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

Optimal. Leaf size=1280 \[ -\frac {\sqrt {-b} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}-\frac {\sqrt {-b} (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {-b} d \log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f^2}-\frac {\sqrt {-b} d \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 ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 f^2}-\frac {\sqrt {-b} d \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 ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 f^2}-\frac {\sqrt {-b} d \log \left (\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right ) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f^2}+\frac {\sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}+\frac {\sqrt {b} (c+d x) \tanh ^{-1}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} d \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 )}{f^2}+\frac {\sqrt {b} d \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 )}{f^2}-\frac {\sqrt {b} d \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 )}{2 f^2}-\frac {\sqrt {b} d \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 )}{2 f^2}-\frac {\sqrt {b} d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {\sqrt {b} d \text {Li}_2\left (1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {\sqrt {b} d \text {Li}_2\left (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 )}{4 f^2}+\frac {\sqrt {b} d \text {Li}_2\left (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 )}{4 f^2}+\frac {\sqrt {-b} d \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {\sqrt {-b} d \text {Li}_2\left (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 )}{4 f^2}-\frac {\sqrt {-b} d \text {Li}_2\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 )}+1\right )}{4 f^2}+\frac {\sqrt {-b} d \text {Li}_2\left (1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{2 f^2} \]

[Out]

-(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*(-b)^(1/2)/f-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2)
)^2*(-b)^(1/2)/f^2+d*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)
^(1/2)/f^2-1/2*d*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)^(1/2)/f^2-1/2*d*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)^(1/2)/f^2-
d*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)^(1/2)/f^2+1/2*d*po
lylog(2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))*(-b)^(1/2)/f^2-1/4*d*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)^(1/2)/f^2-1/4*d*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)^(1/2)/f^2+1/2*d*polylog
(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))*(-b)^(1/2)/f^2+(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*b^(
1/2)/f+1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))^2*b^(1/2)/f^2-d*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)/f^2+d*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)/f^2-1/2*d*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^(1/2)/f^2-1/2*d*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*t
anh(f*x+e))^(1/2)))*b^(1/2)/f^2-1/2*d*polylog(2,1-2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))*b^(1/2)/f^2-1/2*d
*polylog(2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))*b^(1/2)/f^2+1/4*d*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^(1/2)/f^2+1/4*d*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^(1/2)/f^2

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Rubi [F]  time = 0.03, 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 = {} \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx \]

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*}

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Mathematica [C]  time = 5.32, 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 {Li}_2\left (\frac {1}{2} \left (1-\sqrt {\tanh (e+f x)}\right )\right )+2 \text {Li}_2\left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tanh (e+f x)}-1\right )\right )+2 \text {Li}_2\left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tanh (e+f x)}-1\right )\right )+2 \text {Li}_2\left (\frac {1}{2} \left (\sqrt {\tanh (e+f x)}+1\right )\right )-2 \text {Li}_2\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {\tanh (e+f x)}+1\right )\right )-2 \text {Li}_2\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {\tanh (e+f x)}+1\right )\right )+i \text {Li}_2\left (-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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )}\,{d x} \]

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)

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) \sqrt {b \tanh \left (f x +e \right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}\,\left (c+d\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tanh {\left (e + f x \right )}} \left (c + d x\right )\, dx \]

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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