Optimal. Leaf size=111 \[ -\frac {8 x \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {16 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \]
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Rubi [A]
time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3400, 3377,
2718} \begin {gather*} \frac {16 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f^3}-\frac {8 x \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3400
Rubi steps
\begin {align*} \int x^2 \sqrt {a+i a \sinh (e+f x)} \, dx &=\left (\text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int x^2 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx\\ &=\frac {2 x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}-\frac {\left (4 \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int x \cosh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f}\\ &=-\frac {8 x \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}-\frac {\left (8 i \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f^2}\\ &=-\frac {8 x \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {16 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 105, normalized size = 0.95 \begin {gather*} \frac {2 \left (i \left (8+4 i f x+f^2 x^2\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (8-4 i f x+f^2 x^2\right ) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+i a \sinh (e+f x)}}{f^3 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 128, normalized size = 1.15
method | result | size |
risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 f x +2 e}+2 \,{\mathrm e}^{f x +e}-i\right ) {\mathrm e}^{-f x -e}}\, \left (i x^{2} f^{2}+f^{2} x^{2} {\mathrm e}^{f x +e}+4 i x f -4 f x \,{\mathrm e}^{f x +e}+8 i+8 \,{\mathrm e}^{f x +e}\right ) \left ({\mathrm e}^{f x +e}-i\right )}{\left (i {\mathrm e}^{2 f x +2 e}+2 \,{\mathrm e}^{f x +e}-i\right ) f^{3}}\) | \(128\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 x^{2} \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}\, 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 [B]
time = 0.29, size = 92, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2}\,\sqrt {a\,{\mathrm {e}}^{-e-f\,x}\,{\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}\,\left (8\,{\mathrm {e}}^{e+f\,x}+f\,x\,4{}\mathrm {i}+f^2\,x^2\,1{}\mathrm {i}+f^2\,x^2\,{\mathrm {e}}^{e+f\,x}-4\,f\,x\,{\mathrm {e}}^{e+f\,x}+8{}\mathrm {i}\right )}{f^3\,\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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