Optimal. Leaf size=176 \[ -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i B \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 176, 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 = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \cosh (x) (A b-a B)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i (A b-a B) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i B \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx &=-\frac {2 (A b-a B) \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}-\frac {2 \int \frac {\frac {1}{2} (-a A-b B)-\frac {1}{2} (A b-a B) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx}{a^2+b^2}\\ &=-\frac {2 (A b-a B) \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {B \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{b}+\frac {(A b-a B) \int \sqrt {a+b \sinh (x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {2 (A b-a B) \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {\left ((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}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {\left (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}{b \sqrt {a+b \sinh (x)}}\\ &=-\frac {2 (A b-a B) \cosh (x)}{\left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i (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)}}{b \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i 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}}}{b \sqrt {a+b \sinh (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.65, size = 159, normalized size = 0.90 \[ \frac {2 i B \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )+2 b \cosh (x) (a B-A b)+\frac {2 i (A b-a B) (a+b \sinh (x)) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}}{b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \sinh \relax (x) + A\right )} \sqrt {b \sinh \relax (x) + a}}{b^{2} \sinh \relax (x)^{2} + 2 \, a b \sinh \relax (x) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \sinh \relax (x) + A}{{\left (b \sinh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.24, size = 517, normalized size = 2.94 \[ \frac {\sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}\, \left (\frac {2 B \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{b \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {\left (A b -a B \right ) \left (-\frac {2 b \left (\cosh ^{2}\relax (x )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {2 a \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {2 b \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \left (\left (-\frac {a}{b}-i\right ) \EllipticE \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}\right )}{b}\right )}{\cosh \relax (x ) \sqrt {a +b \sinh \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \sinh \relax (x) + A}{{\left (b \sinh \relax (x) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,\mathrm {sinh}\relax (x)}{{\left (a+b\,\mathrm {sinh}\relax (x)\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________