3.24 \(\int \frac{(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=1342 \[ \text{result too large to display} \]
[Out]
(4*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/((-b)^(3/2)*f^2) + (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]
]/Sqrt[-b]]^2)/((-b)^(3/2)*f^3) + (4*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f^2) + (2*d^
2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(b^(3/2)*f^3) - (4*d^2*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^3) + (4*d^2*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^3) - (2*d^2*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]])
)])/(b^(3/2)*f^3) - (2*d^2*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]]))])/(b^(3/2)*f^3) - (4*d^2*ArcTanh[Sqrt[b*Tanh[e
+ f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) + (2*d^2*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]))])/((-b)^(3/2)*f^3) + (2*d^2*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]))])/((-b)^(3/2)*f^3) + (4*d^2*ArcTanh[S
qrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) - (2*d^2*PolyLog[
2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^3) - (2*d^2*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt
[b] + Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^3) + (d^2*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]]))])/(b^(3/2)*f^3) + (d^2*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]]))])/(b^(3/2)*f^3) -
(2*d^2*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) + (d^2*PolyLog[2, 1 - (2*(Sqr
t[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/((-b)^(3/2)*f^3)
+ (d^2*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/S
qrt[-b]))])/((-b)^(3/2)*f^3) - (2*d^2*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3)
- (2*(c + d*x)^2)/(b*f*Sqrt[b*Tanh[e + f*x]]) + Unintegrable[(c + d*x)^2*Sqrt[b*Tanh[e + f*x]], x]/b^2
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Rubi [A] time = 0.155313, 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 = {}
\[ \int \frac{(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(c + d*x)^2/(b*Tanh[e + f*x])^(3/2),x]
[Out]
(-2*(c + d*x)^2)/(b*f*Sqrt[b*Tanh[e + f*x]]) + (4*d*Defer[Int][(c + d*x)/Sqrt[b*Tanh[e + f*x]], x])/(b*f) + De
fer[Int][(c + d*x)^2*Sqrt[b*Tanh[e + f*x]], x]/b^2
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx &=-\frac{2 (c+d x)^2}{b f \sqrt{b \tanh (e+f x)}}+\frac{\int (c+d x)^2 \sqrt{b \tanh (e+f x)} \, dx}{b^2}+\frac{(4 d) \int \frac{c+d x}{\sqrt{b \tanh (e+f x)}} \, dx}{b f}\\ \end{align*}
Mathematica [A] time = 32.9335, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(c + d*x)^2/(b*Tanh[e + f*x])^(3/2),x]
[Out]
Integrate[(c + d*x)^2/(b*Tanh[e + f*x])^(3/2), x]
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Maple [A] time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{2} \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)^2/(b*tanh(f*x+e))^(3/2),x)
[Out]
int((d*x+c)^2/(b*tanh(f*x+e))^(3/2),x)
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{\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)^2/(b*tanh(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
integrate((d*x + c)^2/(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)^2/(b*tanh(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
Exception raised: UnboundLocalError
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x\right )^{2}}{\left (b \tanh{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**2/(b*tanh(f*x+e))**(3/2),x)
[Out]
Integral((c + d*x)**2/(b*tanh(e + f*x))**(3/2), x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{\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)^2/(b*tanh(f*x+e))^(3/2),x, algorithm="giac")
[Out]
integrate((d*x + c)^2/(b*tanh(f*x + e))^(3/2), x)