Optimal. Leaf size=164 \[ -\frac {2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \sqrt {a+b \sinh (x)}}+\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \sqrt {a+b \sinh (x)}}+\frac {2 i (a B+3 A b) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \sqrt {a+b \sinh (x)} (A+B \sinh (x)) \, dx &=\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{3} \int \frac {\frac {1}{2} (3 a A-b B)+\frac {1}{2} (3 A b+a B) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx\\ &=\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}-\frac {\left (\left (a^2+b^2\right ) B\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{3 b}+\frac {(3 A b+a B) \int \sqrt {a+b \sinh (x)} \, dx}{3 b}\\ &=\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {\left ((3 A b+a B) \sqrt {a+b \sinh (x)}\right ) \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (\left (a^2+b^2\right ) B \sqrt {\frac {a+b \sinh (x)}{a-i b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{3 b \sqrt {a+b \sinh (x)}}\\ &=\frac {2}{3} B \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2 i (3 A b+a B) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) B F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \sqrt {a+b \sinh (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.41, size = 151, normalized size = 0.92 \[ \frac {-2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )+2 (b+i a) (a B+3 A b) \sqrt {\frac {a+b \sinh (x)}{a-i b}} E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )+2 b B \cosh (x) (a+b \sinh (x))}{3 b \sqrt {a+b \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sinh \relax (x) + A\right )} \sqrt {b \sinh \relax (x) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sinh \relax (x) + A\right )} \sqrt {b \sinh \relax (x) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.14, size = 897, normalized size = 5.47 \[ \frac {\frac {2 i B \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b}{3}+\frac {2 i B \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{3}}{3}+2 A \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b +2 A \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{3}-2 A \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticE \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b -2 A \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticE \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{3}-\frac {2 B \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticE \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{3}}{3}-\frac {2 B \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticE \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \,b^{2}}{3}+\frac {2 B \,b^{3} \left (\sinh ^{3}\relax (x )\right )}{3}+\frac {2 B a \,b^{2} \left (\sinh ^{2}\relax (x )\right )}{3}+\frac {2 B \,b^{3} \sinh \relax (x )}{3}+\frac {2 B a \,b^{2}}{3}}{b^{2} \cosh \relax (x ) \sqrt {a +b \sinh \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sinh \relax (x) + A\right )} \sqrt {b \sinh \relax (x) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\mathrm {sinh}\relax (x)\right )\,\sqrt {a+b\,\mathrm {sinh}\relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \sinh {\relax (x )}\right ) \sqrt {a + b \sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________