3.1.24 \(\int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx\) [24]

Optimal. Leaf size=1343 \[ \frac {4 d (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {2 d^2 \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{(-b)^{3/2} f^3}+\frac {4 d (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {2 d^2 \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{b^{3/2} f^3}-\frac {4 d^2 \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^3}+\frac {4 d^2 \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^3}-\frac {2 d^2 \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 )}{b^{3/2} f^3}-\frac {2 d^2 \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 )}{b^{3/2} f^3}-\frac {4 d^2 \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^3}+\frac {2 d^2 \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 )}{(-b)^{3/2} f^3}+\frac {2 d^2 \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 )}{(-b)^{3/2} f^3}+\frac {4 d^2 \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^3}-\frac {2 d^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}-\frac {2 d^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}+\frac {d^2 \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 )}{b^{3/2} f^3}+\frac {d^2 \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 )}{b^{3/2} f^3}-\frac {2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}+\frac {d^2 \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 )}{(-b)^{3/2} f^3}+\frac {d^2 \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 )}{(-b)^{3/2} f^3}-\frac {2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\text {Int}\left ((c+d x)^2 \sqrt {b \tanh (e+f x)},x\right )}{b^2} \]

[Out]

4*d*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/(-b)^(3/2)/f^2+2*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^
(1/2))^2/(-b)^(3/2)/f^3+4*d*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/b^(3/2)/f^2+2*d^2*arctanh((b*tanh(f
*x+e))^(1/2)/b^(1/2))^2/b^(3/2)/f^3-4*d^2*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^3+4*d^2*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^3-2*d^2*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^3-2*d^2*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^3-4*d^2*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^3+2*d^2*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^3+2*d^2*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^3+4*d^2
*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^3-2*d^2*pol
ylog(2,1-2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/b^(3/2)/f^3-2*d^2*polylog(2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f
*x+e))^(1/2)))/b^(3/2)/f^3+d^2*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^3+d^2*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^3-2*d^2*polylog(2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)
^(1/2)))/(-b)^(3/2)/f^3+d^2*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^3+d^2*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^3-2*d^2*polylog(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/
(-b)^(3/2)/f^3-2*(d*x+c)^2/b/f/(b*tanh(f*x+e))^(1/2)+Unintegrable((d*x+c)^2*(b*tanh(f*x+e))^(1/2),x)/b^2

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Rubi [A]
time = 1.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[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]]) + Defer[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*}

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Mathematica [A]
time = 19.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{2}}{\left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^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 = 0.00, 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)^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 = 0.00, 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)^2/(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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{2}}{\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)**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 = 0.00, 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)^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)

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

[Out]

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

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