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

Optimal. Leaf size=1363 \[ \text{result too large to display} \]

[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

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Rubi [F]  time = 0.105962, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx \]

Verification is Not applicable to the result.

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

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [F]  time = 27.2572, size = 0, normalized size = 0. \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx \]

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 = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) \left ( b\tanh \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \left (b \tanh \left (f x + e\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \tanh{\left (e + f x \right )}\right )^{\frac{3}{2}} \left (c + d x\right )\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \left (b \tanh \left (f x + e\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}

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)