Integrand size = 21, antiderivative size = 181 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=-\frac {384 x \sqrt {a+i a \sinh (e+f x)}}{f^4}-\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {768 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^5}+\frac {96 x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^3}+\frac {2 x^4 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \]
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Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3400, 3377, 2718} \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {768 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f^5}-\frac {384 x \sqrt {a+i a \sinh (e+f x)}}{f^4}+\frac {96 x^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f^3}-\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x^4 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f} \]
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Rule 2718
Rule 3377
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \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^4 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx \\ & = \frac {2 x^4 \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 \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^3 \cosh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f} \\ & = -\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x^4 \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 (48 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 x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f^2} \\ & = -\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {96 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^3}+\frac {2 x^4 \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 (192 \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^3} \\ & = -\frac {384 x \sqrt {a+i a \sinh (e+f x)}}{f^4}-\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {96 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^3}+\frac {2 x^4 \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 (384 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^4} \\ & = -\frac {384 x \sqrt {a+i a \sinh (e+f x)}}{f^4}-\frac {16 x^3 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {768 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f^5}+\frac {96 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^3}+\frac {2 x^4 \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*}
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {2 \left (i \left (384+192 i f x+48 f^2 x^2+8 i f^3 x^3+f^4 x^4\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (384-192 i f x+48 f^2 x^2-8 i f^3 x^3+f^4 x^4\right ) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+i a \sinh (e+f x)}}{f^5 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 0.90 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) {\mathrm e}^{-f x -e}}\, \left (i x^{4} f^{4}+f^{4} x^{4} {\mathrm e}^{f x +e}+8 i x^{3} f^{3}-8 \,{\mathrm e}^{f x +e} f^{3} x^{3}+48 i x^{2} f^{2}+48 f^{2} x^{2} {\mathrm e}^{f x +e}+192 i x f -192 f x \,{\mathrm e}^{f x +e}+384 i+384 \,{\mathrm e}^{f x +e}\right ) \left ({\mathrm e}^{f x +e}-i\right )}{\left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) f^{5}}\) | \(174\) |
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Exception generated. \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\int x^{4} \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}\, dx \]
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\[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{4} \,d x } \]
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\[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x^{4} \,d x } \]
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Time = 0.80 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int x^4 \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{e+f\,x}+1{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-e-f\,x}\,{\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}\,\left (384\,{\mathrm {e}}^{e+f\,x}+f\,x\,192{}\mathrm {i}+f^2\,x^2\,48{}\mathrm {i}+f^3\,x^3\,8{}\mathrm {i}+f^4\,x^4\,1{}\mathrm {i}+48\,f^2\,x^2\,{\mathrm {e}}^{e+f\,x}-8\,f^3\,x^3\,{\mathrm {e}}^{e+f\,x}+f^4\,x^4\,{\mathrm {e}}^{e+f\,x}-192\,f\,x\,{\mathrm {e}}^{e+f\,x}+384{}\mathrm {i}\right )}{f^5\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \]
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