Optimal. Leaf size=79 \[ \frac {(i A+3 B) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2829, 2728,
212} \begin {gather*} \frac {(3 B+i A) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(-B+i A) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2728
Rule 2829
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx &=\frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}}+\frac {(A-3 i B) \int \frac {1}{\sqrt {a+i a \sinh (x)}} \, dx}{4 a}\\ &=\frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}}+\frac {(i A+3 B) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (x)}{\sqrt {a+i a \sinh (x)}}\right )}{2 a}\\ &=\frac {(i A+3 B) \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 105, normalized size = 1.33 \begin {gather*} \frac {\left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right ) \left (i (A+i B) \cosh \left (\frac {x}{2}\right )+(A+i B) \sinh \left (\frac {x}{2}\right )+(1+i) \sqrt [4]{-1} (A-3 i B) \text {ArcTan}\left (\frac {i+\tanh \left (\frac {x}{4}\right )}{\sqrt {2}}\right ) (-i+\sinh (x))\right )}{2 (a+i a \sinh (x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.85, size = 0, normalized size = 0.00 \[\int \frac {A +B \sinh \left (x \right )}{\left (a +i a \sinh \left (x \right )\right )^{\frac {3}{2}}}\, dx\]
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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 264 vs. \(2 (54) = 108\).
time = 0.46, size = 264, normalized size = 3.34 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\left (a^{2} e^{\left (2 \, x\right )} - 2 i \, a^{2} e^{x} - a^{2}\right )} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} \log \left (\frac {\sqrt {\frac {1}{2}} a^{2} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A + 3 \, B\right )}}{i \, A + 3 \, B}\right ) - \sqrt {\frac {1}{2}} {\left (a^{2} e^{\left (2 \, x\right )} - 2 i \, a^{2} e^{x} - a^{2}\right )} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} \log \left (-\frac {\sqrt {\frac {1}{2}} a^{2} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} - \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A + 3 \, B\right )}}{i \, A + 3 \, B}\right ) - 2 \, {\left ({\left (i \, A - B\right )} e^{\left (2 \, x\right )} - {\left (A + i \, B\right )} e^{x}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}}}{2 \, {\left (a^{2} e^{\left (2 \, x\right )} - 2 i \, a^{2} e^{x} - a^{2}\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 {A + B \sinh {\left (x \right )}}{\left (i a \left (\sinh {\left (x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {A+B\,\mathrm {sinh}\left (x\right )}{{\left (a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________