Optimal. Leaf size=60 \[ -\frac {i \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3178, 3177} \[ -\frac {i \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3177
Rule 3178
Rubi steps
\begin {align*} \int \sqrt {a+b \sinh ^2(e+f x)} \, dx &=\frac {\sqrt {a+b \sinh ^2(e+f x)} \int \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \, dx}{\sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}\\ &=-\frac {i E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 69, normalized size = 1.15 \[ -\frac {i a \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{f \sqrt {2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sinh \left (f x + e\right )^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
[Out]
________________________________________________________________________________________
maple [A] time = 0.22, size = 140, normalized size = 2.33 \[ \frac {\sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (a \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-b \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+b \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sinh \left (f x + e\right )^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________