Optimal. Leaf size=41 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3273, 65, 214}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{f \sqrt {a-b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 214
Rule 3273
Rubi steps
\begin {align*} \int \frac {\tanh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 44, normalized size = 1.07 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b+b \cosh ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.96, size = 41, normalized size = 1.00
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sinh \left (f x +e \right )}{\cosh \left (f x +e \right )^{2} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs.
\(2 (35) = 70\).
time = 0.52, size = 433, normalized size = 10.56 \begin {gather*} \left [\frac {\log \left (\frac {b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (4 \, a - 3 \, b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (f x + e\right )^{2} - 4 \, \sqrt {2} \sqrt {a - b} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b}{\cosh \left (f x + e\right )^{4} + 4 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} + 1\right )} \sinh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 1}\right )}{2 \, \sqrt {a - b} f}, -\frac {\sqrt {-a + b} \arctan \left (-\frac {\sqrt {2} \sqrt {-a + b} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left ({\left (a - b\right )} \cosh \left (f x + e\right ) + {\left (a - b\right )} \sinh \left (f x + e\right )\right )}}\right )}{{\left (a - b\right )} f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {tanh}\left (e+f\,x\right )}{\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________