{\left (i \, {\left (c + 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} + c^{2} + i \, {\left (c + 1\right )} d + {\left (i \, c^{2} + i \, d^{2} + 2 i \, c + i\right )} \tan \left (b x + a\right ) + 2 \, c + 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 2 \, a \log \left (\frac {{\left (i \, {\left (c - 1\right )} d + d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} + i \, {\left (c - 1\right )} d + {\left (i \, c^{2} + i \, d^{2} - 2 i \, c + i\right )} \tan \left (b x + a\right ) + 2 \, c - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 2 \, a \log \left (\frac {{\left (i \, {\left (c - 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} + c^{2} + i \, {\left (c - 1\right )} d + {\left (i \, c^{2} + i \, d^{2} - 2 i \, c + i\right )} \tan \left (b x + a\right ) - 2 \, c + 1}{\tan \left (b x + a\right )^{2} + 1}\right ) - i \, {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, {\left (c + 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} - i \, {\left (c + 1\right )} d + {\left (i \, c^{2} - 2 \, {\left (c + 1\right )} d - i \, d^{2} + 2 i \, c + i\right )} \tan \left (b x + a\right ) - 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} + 2 \, c + 1} + 1\right ) + i \, {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, {\left (c + 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} + i \, {\left (c + 1\right )} d + {\left (-i \, c^{2} - 2 \, {\left (c + 1\right )} d + i \, d^{2} - 2 i \, c - i\right )} \tan \left (b x + a\right ) - 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} + 2 \, c + 1} + 1\right ) + i \, {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, {\left (c - 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} - i \, {\left (c - 1\right )} d + {\left (i \, c^{2} - 2 \, {\left (c - 1\right )} d - i \, d^{2} - 2 i \, c + i\right )} \tan \left (b x + a\right ) + 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} - 2 \, c + 1} + 1\right ) - i \, {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, {\left (c - 1\right )} d - d^{2}\right )} \tan \left (b x + a\right )^{2} - c^{2} + i \, {\left (c - 1\right )} d + {\left (-i \, c^{2} - 2 \, {\left (c - 1\right )} d + i \, d^{2} + 2 i \, c - i\right )} \tan \left (b x + a\right ) + 2 \, c - 1\right )}}{{\left (c^{2} + d^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + c^{2} + d^{2} - 2 \, c + 1} + 1\right )}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(c+d*tan(b*x+a)),x, algorithm="fricas")

[Out]

1/8*(4*b*x*log(-(d*tan(b*x + a) + c + 1)/(d*tan(b*x + a) + c - 1)) - 2*(b*x + a)*log(-2*((I*(c + 1)*d - d^2)*t
an(b*x + a)^2 - c^2 - I*(c + 1)*d + (I*c^2 - 2*(c + 1)*d - I*d^2 + 2*I*c + I)*tan(b*x + a) - 2*c - 1)/((c^2 +
d^2 + 2*c + 1)*tan(b*x + a)^2 + c^2 + d^2 + 2*c + 1)) - 2*(b*x + a)*log(-2*((-I*(c + 1)*d - d^2)*tan(b*x + a)^
2 - c^2 + I*(c + 1)*d + (-I*c^2 - 2*(c + 1)*d + I*d^2 - 2*I*c - I)*tan(b*x + a) - 2*c - 1)/((c^2 + d^2 + 2*c +
1)*tan(b*x + a)^2 + c^2 + d^2 + 2*c + 1)) + 2*(b*x + a)*log(-2*((I*(c - 1)*d - d^2)*tan(b*x + a)^2 - c^2 - I*
(c - 1)*d + (I*c^2 - 2*(c - 1)*d - I*d^2 - 2*I*c + I)*tan(b*x + a) + 2*c - 1)/((c^2 + d^2 - 2*c + 1)*tan(b*x +
a)^2 + c^2 + d^2 - 2*c + 1)) + 2*(b*x + a)*log(-2*((-I*(c - 1)*d - d^2)*tan(b*x + a)^2 - c^2 + I*(c - 1)*d +
(-I*c^2 - 2*(c - 1)*d + I*d^2 + 2*I*c - I)*tan(b*x + a) + 2*c - 1)/((c^2 + d^2 - 2*c + 1)*tan(b*x + a)^2 + c^2
+ d^2 - 2*c + 1)) + 2*a*log(((I*(c + 1)*d + d^2)*tan(b*x + a)^2 - c^2 + I*(c + 1)*d + (I*c^2 + I*d^2 + 2*I*c
+ I)*tan(b*x + a) - 2*c - 1)/(tan(b*x + a)^2 + 1)) + 2*a*log(((I*(c + 1)*d - d^2)*tan(b*x + a)^2 + c^2 + I*(c
+ 1)*d + (I*c^2 + I*d^2 + 2*I*c + I)*tan(b*x + a) + 2*c + 1)/(tan(b*x + a)^2 + 1)) - 2*a*log(((I*(c - 1)*d + d
^2)*tan(b*x + a)^2 - c^2 + I*(c - 1)*d + (I*c^2 + I*d^2 - 2*I*c + I)*tan(b*x + a) + 2*c - 1)/(tan(b*x + a)^2 +
1)) - 2*a*log(((I*(c - 1)*d - d^2)*tan(b*x + a)^2 + c^2 + I*(c - 1)*d + (I*c^2 + I*d^2 - 2*I*c + I)*tan(b*x +
a) - 2*c + 1)/(tan(b*x + a)^2 + 1)) - I*dilog(2*((I*(c + 1)*d - d^2)*tan(b*x + a)^2 - c^2 - I*(c + 1)*d + (I*
c^2 - 2*(c + 1)*d - I*d^2 + 2*I*c + I)*tan(b*x + a) - 2*c - 1)/((c^2 + d^2 + 2*c + 1)*tan(b*x + a)^2 + c^2 + d
^2 + 2*c + 1) + 1) + I*dilog(2*((-I*(c + 1)*d - d^2)*tan(b*x + a)^2 - c^2 + I*(c + 1)*d + (-I*c^2 - 2*(c + 1)*
d + I*d^2 - 2*I*c - I)*tan(b*x + a) - 2*c - 1)/((c^2 + d^2 + 2*c + 1)*tan(b*x + a)^2 + c^2 + d^2 + 2*c + 1) +
1) + I*dilog(2*((I*(c - 1)*d - d^2)*tan(b*x + a)^2 - c^2 - I*(c - 1)*d + (I*c^2 - 2*(c - 1)*d - I*d^2 - 2*I*c
+ I)*tan(b*x + a) + 2*c - 1)/((c^2 + d^2 - 2*c + 1)*tan(b*x + a)^2 + c^2 + d^2 - 2*c + 1) + 1) - I*dilog(2*((-
I*(c - 1)*d - d^2)*tan(b*x + a)^2 - c^2 + I*(c - 1)*d + (-I*c^2 - 2*(c - 1)*d + I*d^2 + 2*I*c - I)*tan(b*x + a
) + 2*c - 1)/((c^2 + d^2 - 2*c + 1)*tan(b*x + a)^2 + c^2 + d^2 - 2*c + 1) + 1))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {atanh}{\left (c + d \tan {\left (a + b x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(c+d*tan(b*x+a)),x)

[Out]

Integral(atanh(c + d*tan(a + b*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(c+d*tan(b*x+a)),x, algorithm="giac")

[Out]

integrate(arctanh(d*tan(b*x + a) + c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {atanh}\left (c+d\,\mathrm {tan}\left (a+b\,x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(c + d*tan(a + b*x)),x)

[Out]

int(atanh(c + d*tan(a + b*x)), x)

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