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

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

[Out]

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

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

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

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Mathematica [C]  time = 4.39, size = 556, normalized size = 0.43 \[ \frac {\sqrt {\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 {b \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[Tanh[e + f*x]])/(8*f^2*Sqrt[b*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 \[ \int \frac {d x + c}{\sqrt {b \tanh \left (f x + e\right )}}\,{d x} \]

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 = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {d x +c}{\sqrt {b \tanh \left (f x +e \right )}}\, dx \]

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 \[ \int \frac {d x + c}{\sqrt {b \tanh \left (f x + e\right )}}\,{d x} \]

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 \[ \int \frac {c+d\,x}{\sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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((c + d*x)/sqrt(b*tanh(e + f*x)), x)

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