Optimal. Leaf size=364 \[ \frac {3 i a^{3/2} \sqrt {e} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {3 i a^{3/2} \sqrt {e} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {3 i a^{3/2} \sqrt {e} \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} d}+\frac {3 i a^{3/2} \sqrt {e} \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} d}+\frac {i a \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.32, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3498, 3495, 297, 1162, 617, 204, 1165, 628} \[ \frac {3 i a^{3/2} \sqrt {e} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {3 i a^{3/2} \sqrt {e} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {3 i a^{3/2} \sqrt {e} \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} d}+\frac {3 i a^{3/2} \sqrt {e} \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{2 \sqrt {2} d}+\frac {i a \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3495
Rule 3498
Rubi steps
\begin {align*} \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^{3/2} \, dx &=\frac {i a \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}+\frac {1}{2} (3 a) \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {i a \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\left (6 i a^2 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d}\\ &=\frac {i a \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}+\frac {\left (3 i a^2 e\right ) \operatorname {Subst}\left (\int \frac {a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d}-\frac {\left (3 i a^2 e\right ) \operatorname {Subst}\left (\int \frac {a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d}\\ &=\frac {i a \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 d}-\frac {\left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 d}-\frac {\left (3 i a^{3/2} \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}+2 x}{-\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 \sqrt {2} d}-\frac {\left (3 i a^{3/2} \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}-2 x}{-\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{2 \sqrt {2} d}\\ &=-\frac {3 i a^{3/2} \sqrt {e} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} d}+\frac {3 i a^{3/2} \sqrt {e} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} d}+\frac {i a \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}-\frac {\left (3 i a^{3/2} \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}+\frac {\left (3 i a^{3/2} \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}\\ &=\frac {3 i a^{3/2} \sqrt {e} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {3 i a^{3/2} \sqrt {e} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {3 i a^{3/2} \sqrt {e} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} d}+\frac {3 i a^{3/2} \sqrt {e} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{2 \sqrt {2} d}+\frac {i a \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}\\ \end {align*}
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Mathematica [A] time = 3.35, size = 338, normalized size = 0.93 \[ \frac {a e \sqrt {a+i a \tan (c+d x)} \left (-3 \sqrt {-\sin (c)-i \cos (c)-1} \sqrt {-\sin (c)+i \cos (c)+1} \sqrt {\tan \left (\frac {d x}{2}\right )+i} \tanh ^{-1}\left (\frac {\sqrt {\sin (c)-i \cos (c)+1} \sqrt {-\tan \left (\frac {d x}{2}\right )+i}}{\sqrt {-\sin (c)-i \cos (c)-1} \sqrt {\tan \left (\frac {d x}{2}\right )+i}}\right )+3 \sqrt {\sin (c)-i \cos (c)+1} \sqrt {\sin (c)+i \cos (c)-1} \sqrt {\tan \left (\frac {d x}{2}\right )+i} \tanh ^{-1}\left (\frac {\sqrt {-\sin (c)+i \cos (c)+1} \sqrt {-\tan \left (\frac {d x}{2}\right )+i}}{\sqrt {\sin (c)+i \cos (c)-1} \sqrt {\tan \left (\frac {d x}{2}\right )+i}}\right )+i \sqrt {i \sin (2 c)+\cos (2 c)+1} \sqrt {-\tan \left (\frac {d x}{2}\right )+i} \sec (c+d x)\right )}{d \sqrt {i \sin (2 c)+\cos (2 c)+1} \sqrt {-\tan \left (\frac {d x}{2}\right )+i} \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 420, normalized size = 1.15 \[ \frac {4 i \, a \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {3}{2} i \, d x + \frac {3}{2} i \, c\right )} + \sqrt {\frac {9 i \, a^{3} e}{d^{2}}} d \log \left (\frac {2 \, {\left (3 \, {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + \sqrt {\frac {9 i \, a^{3} e}{d^{2}}} d\right )}}{3 \, a}\right ) - \sqrt {\frac {9 i \, a^{3} e}{d^{2}}} d \log \left (\frac {2 \, {\left (3 \, {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - \sqrt {\frac {9 i \, a^{3} e}{d^{2}}} d\right )}}{3 \, a}\right ) - \sqrt {-\frac {9 i \, a^{3} e}{d^{2}}} d \log \left (\frac {2 \, {\left (3 \, {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + \sqrt {-\frac {9 i \, a^{3} e}{d^{2}}} d\right )}}{3 \, a}\right ) + \sqrt {-\frac {9 i \, a^{3} e}{d^{2}}} d \log \left (\frac {2 \, {\left (3 \, {\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - \sqrt {-\frac {9 i \, a^{3} e}{d^{2}}} d\right )}}{3 \, a}\right )}{2 \, d} \]
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 \[ \int \sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.30, size = 304, normalized size = 0.84 \[ -\frac {\sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right ) \left (3 i \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )-3 i \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right )}{2}\right )-2 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-3 \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )-3 \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right )}{2}\right )-2 \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-2 \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\right ) a}{2 d \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.13, size = 1881, normalized size = 5.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {e \sec {\left (c + d x \right )}} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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