Optimal. Leaf size=69 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}+\frac {1}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3273, 53, 65,
214} \begin {gather*} \frac {1}{f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 53
Rule 65
Rule 214
Rule 3273
Rubi steps
\begin {align*} \int \frac {\tanh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=\frac {1}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{2 (a-b) f}\\ &=\frac {1}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\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 )}{(a-b) b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}+\frac {1}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 58, normalized size = 0.84 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \cosh ^2(e+f x)}{a-b}\right )}{(-a+b) f \sqrt {a-b+b \cosh ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.02, size = 93, normalized size = 1.35
method | result | size |
default | \(\frac {\mathit {`\,int/indef0`\,}\left (-\frac {\sinh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{-b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+\left (-2 a b -b^{2}\right ) \left (\sinh ^{4}\left (f x +e \right )\right )+\left (-a^{2}-2 a b \right ) \left (\sinh ^{2}\left (f x +e \right )\right )-a^{2}}, \sinh \left (f x +e \right )\right )}{f}\) | \(93\) |
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. 587 vs.
\(2 (61) = 122\).
time = 0.52, size = 1370, normalized size = 19.86 \begin {gather*} \left [-\frac {{\left (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 (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b\right )} \sqrt {a - b} \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 ) - 4 \, \sqrt {2} {\left ({\left (a - b\right )} \cosh \left (f x + e\right ) + {\left (a - b\right )} \sinh \left (f x + e\right )\right )} \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^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right )^{4} + 4 \, {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{3} - 5 \, a^{2} b + 4 \, a b^{2} - b^{3}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} - 5 \, a^{2} b + 4 \, a b^{2} - b^{3}\right )} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f + 4 \, {\left ({\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{3} - 5 \, a^{2} b + 4 \, a b^{2} - b^{3}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )\right )}}, -\frac {{\left (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 (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b\right )} \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 ) - 2 \, \sqrt {2} {\left ({\left (a - b\right )} \cosh \left (f x + e\right ) + {\left (a - b\right )} \sinh \left (f x + e\right )\right )} \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 (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right )^{4} + 4 \, {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{3} - 5 \, a^{2} b + 4 \, a b^{2} - b^{3}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right )^{2} + {\left (2 \, a^{3} - 5 \, a^{2} b + 4 \, a b^{2} - b^{3}\right )} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f + 4 \, {\left ({\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \cosh \left (f x + e\right )^{3} + {\left (2 \, a^{3} - 5 \, a^{2} b + 4 \, a b^{2} - b^{3}\right )} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}\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 )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {tanh}\left (e+f\,x\right )}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________