Optimal. Leaf size=87 \[ \frac {i \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2729, 2728,
212} \begin {gather*} \frac {i \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx &=\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx}{4 a}\\ &=\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}+\frac {i \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (c+d x)}{\sqrt {a+i a \sinh (c+d x)}}\right )}{2 a d}\\ &=\frac {i \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 156, normalized size = 1.79 \begin {gather*} \frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \left ((1-i) \sqrt [4]{-1} \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 a d (-i+\sinh (c+d x)) \sqrt {a+i a \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.66, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 235 vs. \(2 (64) = 128\).
time = 0.37, size = 235, normalized size = 2.70 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} d e^{\left (d x + c\right )} - i \, a^{2} d\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (\sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) + \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{2} d e^{\left (d x + c\right )} + i \, a^{2} d\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (-\sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )}\right )}}{2 \, {\left (a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a^{2} d e^{\left (d x + c\right )} - a^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i a \sinh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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