\(\int (c+d x) \sqrt {b \tanh (e+f x)} \, dx\) [18]

Optimal result

Integrand size = 18, antiderivative size = 1280 \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {\sqrt {-b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {\sqrt {b} d \text {arctanh}\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 )}{f^2}+\frac {\sqrt {b} d \text {arctanh}\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 )}{f^2}-\frac {\sqrt {b} d \text {arctanh}\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 f^2}-\frac {\sqrt {b} d \text {arctanh}\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 f^2}+\frac {\sqrt {-b} d \text {arctanh}\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 )}{f^2}-\frac {\sqrt {-b} d \text {arctanh}\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 f^2}-\frac {\sqrt {-b} d \text {arctanh}\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 f^2}-\frac {\sqrt {-b} d \text {arctanh}\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 )}{f^2}-\frac {\sqrt {b} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {\sqrt {b} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {\sqrt {b} d \operatorname {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 f^2}+\frac {\sqrt {b} d \operatorname {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 f^2}+\frac {\sqrt {-b} d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {\sqrt {-b} d \operatorname {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 f^2}-\frac {\sqrt {-b} d \operatorname {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 f^2}+\frac {\sqrt {-b} d \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 1280, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3817, 213, 281, 6857, 6139, 6057, 2449, 2352, 2497, 6131, 6055, 212} \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=-\frac {\sqrt {-b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {-b} d \log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f^2}-\frac {\sqrt {-b} 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 ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 f^2}-\frac {\sqrt {-b} 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 ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 f^2}-\frac {\sqrt {-b} d \log \left (\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f^2}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} d \text {arctanh}\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 )}{f^2}+\frac {\sqrt {b} d \text {arctanh}\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 )}{f^2}-\frac {\sqrt {b} d \text {arctanh}\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 f^2}-\frac {\sqrt {b} d \text {arctanh}\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 f^2}-\frac {\sqrt {b} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {\sqrt {b} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {\sqrt {b} d \operatorname {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 f^2}+\frac {\sqrt {b} d \operatorname {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 f^2}+\frac {\sqrt {-b} d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {\sqrt {-b} d \operatorname {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 f^2}-\frac {\sqrt {-b} d \operatorname {PolyLog}\left (2,\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 f^2}+\frac {\sqrt {-b} d \operatorname {PolyLog}\left (2,1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{2 f^2} \]

[In]

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

[Out]

-((Sqrt[-b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/
Sqrt[-b]]^2)/(2*f^2) + (Sqrt[b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f + (Sqrt[b]*d*ArcTanh[Sqrt[
b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*f^2) - (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt
[b] - Sqrt[b*Tanh[e + f*x]])])/f^2 + (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b
] + Sqrt[b*Tanh[e + f*x]])])/f^2 - (Sqrt[b]*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*f^2) - (Sqrt[b]*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*f^2) + (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sq
rt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sq
rt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - (Sqrt[-b]*d*ArcT
anh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(-2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt
[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b
*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2
*f^2) - (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*f^2) + (Sqrt[b]*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*f^2) + (Sqrt[b]*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*f^2) + (Sqrt[-b]*d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]
)])/(2*f^2) - (Sqrt[-b]*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*f^2) - (Sqrt[-b]*d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sq
rt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) + (Sqrt[-b]*d*PolyLog[2, 1 - 2/(1 + Sqrt[b*T
anh[e + f*x]]/Sqrt[-b])])/(2*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 3817

Int[((c_.) + (d_.)*(x_))*Sqrt[(a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(-I)*Rt[a - I*b, 2]*((
c + d*x)/f)*ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x] + (Dist[I*d*(Rt[a - I*b, 2]/f), Int[ArcTanh[S
qrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x], x] - Dist[I*d*(Rt[a + I*b, 2]/f), Int[ArcTanh[Sqrt[a + b*Tan[e +
f*x]]/Rt[a + I*b, 2]], x], x] + Simp[I*Rt[a + I*b, 2]*((c + d*x)/f)*ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[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*} \text {integral}& = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {\left (\sqrt {-b} d\right ) \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \, dx}{f}-\frac {\left (\sqrt {b} d\right ) \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \, dx}{f} \\ & = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {\left (\sqrt {-b} d\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b \sqrt {b x}}{(-b)^{3/2}}\right )}{-1+x^2} \, dx,x,\tanh (e+f x)\right )}{f^2}-\frac {\left (\sqrt {b} d\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {\sqrt {b x}}{\sqrt {b}}\right )}{1-x^2} \, dx,x,\tanh (e+f x)\right )}{f^2} \\ & = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {(2 d) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{\sqrt {-b} f^2}-\frac {(2 d) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{1-\frac {x^4}{b^2}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{\sqrt {b} f^2} \\ & = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {(2 d) \text {Subst}\left (\int \left (-\frac {b x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (b-x^2\right )}-\frac {b x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (b+x^2\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{\sqrt {-b} f^2}-\frac {(2 d) \text {Subst}\left (\int \left (\frac {b x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (b-x^2\right )}+\frac {b x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (b+x^2\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{\sqrt {b} f^2} \\ & = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\left (\sqrt {-b} d\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{b-x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (\sqrt {-b} d\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{b+x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (\sqrt {b} d\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{b-x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (\sqrt {b} d\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{b+x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2} \\ & = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {\sqrt {-b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {d \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{1-\frac {x}{\sqrt {b}}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {d \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{1-\frac {b x}{(-b)^{3/2}}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (\sqrt {-b} d\right ) \text {Subst}\left (\int \left (\frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (\sqrt {b}-x\right )}-\frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (\sqrt {b}+x\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (\sqrt {b} d\right ) \text {Subst}\left (\int \left (-\frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (\sqrt {-b}-x\right )}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (\sqrt {-b}+x\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2} \\ & = -\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {\sqrt {-b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {\sqrt {b} d \text {arctanh}\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 )}{f^2}-\frac {\sqrt {-b} d \text {arctanh}\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 )}{f^2}+\frac {d \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {b}}}\right )}{1-\frac {x^2}{b}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}+\frac {d \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {b x}{(-b)^{3/2}}}\right )}{1+\frac {x^2}{b}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (\sqrt {-b} d\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{\sqrt {b}-x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{2 f^2}+\frac {\left (\sqrt {-b} d\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{\sqrt {b}+x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{2 f^2}+\frac {\left (\sqrt {b} d\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{\sqrt {-b}-x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{2 f^2}-\frac {\left (\sqrt {b} d\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{\sqrt {-b}+x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{2 f^2} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.71 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.43 \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\frac {\left (-4 f (c+d x) \left (2 \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 \arctan \left (\sqrt {\tanh (e+f x)}\right )^2-4 \arctan \left (\sqrt {\tanh (e+f x)}\right ) \log \left (1+e^{4 i \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 \operatorname {PolyLog}\left (2,-e^{4 i \arctan \left (\sqrt {\tanh (e+f x)}\right )}\right )-2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\tanh (e+f x)}\right )\right )+2 \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\tanh (e+f x)}\right )\right )+2 \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\tanh (e+f x)}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right )\right )\right ) \sqrt {b \tanh (e+f x)}}{8 f^2 \sqrt {\tanh (e+f x)}} \]

[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[b*Tanh[e + f*x]])/(8*f^2*Sqrt[Tanh[e + f*x]])

Maple [F]

\[\int \left (d x +c \right ) \sqrt {b \tanh \left (f x +e \right )}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int \sqrt {b \tanh {\left (e + f x \right )}} \left (c + d x\right )\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int { {\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )} \,d x } \]

[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)

Giac [F]

\[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int { {\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int \sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}\,\left (c+d\,x\right ) \,d x \]

[In]

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

[Out]

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