Optimal. Leaf size=122 \[ \frac {3 i \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {3 i \cosh (c+d x)}{16 a d (a+i a \sinh (c+d x))^{3/2}}+\frac {i \cosh (c+d x)}{4 d (a+i a \sinh (c+d x))^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2650, 2649, 206} \[ \frac {3 i \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {3 i \cosh (c+d x)}{16 a d (a+i a \sinh (c+d x))^{3/2}}+\frac {i \cosh (c+d x)}{4 d (a+i a \sinh (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \sinh (c+d x))^{5/2}} \, dx &=\frac {i \cosh (c+d x)}{4 d (a+i a \sinh (c+d x))^{5/2}}+\frac {3 \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx}{8 a}\\ &=\frac {i \cosh (c+d x)}{4 d (a+i a \sinh (c+d x))^{5/2}}+\frac {3 i \cosh (c+d x)}{16 a d (a+i a \sinh (c+d x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx}{32 a^2}\\ &=\frac {i \cosh (c+d x)}{4 d (a+i a \sinh (c+d x))^{5/2}}+\frac {3 i \cosh (c+d x)}{16 a d (a+i a \sinh (c+d x))^{3/2}}+\frac {(3 i) \operatorname {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 )}{16 a^2 d}\\ &=\frac {3 i \tanh ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {i \cosh (c+d x)}{4 d (a+i a \sinh (c+d x))^{5/2}}+\frac {3 i \cosh (c+d x)}{16 a d (a+i a \sinh (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 210, normalized size = 1.72 \[ \frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (4 \sinh \left (\frac {1}{2} (c+d x)\right )+4 i \cosh \left (\frac {1}{2} (c+d x)\right )+6 \sinh \left (\frac {1}{2} (c+d x)\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+3 \left (\sinh \left (\frac {1}{2} (c+d x)\right )-i \cosh \left (\frac {1}{2} (c+d x)\right )\right )^3+(3-3 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^4\right )}{16 d (a+i a \sinh (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 348, normalized size = 2.85 \[ \frac {\sqrt {\frac {1}{2}} {\left (12 i \, a^{3} d e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a^{3} d e^{\left (3 \, d x + 3 \, c\right )} - 72 i \, a^{3} d e^{\left (2 \, d x + 2 \, c\right )} - 48 \, a^{3} d e^{\left (d x + c\right )} + 12 i \, a^{3} d\right )} \sqrt {\frac {1}{a^{5} d^{2}}} \log \left (\sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) + \sqrt {\frac {1}{2}} {\left (-12 i \, a^{3} d e^{\left (4 \, d x + 4 \, c\right )} - 48 \, a^{3} d e^{\left (3 \, d x + 3 \, c\right )} + 72 i \, a^{3} d e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a^{3} d e^{\left (d x + c\right )} - 12 i \, a^{3} d\right )} \sqrt {\frac {1}{a^{5} d^{2}}} \log \left (-\sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {1}{a^{5} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) + 8 \, \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (-3 i \, e^{\left (4 \, d x + 4 \, c\right )} - 11 \, e^{\left (3 \, d x + 3 \, c\right )} - 11 i \, e^{\left (2 \, d x + 2 \, c\right )} - 3 \, e^{\left (d x + c\right )}\right )}}{8 \, {\left (8 \, a^{3} d e^{\left (4 \, d x + 4 \, c\right )} - 32 i \, a^{3} d e^{\left (3 \, d x + 3 \, c\right )} - 48 \, a^{3} d e^{\left (2 \, d x + 2 \, c\right )} + 32 i \, a^{3} d e^{\left (d x + c\right )} + 8 \, a^{3} d\right )}} \]
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 \[ \int \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (i a \sinh {\left (c + d x \right )} + a\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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