3.1.17 \(\int (c+d x) (b \tanh (e+f x))^{3/2} \, dx\) [17]

Optimal. Leaf size=1363 \[ -\frac {2 b^{3/2} d \text {ArcTan}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {(-b)^{3/2} (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {(-b)^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {2 b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {b^{3/2} 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 )}{f^2}+\frac {b^{3/2} 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 )}{f^2}-\frac {b^{3/2} 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 f^2}-\frac {b^{3/2} 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 f^2}+\frac {(-b)^{3/2} 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 )}{f^2}-\frac {(-b)^{3/2} 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 f^2}-\frac {(-b)^{3/2} 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 f^2}-\frac {(-b)^{3/2} 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 )}{f^2}-\frac {b^{3/2} d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {b^{3/2} d \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {b^{3/2} 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 f^2}+\frac {b^{3/2} 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 f^2}+\frac {(-b)^{3/2} d \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {(-b)^{3/2} 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 f^2}-\frac {(-b)^{3/2} 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 f^2}+\frac {(-b)^{3/2} d \text {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f} \]

[Out]

-2*b^(3/2)*d*arctan((b*tanh(f*x+e))^(1/2)/b^(1/2))/f^2-(-b)^(3/2)*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(
1/2))/f-1/2*(-b)^(3/2)*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2/f^2+2*b^(3/2)*d*arctanh((b*tanh(f*x+e))^(
1/2)/b^(1/2))/f^2+b^(3/2)*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/f+1/2*b^(3/2)*d*arctanh((b*tanh(f*x+e
))^(1/2)/b^(1/2))^2/f^2-b^(3/2)*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^(3/2)*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^
2-1/2*b^(3/2)*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^2-1/2*b^(3/2)*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)^(3/2)
*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^2-1/2*(-b)^(3/2)*d*a
rctanh((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^2-1/2*(-b)^(3/2)*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(-2*(b^(1/2)+(b*ta
nh(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)^(3/2)*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^2-1/2*b^(3/2)*d*polylog(2,1-2*b^(1/2)/(
b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^2-1/2*b^(3/2)*d*polylog(2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2+1
/4*b^(3/2)*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+1/4*b^(3/2)*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+1/2*(-b)^(3/2)*d*polylog(2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2-
1/4*(-b)^(3/2)*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^2-1/4*(-b)^(3/2)*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+1/2*(-b)^(3/2)*d*polylog(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2-2*b*(
d*x+c)*(b*tanh(f*x+e))^(1/2)/f

________________________________________________________________________________________

Rubi [A]
time = 1.16, antiderivative size = 1363, 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 = {3801, 3557, 335, 304, 209, 212, 3819, 213, 281, 6857, 6139, 6057, 2449, 2352, 2497, 6131, 6055} \begin {gather*} -\frac {(-b)^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}-\frac {(-b)^{3/2} (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {(-b)^{3/2} 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 )}{f^2}-\frac {(-b)^{3/2} 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 f^2}-\frac {(-b)^{3/2} 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 f^2}-\frac {(-b)^{3/2} 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 )}{f^2}+\frac {b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {2 b^{3/2} d \text {ArcTan}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {2 b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {b^{3/2} 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 )}{f^2}+\frac {b^{3/2} 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 )}{f^2}-\frac {b^{3/2} 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 f^2}-\frac {b^{3/2} 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 f^2}-\frac {b^{3/2} d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {b^{3/2} d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {b^{3/2} 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 f^2}+\frac {b^{3/2} 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 f^2}+\frac {(-b)^{3/2} d \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {(-b)^{3/2} 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 f^2}-\frac {(-b)^{3/2} 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 f^2}+\frac {(-b)^{3/2} d \text {Li}_2\left (1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{2 f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^(3/2)*d*ArcTan[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f^2 - ((-b)^(3/2)*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]
/Sqrt[-b]])/f - ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*f^2) + (2*b^(3/2)*d*ArcTanh[Sqrt[b
*Tanh[e + f*x]]/Sqrt[b]])/f^2 + (b^(3/2)*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f + (b^(3/2)*d*ArcT
anh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*f^2) - (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[
b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^2 + (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]
)/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^2 - (b^(3/2)*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*f^2) - (b^(3/
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] + Sq
rt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) + ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Lo
g[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - ((-b)^(3/2)*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*f^2) - ((-
b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(-2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqr
t[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*
Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (b^(3/2)*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tan
h[e + f*x]])])/(2*f^2) - (b^(3/2)*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*f^2) + (
b^(3/2)*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*f^2) + (b^(3/2)*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*f^2) + ((-b)^(3/2)*d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh
[e + f*x]]/Sqrt[-b])])/(2*f^2) - ((-b)^(3/2)*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*f^2) - ((-b)^(3/2)*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*f^2) + ((-b)^(3/2)*d*PolyL
og[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2) - (2*b*(c + d*x)*Sqrt[b*Tanh[e + f*x]])/f

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 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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

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 3801

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

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

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Mathematica [F]
time = 20.11, size = 0, normalized size = 0.00 \begin {gather*} \int (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 = 1.09, size = 0, normalized size = 0.00 \[\int \left (d x +c \right ) \left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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