3.21 \(\int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx\)
Optimal. Leaf size=1340 \[ \text{result too large to display} \]
[Out]
(4*(-b)^(3/2)*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f^2 + (2*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh
[e + f*x]]/Sqrt[-b]]^2)/f^3 + (4*b^(3/2)*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f^2 + (2*b^(3/2)*
d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/f^3 - (4*b^(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]])])/f^3 + (4*b^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Lo
g[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^3 - (2*b^(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]]))])/
f^3 - (2*b^(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]]))])/f^3 - (4*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*
x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 + (2*(-b)^(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]))])/f^3 + (2*(-b)^(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]))])/f^3 + (4*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[
e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 - (2*b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b
])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^3 - (2*b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[
e + f*x]])])/f^3 + (b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqr
t[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/f^3 + (b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tan
h[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/f^3 - (2*(-b)^(3/2)*d^2*PolyLog[2, 1
- 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 + ((-b)^(3/2)*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]]/Sqrt[-b]))])/f^3 + ((-b)^(3/2)*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]]/Sqrt[-b]))])/f^3 - (2*(-b)
^(3/2)*d^2*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 - (2*b*(c + d*x)^2*Sqrt[b*Tanh[e + f*x]
])/f + b^2*Unintegrable[(c + d*x)^2/Sqrt[b*Tanh[e + f*x]], x]
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Rubi [A] time = 0.148901, 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 = {}
\[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2),x]
[Out]
(-2*b*(c + d*x)^2*Sqrt[b*Tanh[e + f*x]])/f + b^2*Defer[Int][(c + d*x)^2/Sqrt[b*Tanh[e + f*x]], x] + (4*b*d*Def
er[Int][(c + d*x)*Sqrt[b*Tanh[e + f*x]], x])/f
Rubi steps
\begin{align*} \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx &=-\frac{2 b (c+d x)^2 \sqrt{b \tanh (e+f x)}}{f}+b^2 \int \frac{(c+d x)^2}{\sqrt{b \tanh (e+f x)}} \, dx+\frac{(4 b d) \int (c+d x) \sqrt{b \tanh (e+f x)} \, dx}{f}\\ \end{align*}
Mathematica [A] time = 27.2827, size = 0, normalized size = 0. \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx \]
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 = 0.076, 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{\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 \left (b \tanh{\left (e + f x \right )}\right )^{\frac{3}{2}} \left (c + d x\right )^{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((b*tanh(e + f*x))**(3/2)*(c + d*x)**2, x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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)