3.1.19 \(\int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx\) [19]

Optimal. Leaf size=1280 \[ -\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 \sqrt {-b} f^2}+\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 \sqrt {b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{\sqrt {b} f^2}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{\sqrt {b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{2 \sqrt {b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{2 \sqrt {b} f^2}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{\sqrt {-b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{2 \sqrt {-b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (-\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{2 \sqrt {-b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{\sqrt {-b} f^2}-\frac {d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 \sqrt {b} f^2}-\frac {d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 \sqrt {b} f^2}+\frac {d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{4 \sqrt {b} f^2}+\frac {d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{4 \sqrt {b} f^2}+\frac {d \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 \sqrt {-b} f^2}-\frac {d \text {PolyLog}\left (2,1-\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{4 \sqrt {-b} f^2}-\frac {d \text {PolyLog}\left (2,1+\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{4 \sqrt {-b} f^2}+\frac {d \text {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 \sqrt {-b} f^2} \]

[Out]

-(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/f/(-b)^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2)
)^2/f^2/(-b)^(1/2)+d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/
(-b)^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(2*(b^(1/2)-(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^
(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln
(-2*(b^(1/2)+(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-
d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)+1/2*d*po
lylog(2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-1/4*d*polylog(2,1-2*(b^(1/2)-(b*tanh(f*x+e))^
(1/2))/((-b)^(1/2)+b^(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-1/4*d*polylog(2,1+2*(b^(1/2)+
(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)+1/2*d*polylog
(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)+(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/f/b
^(1/2)+1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))^2/f^2/b^(1/2)-d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*ln(
2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)+d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*ln(2*b^(1/2)/(
b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*ln(2*b^(1/2)*((-b)^(1
/2)-(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)-1/2*d*arctanh((b*
tanh(f*x+e))^(1/2)/b^(1/2))*ln(2*b^(1/2)*((-b)^(1/2)+(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^(1/2))/(b^(1/2)+(b*t
anh(f*x+e))^(1/2)))/f^2/b^(1/2)-1/2*d*polylog(2,1-2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)-1/2*d
*polylog(2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)+1/4*d*polylog(2,1-2*b^(1/2)*((-b)^(1/2)-(b
*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)+1/4*d*polylog(2,1-2*b^(
1/2)*((-b)^(1/2)+(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^(1/2))/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2/b^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.99, antiderivative size = 1280, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3819, 213, 281, 6857, 6139, 6057, 2449, 2352, 2497, 6131, 6055, 212} \begin {gather*} -\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 \sqrt {-b} f^2}-\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {d \log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f^2}-\frac {d \log \left (\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 \sqrt {-b} f^2}-\frac {d \log \left (-\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 \sqrt {-b} f^2}-\frac {d \log \left (\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f^2}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 \sqrt {b} f^2}+\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{\sqrt {b} f^2}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{\sqrt {b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{2 \sqrt {b} f^2}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{2 \sqrt {b} f^2}-\frac {d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 \sqrt {b} f^2}-\frac {d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 \sqrt {b} f^2}+\frac {d \text {Li}_2\left (1-\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{4 \sqrt {b} f^2}+\frac {d \text {Li}_2\left (1-\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{4 \sqrt {b} f^2}+\frac {d \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 \sqrt {-b} f^2}-\frac {d \text {Li}_2\left (1-\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{4 \sqrt {-b} f^2}-\frac {d \text {Li}_2\left (\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}+1\right )}{4 \sqrt {-b} f^2}+\frac {d \text {Li}_2\left (1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{2 \sqrt {-b} f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/Sqrt[b*Tanh[e + f*x]],x]

[Out]

-(((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/(Sqrt[-b]*f)) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b
]]^2)/(2*Sqrt[-b]*f^2) + ((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(Sqrt[b]*f) + (d*ArcTanh[Sqrt[b*Ta
nh[e + f*x]]/Sqrt[b]]^2)/(2*Sqrt[b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b]
- Sqrt[b*Tanh[e + f*x]])])/(Sqrt[b]*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b]
+ Sqrt[b*Tanh[e + f*x]])])/(Sqrt[b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b]
- Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*Sqrt[b]*f^2) - (d*ArcT
anh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(S
qrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*Sqrt[b]*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqr
t[b*Tanh[e + f*x]]/Sqrt[-b])])/(Sqrt[-b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - S
qrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*Sqrt[-b]*f^2) - (d*Arc
Tanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(-2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqr
t[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*Sqrt[-b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[
b*Tanh[e + f*x]]/Sqrt[-b])])/(Sqrt[-b]*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])]
)/(2*Sqrt[b]*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*Sqrt[b]*f^2) + (d*Pol
yLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*
x]]))])/(4*Sqrt[b]*f^2) + (d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b
])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*Sqrt[b]*f^2) + (d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[
-b])])/(2*Sqrt[-b]*f^2) - (d*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - S
qrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*Sqrt[-b]*f^2) - (d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/(
(Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*Sqrt[-b]*f^2) + (d*PolyLog[2, 1 - 2/(1 + Sqrt[
b*Tanh[e + f*x]]/Sqrt[-b])])/(2*Sqrt[-b]*f^2)

Rule 212

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

Rule 213

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 3819

Int[((c_.) + (d_.)*(x_))/Sqrt[(a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(-I)*((c + d*x)/(f*Rt[
a - I*b, 2]))*ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x] + (Dist[I*(d/(f*Rt[a - I*b, 2])), Int[ArcTa
nh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x], x] - Dist[I*(d/(f*Rt[a + I*b, 2])), Int[ArcTanh[Sqrt[a + b*Ta
n[e + f*x]]/Rt[a + I*b, 2]], x], x] + Simp[I*((c + d*x)/(f*Rt[a + I*b, 2]))*ArcTanh[Sqrt[a + b*Tan[e + f*x]]/R
t[a + I*b, 2]], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^
2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 6057

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcTanh[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 6131

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 6139

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx &=\int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.10, size = 556, normalized size = 0.43 \begin {gather*} \frac {\left (4 f (c+d x) \left (2 \text {ArcTan}\left (\sqrt {\tanh (e+f x)}\right )-\log \left (1-\sqrt {\tanh (e+f x)}\right )+\log \left (1+\sqrt {\tanh (e+f x)}\right )\right )+d \left (-4 i \text {ArcTan}\left (\sqrt {\tanh (e+f x)}\right )^2+4 \text {ArcTan}\left (\sqrt {\tanh (e+f x)}\right ) \log \left (1+e^{4 i \text {ArcTan}\left (\sqrt {\tanh (e+f x)}\right )}\right )-\log ^2\left (1-\sqrt {\tanh (e+f x)}\right )+2 \log \left (1-\sqrt {\tanh (e+f x)}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {\tanh (e+f x)}\right )\right )+2 \log \left (1-\sqrt {\tanh (e+f x)}\right ) \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\tanh (e+f x)}\right )\right )-2 \log \left (1-\sqrt {\tanh (e+f x)}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \log \left (1-\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right ) \log \left (1+\sqrt {\tanh (e+f x)}\right )+2 \log \left (\frac {1}{2} \left (1-\sqrt {\tanh (e+f x)}\right )\right ) \log \left (1+\sqrt {\tanh (e+f x)}\right )-2 \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\tanh (e+f x)}\right )\right ) \log \left (1+\sqrt {\tanh (e+f x)}\right )+\log ^2\left (1+\sqrt {\tanh (e+f x)}\right )-i \text {PolyLog}\left (2,-e^{4 i \text {ArcTan}\left (\sqrt {\tanh (e+f x)}\right )}\right )-2 \text {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\tanh (e+f x)}\right )\right )+2 \text {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\tanh (e+f x)}\right )\right )+2 \text {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\tanh (e+f x)}\right )\right )+2 \text {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \text {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \text {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right )\right )\right ) \sqrt {\tanh (e+f x)}}{8 f^2 \sqrt {b \tanh (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/Sqrt[b*Tanh[e + f*x]],x]

[Out]

((4*f*(c + d*x)*(2*ArcTan[Sqrt[Tanh[e + f*x]]] - Log[1 - Sqrt[Tanh[e + f*x]]] + Log[1 + Sqrt[Tanh[e + f*x]]])
+ d*((-4*I)*ArcTan[Sqrt[Tanh[e + f*x]]]^2 + 4*ArcTan[Sqrt[Tanh[e + f*x]]]*Log[1 + E^((4*I)*ArcTan[Sqrt[Tanh[e
+ f*x]]])] - Log[1 - Sqrt[Tanh[e + f*x]]]^2 + 2*Log[1 - Sqrt[Tanh[e + f*x]]]*Log[(1/2 + I/2)*(-I + Sqrt[Tanh[e
+ f*x]])] + 2*Log[1 - Sqrt[Tanh[e + f*x]]]*Log[(1/2 - I/2)*(I + Sqrt[Tanh[e + f*x]])] - 2*Log[1 - Sqrt[Tanh[e
+ f*x]]]*Log[(1 + Sqrt[Tanh[e + f*x]])/2] - 2*Log[1 - (1/2 - I/2)*(1 + Sqrt[Tanh[e + f*x]])]*Log[1 + Sqrt[Tan
h[e + f*x]]] + 2*Log[(1 - Sqrt[Tanh[e + f*x]])/2]*Log[1 + Sqrt[Tanh[e + f*x]]] - 2*Log[(-1/2 - I/2)*(I + Sqrt[
Tanh[e + f*x]])]*Log[1 + Sqrt[Tanh[e + f*x]]] + Log[1 + Sqrt[Tanh[e + f*x]]]^2 - I*PolyLog[2, -E^((4*I)*ArcTan
[Sqrt[Tanh[e + f*x]]])] - 2*PolyLog[2, (1 - Sqrt[Tanh[e + f*x]])/2] + 2*PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[Tan
h[e + f*x]])] + 2*PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[Tanh[e + f*x]])] + 2*PolyLog[2, (1 + Sqrt[Tanh[e + f*x]])
/2] - 2*PolyLog[2, (1/2 - I/2)*(1 + Sqrt[Tanh[e + f*x]])] - 2*PolyLog[2, (1/2 + I/2)*(1 + Sqrt[Tanh[e + f*x]])
]))*Sqrt[Tanh[e + f*x]])/(8*f^2*Sqrt[b*Tanh[e + f*x]])

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Maple [F]
time = 1.38, size = 0, normalized size = 0.00 \[\int \frac {d x +c}{\sqrt {b \tanh \left (f x +e \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(b*tanh(f*x+e))^(1/2),x)

[Out]

int((d*x+c)/(b*tanh(f*x+e))^(1/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)/sqrt(b*tanh(f*x + e)), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c + d x}{\sqrt {b \tanh {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*tanh(f*x+e))**(1/2),x)

[Out]

Integral((c + d*x)/sqrt(b*tanh(e + f*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((d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((d*x + c)/sqrt(b*tanh(f*x + e)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c+d\,x}{\sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(b*tanh(e + f*x))^(1/2),x)

[Out]

int((c + d*x)/(b*tanh(e + f*x))^(1/2), x)

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