Optimal. Leaf size=524 \[ -\frac {i a^{3/2} \sec (c+d x) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} \sec (c+d x) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} \sec (c+d x) \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {e}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i a^{3/2} \sec (c+d x) \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {e}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a}{d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.59, antiderivative size = 620, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3515, 3498, 3499, 3495, 297, 1162, 617, 204, 1165, 628} \[ -\frac {i a^{3/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}+\frac {i a^{3/2} e^{3/2} \sec (c+d x) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}+\frac {i a^{3/2} e^{3/2} \sec (c+d x) \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}-\frac {i a^{3/2} e^{3/2} \sec (c+d x) \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{2 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}+\frac {i a}{d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3495
Rule 3498
Rule 3499
Rule 3515
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {\int (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {a \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{2 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {(a e \sec (c+d x)) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)} \, dx}{2 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (2 i a^2 e^3 \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {\left (i a^2 e^2 \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i a^2 e^2 \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i a^2 e \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i a^2 e \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i a^{3/2} e^{3/2} \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i a^{3/2} e^{3/2} \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} e^{3/2} \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 ) \sec (c+d x)}{2 \sqrt {2} d (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i a^{3/2} e^{3/2} \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 ) \sec (c+d x)}{2 \sqrt {2} d (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i a^{3/2} e^{3/2} \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {\left (i a^{3/2} e^{3/2} \sec (c+d x)\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 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{d (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {i a^{3/2} e^{3/2} \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 ) \sec (c+d x)}{\sqrt {2} d (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} e^{3/2} \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 ) \sec (c+d x)}{\sqrt {2} d (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i a^{3/2} e^{3/2} \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 ) \sec (c+d x)}{2 \sqrt {2} d (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i a^{3/2} e^{3/2} \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 ) \sec (c+d x)}{2 \sqrt {2} d (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 3.85, size = 274, normalized size = 0.52 \[ \frac {i e^{-\frac {1}{2} i (c+d x)} \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)} (\cos (c+d x)+i \sin (c+d x)) \left (-2 i \sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )+2 \sqrt {2} \cos \left (\frac {1}{2} (c+d x)\right )+\log \left (-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}+1\right ) \cos (c+d x)-\log \left (\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}+1\right ) \cos (c+d x)+2 \cos (c+d x) \tan ^{-1}\left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )-2 \cos (c+d x) \tan ^{-1}\left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )\right )}{\sqrt {2} d \left (1+e^{2 i (c+d x)}\right ) (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 470, normalized size = 0.90 \[ \frac {4 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {\frac {i \, a}{d^{2} e^{3}}} \log \left (d e^{2} \sqrt {\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {\frac {i \, a}{d^{2} e^{3}}} \log \left (-d e^{2} \sqrt {\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} \log \left (d e^{2} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) - {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} \log \left (-d e^{2} \sqrt {-\frac {i \, a}{d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )}{2 \, {\left (d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{2}\right )}} \]
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 \[ \int \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.53, size = 308, normalized size = 0.59 \[ \frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \left (-1+\cos \left (d x +c \right )\right )^{2} \left (i \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (-\cos \left (d x +c \right )-1+\sin \left (d x +c \right )\right )}{2}\right )+i \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )+2 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-\cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (-\cos \left (d x +c \right )-1+\sin \left (d x +c \right )\right )}{2}\right )+\cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )-2 \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-2 \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\right )}{2 d \sin \left (d x +c \right )^{3} \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )-1\right ) \left (\frac {1}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 1834, normalized size = 3.50 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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