Optimal. Leaf size=72 \[ \frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3775, 21, 3774, 203} \[ \frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}+\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 203
Rule 3774
Rule 3775
Rubi steps
\begin {align*} \int (a+i a \text {csch}(c+d x))^{3/2} \, dx &=\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}+(2 a) \int \frac {\frac {a}{2}+\frac {1}{2} i a \text {csch}(c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx\\ &=\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}+a \int \sqrt {a+i a \text {csch}(c+d x)} \, dx\\ &=\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}-\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {i a \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}\\ &=\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 100, normalized size = 1.39 \[ -\frac {2 i a \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)} \left (\sqrt {\text {csch}(c+d x)+i}-\sqrt [4]{-1} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\text {csch}(c+d x)+i}\right )\right )}{d (\text {csch}(c+d x)-i) \sqrt {\text {csch}(c+d x)+i}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 467, normalized size = 6.49 \[ \frac {2 \, \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {{\left (2 \, a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2} + 2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) - 2 \, \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {{\left (2 \, a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2} - 2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) + 2 \, \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {{\left (2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} + {\left (2 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{2} e^{\left (d x + c\right )} + 4 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - 2 \, \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (-\frac {{\left (2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} - {\left (2 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{2} e^{\left (d x + c\right )} + 4 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) + {\left (8 \, a e^{\left (d x + c\right )} - 8 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}}{4 \, d} \]
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 \[ \int {\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \[ \int \left (a +i a \,\mathrm {csch}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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 \[ \int \left (i a \operatorname {csch}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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