3.1.20 \(\int \frac {c+d x}{(b \tanh (e+f x))^{3/2}} \, dx\) [20]

Optimal. Leaf size=1365 \[ \frac {2 d \text {ArcTan}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 (-b)^{3/2} f^2}+\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 b^{3/2} 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 )}{b^{3/2} 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 )}{b^{3/2} 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 b^{3/2} 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 b^{3/2} 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 )}{(-b)^{3/2} 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 (-b)^{3/2} 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 (-b)^{3/2} 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 )}{(-b)^{3/2} f^2}-\frac {d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 b^{3/2} f^2}-\frac {d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 b^{3/2} 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 b^{3/2} 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 b^{3/2} f^2}+\frac {d \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 (-b)^{3/2} 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 (-b)^{3/2} 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 (-b)^{3/2} f^2}+\frac {d \text {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 (-b)^{3/2} f^2}-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}} \]

[Out]

2*d*arctan((b*tanh(f*x+e))^(1/2)/b^(1/2))/b^(3/2)/f^2-(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/(-b)^(
3/2)/f-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2/(-b)^(3/2)/f^2+2*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1
/2))/b^(3/2)/f^2+(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/b^(3/2)/f+1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/
b^(1/2))^2/b^(3/2)/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^(3/2)/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^(3/2)/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^(3/2)/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^(3/2)/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)^(3/2)/f^2-1/2*d*arctanh((b*ta
nh(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)^(3/2)/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)^(3/2)/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)^(3/2)/f^2-1/2*d*polylog(2,1-2*b^(1/2)/(b^(1/2)-(
b*tanh(f*x+e))^(1/2)))/b^(3/2)/f^2-1/2*d*polylog(2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/b^(3/2)/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^(3/2)/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^(3/2)/f^2+1/2*d*polylog(2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/(-b)^(3/2)/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)^(3/2)/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)^(3/2)/f^2+1/2*d*polylog(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/(-b)^(3/2)/f^2-2*(d*x
+c)/b/f/(b*tanh(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 1365, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 17, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.944, Rules used = {3802, 3557, 335, 218, 212, 209, 3817, 213, 281, 6857, 6139, 6057, 2449, 2352, 2497, 6131, 6055} \begin {gather*} -\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 (-b)^{3/2} f^2}-\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} 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 )}{(-b)^{3/2} 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 (-b)^{3/2} 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 (-b)^{3/2} 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 )}{(-b)^{3/2} f^2}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 b^{3/2} f^2}+\frac {2 d \text {ArcTan}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {(c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f}+\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} 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 )}{b^{3/2} 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 )}{b^{3/2} 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 b^{3/2} 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 b^{3/2} f^2}-\frac {d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 b^{3/2} f^2}-\frac {d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 b^{3/2} 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 b^{3/2} 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 b^{3/2} f^2}+\frac {d \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 (-b)^{3/2} 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 (-b)^{3/2} 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 (-b)^{3/2} f^2}+\frac {d \text {Li}_2\left (1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{2 (-b)^{3/2} f^2}-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(b*Tanh[e + f*x])^(3/2),x]

[Out]

(2*d*ArcTan[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f^2) - ((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]
)/((-b)^(3/2)*f) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*(-b)^(3/2)*f^2) + (2*d*ArcTanh[Sqrt[b*Tanh
[e + f*x]]/Sqrt[b]])/(b^(3/2)*f^2) + ((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f) + (d*ArcTa
nh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*b^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b
])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b
])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*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*b^(3/2)*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*b^(3/2)*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Lo
g[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^2) - (d*ArcTanh[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*(-b)^(3/
2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(-2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - S
qrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*(-b)^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]
]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqr
t[b*Tanh[e + f*x]])])/(2*b^(3/2)*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*b
^(3/2)*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*b^(3/2)*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*b^(3/2)*f^2) + (d*PolyLog[2, 1 - 2/(1 - Sqrt[b*
Tanh[e + f*x]]/Sqrt[-b])])/(2*(-b)^(3/2)*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*(-b)^(3/2)*f^2) - (d*PolyLog[2, 1 + (2*(Sqrt[b] + S
qrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*(-b)^(3/2)*f^2) + (d*P
olyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*(-b)^(3/2)*f^2) - (2*(c + d*x))/(b*f*Sqrt[b*Tanh[e +
f*x]])

Rule 209

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

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[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 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 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3802

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)^m*((b*Tan[e +
f*x])^(n + 1)/(b*f*(n + 1))), x] + (-Dist[d*(m/(b*f*(n + 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1)
, x], x] - Dist[1/b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n
, -1] && GtQ[m, 0]

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*} \int \frac {c+d x}{(b \tanh (e+f x))^{3/2}} \, dx &=-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x) \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {(2 d) \int \frac {1}{\sqrt {b \tanh (e+f x)}} \, dx}{b f}\\ &=-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x) \sqrt {b \tanh (e+f x)} \, dx}{b^2}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2+x^2\right )} \, dx,x,b \tanh (e+f x)\right )}{f^2}\\ &=-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x) \sqrt {b \tanh (e+f x)} \, dx}{b^2}-\frac {(4 d) \text {Subst}\left (\int \frac {1}{-b^2+x^4} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}\\ &=-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x) \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {(2 d) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b f^2}+\frac {(2 d) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b f^2}\\ &=\frac {2 d \tan ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {2 (c+d x)}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x) \sqrt {b \tanh (e+f x)} \, dx}{b^2}\\ \end {align*}

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Mathematica [F]
time = 17.79, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x}{(b \tanh (e+f x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(c + d*x)/(b*Tanh[e + f*x])^(3/2),x]

[Out]

Integrate[(c + d*x)/(b*Tanh[e + f*x])^(3/2), x]

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Maple [F]
time = 1.26, size = 0, normalized size = 0.00 \[\int \frac {d x +c}{\left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(b*tanh(f*x+e))^(3/2),x)

[Out]

int((d*x+c)/(b*tanh(f*x+e))^(3/2),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))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)/(b*tanh(f*x + e))^(3/2), 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))^(3/2),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}{\left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*tanh(f*x+e))**(3/2),x)

[Out]

Integral((c + d*x)/(b*tanh(e + f*x))**(3/2), 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))^(3/2),x, algorithm="giac")

[Out]

integrate((d*x + c)/(b*tanh(f*x + e))^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c+d\,x}{{\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(b*tanh(e + f*x))^(3/2),x)

[Out]

int((c + d*x)/(b*tanh(e + f*x))^(3/2), x)

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