3.18 \(\int (c+d x) \sqrt{b \tanh (e+f x)} \, dx\)

Optimal. Leaf size=1280 \[ \text{result too large to display} \]

[Out]

-((Sqrt[-b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/
Sqrt[-b]]^2)/(2*f^2) + (Sqrt[b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f + (Sqrt[b]*d*ArcTanh[Sqrt[
b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*f^2) - (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt
[b] - Sqrt[b*Tanh[e + f*x]])])/f^2 + (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b
] + Sqrt[b*Tanh[e + f*x]])])/f^2 - (Sqrt[b]*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) - (Sqrt[b]*d*ArcT
anh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(S
qrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) + (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sq
rt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sq
rt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - (Sqrt[-b]*d*ArcT
anh[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) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b
*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2
*f^2) - (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*f^2) + (Sqrt[b]*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) + (Sqrt[b]*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) + (Sqrt[-b]*d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]
)])/(2*f^2) - (Sqrt[-b]*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) - (Sqrt[-b]*d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sq
rt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) + (Sqrt[-b]*d*PolyLog[2, 1 - 2/(1 + Sqrt[b*T
anh[e + f*x]]/Sqrt[-b])])/(2*f^2)

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Rubi [F]  time = 0.0286067, 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 = {} \[ \int (c+d x) \sqrt{b \tanh (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + d*x)*Sqrt[b*Tanh[e + f*x]],x]

[Out]

Defer[Int][(c + d*x)*Sqrt[b*Tanh[e + f*x]], x]

Rubi steps

\begin{align*} \int (c+d x) \sqrt{b \tanh (e+f x)} \, dx &=\int (c+d x) \sqrt{b \tanh (e+f x)} \, dx\\ \end{align*}

Mathematica [C]  time = 5.11689, size = 556, normalized size = 0.43 \[ \frac{\sqrt{b \tanh (e+f x)} \left (-4 f (c+d x) \left (\log \left (1-\sqrt{\tanh (e+f x)}\right )-\log \left (\sqrt{\tanh (e+f x)}+1\right )+2 \tan ^{-1}\left (\sqrt{\tanh (e+f x)}\right )\right )+d \left (-2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{\tanh (e+f x)}\right )\right )+2 \text{PolyLog}\left (2,\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}-1\right )\right )+2 \text{PolyLog}\left (2,\left (-\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}-1\right )\right )+2 \text{PolyLog}\left (2,\frac{1}{2} \left (\sqrt{\tanh (e+f x)}+1\right )\right )-2 \text{PolyLog}\left (2,\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}+1\right )\right )-2 \text{PolyLog}\left (2,\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}+1\right )\right )+i \text{PolyLog}\left (2,-e^{4 i \tan ^{-1}\left (\sqrt{\tanh (e+f x)}\right )}\right )-\log ^2\left (1-\sqrt{\tanh (e+f x)}\right )+\log ^2\left (\sqrt{\tanh (e+f x)}+1\right )+2 \log \left (1-\sqrt{\tanh (e+f x)}\right ) \log \left (\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}-i\right )\right )+2 \log \left (1-\sqrt{\tanh (e+f x)}\right ) \log \left (\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}+i\right )\right )-2 \log \left (1-\sqrt{\tanh (e+f x)}\right ) \log \left (\frac{1}{2} \left (\sqrt{\tanh (e+f x)}+1\right )\right )-2 \log \left (1-\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}+1\right )\right ) \log \left (\sqrt{\tanh (e+f x)}+1\right )+2 \log \left (\frac{1}{2} \left (1-\sqrt{\tanh (e+f x)}\right )\right ) \log \left (\sqrt{\tanh (e+f x)}+1\right )-2 \log \left (\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\tanh (e+f x)}+i\right )\right ) \log \left (\sqrt{\tanh (e+f x)}+1\right )+4 i \tan ^{-1}\left (\sqrt{\tanh (e+f x)}\right )^2-4 \tan ^{-1}\left (\sqrt{\tanh (e+f x)}\right ) \log \left (1+e^{4 i \tan ^{-1}\left (\sqrt{\tanh (e+f x)}\right )}\right )\right )\right )}{8 f^2 \sqrt{\tanh (e+f x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)*Sqrt[b*Tanh[e + f*x]],x]

[Out]

((-4*f*(c + d*x)*(2*ArcTan[Sqrt[Tanh[e + f*x]]] + Log[1 - Sqrt[Tanh[e + f*x]]] - Log[1 + Sqrt[Tanh[e + f*x]]])
+ d*((4*I)*ArcTan[Sqrt[Tanh[e + f*x]]]^2 - 4*ArcTan[Sqrt[Tanh[e + f*x]]]*Log[1 + E^((4*I)*ArcTan[Sqrt[Tanh[e
+ f*x]]])] - Log[1 - Sqrt[Tanh[e + f*x]]]^2 + 2*Log[1 - Sqrt[Tanh[e + f*x]]]*Log[(1/2 + I/2)*(-I + Sqrt[Tanh[e
+ f*x]])] + 2*Log[1 - Sqrt[Tanh[e + f*x]]]*Log[(1/2 - I/2)*(I + Sqrt[Tanh[e + f*x]])] - 2*Log[1 - Sqrt[Tanh[e
+ f*x]]]*Log[(1 + Sqrt[Tanh[e + f*x]])/2] - 2*Log[1 - (1/2 - I/2)*(1 + Sqrt[Tanh[e + f*x]])]*Log[1 + Sqrt[Tan
h[e + f*x]]] + 2*Log[(1 - Sqrt[Tanh[e + f*x]])/2]*Log[1 + Sqrt[Tanh[e + f*x]]] - 2*Log[(-1/2 - I/2)*(I + Sqrt[
Tanh[e + f*x]])]*Log[1 + Sqrt[Tanh[e + f*x]]] + Log[1 + Sqrt[Tanh[e + f*x]]]^2 + I*PolyLog[2, -E^((4*I)*ArcTan
[Sqrt[Tanh[e + f*x]]])] - 2*PolyLog[2, (1 - Sqrt[Tanh[e + f*x]])/2] + 2*PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[Tan
h[e + f*x]])] + 2*PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[Tanh[e + f*x]])] + 2*PolyLog[2, (1 + Sqrt[Tanh[e + f*x]])
/2] - 2*PolyLog[2, (1/2 - I/2)*(1 + Sqrt[Tanh[e + f*x]])] - 2*PolyLog[2, (1/2 + I/2)*(1 + Sqrt[Tanh[e + f*x]])
]))*Sqrt[b*Tanh[e + f*x]])/(8*f^2*Sqrt[Tanh[e + f*x]])

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) \sqrt{b\tanh \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*(b*tanh(f*x+e))^(1/2),x)

[Out]

int((d*x+c)*(b*tanh(f*x+e))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \sqrt{b \tanh \left (f x + e\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*(b*tanh(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)*sqrt(b*tanh(f*x + e)), 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))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tanh{\left (e + f x \right )}} \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))**(1/2),x)

[Out]

Integral(sqrt(b*tanh(e + f*x))*(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 )} \sqrt{b \tanh \left (f x + e\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*(b*tanh(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((d*x + c)*sqrt(b*tanh(f*x + e)), x)