Optimal. Leaf size=335 \[ \frac {i \sqrt {2} \sqrt {a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}-\frac {i \sqrt {2} \sqrt {a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}-\frac {i \sqrt {a} \log \left (-\sqrt {2} \sqrt {a} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))+a \sqrt {e}\right )}{\sqrt {2} d \sqrt {e}}+\frac {i \sqrt {a} \log \left (\sqrt {2} \sqrt {a} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))+a \sqrt {e}\right )}{\sqrt {2} d \sqrt {e}} \]
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Rubi [A] time = 0.21, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3513, 297, 1162, 617, 204, 1165, 628} \[ \frac {i \sqrt {2} \sqrt {a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}-\frac {i \sqrt {2} \sqrt {a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}-\frac {i \sqrt {a} \log \left (-\sqrt {2} \sqrt {a} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))+a \sqrt {e}\right )}{\sqrt {2} d \sqrt {e}}+\frac {i \sqrt {a} \log \left (\sqrt {2} \sqrt {a} \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))+a \sqrt {e}\right )}{\sqrt {2} d \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3513
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {(4 i a) \operatorname {Subst}\left (\int \frac {x^2}{a^2 e^2+x^4} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=\frac {(2 i a) \operatorname {Subst}\left (\int \frac {a e-x^2}{a^2 e^2+x^4} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {(2 i a) \operatorname {Subst}\left (\int \frac {a e+x^2}{a^2 e^2+x^4} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=-\frac {(i a) \operatorname {Subst}\left (\int \frac {1}{a e-\sqrt {2} \sqrt {a} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {(i a) \operatorname {Subst}\left (\int \frac {1}{a e+\sqrt {2} \sqrt {a} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {\left (i \sqrt {a}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {a} \sqrt {e}+2 x}{-a e-\sqrt {2} \sqrt {a} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{\sqrt {2} d \sqrt {e}}-\frac {\left (i \sqrt {a}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {a} \sqrt {e}-2 x}{-a e+\sqrt {2} \sqrt {a} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}\right )}{\sqrt {2} d \sqrt {e}}\\ &=-\frac {i \sqrt {a} \log \left (a \sqrt {e}-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d \sqrt {e}}+\frac {i \sqrt {a} \log \left (a \sqrt {e}+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d \sqrt {e}}-\frac {\left (i \sqrt {2} \sqrt {a}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}+\frac {\left (i \sqrt {2} \sqrt {a}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}\\ &=\frac {i \sqrt {2} \sqrt {a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}-\frac {i \sqrt {2} \sqrt {a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d \sqrt {e}}-\frac {i \sqrt {a} \log \left (a \sqrt {e}-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d \sqrt {e}}+\frac {i \sqrt {a} \log \left (a \sqrt {e}+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 125, normalized size = 0.37 \[ \frac {i \left (-e^{-2 i c}\right )^{3/4} e^{-\frac {3}{2} i d x} \left (1+e^{2 i (c+d x)}\right ) \sqrt {a+i a \tan (c+d x)} \left (\tan ^{-1}\left (\frac {e^{\frac {i d x}{2}}}{\sqrt [4]{-e^{-2 i c}}}\right )-\tanh ^{-1}\left (\frac {e^{\frac {i d x}{2}}}{\sqrt [4]{-e^{-2 i c}}}\right )\right )}{d \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 313, normalized size = 0.93 \[ -\frac {1}{2} \, \sqrt {\frac {4 i \, a}{d^{2} e}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + \frac {1}{2} i \, d e \sqrt {\frac {4 i \, a}{d^{2} e}}\right ) + \frac {1}{2} \, \sqrt {\frac {4 i \, a}{d^{2} e}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - \frac {1}{2} i \, d e \sqrt {\frac {4 i \, a}{d^{2} e}}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i \, a}{d^{2} e}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + \frac {1}{2} i \, d e \sqrt {-\frac {4 i \, a}{d^{2} e}}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i \, a}{d^{2} e}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - \frac {1}{2} i \, d e \sqrt {-\frac {4 i \, a}{d^{2} e}}\right ) \]
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 \[ \int \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {e \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.54, size = 226, normalized size = 0.67 \[ -\frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \left (-1+\cos \left (d x +c \right )\right ) \left (i \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )+i \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (-\cos \left (d x +c \right )-1+\sin \left (d x +c \right )\right )}{2}\right )-\arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )+\arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (-\cos \left (d x +c \right )-1+\sin \left (d x +c \right )\right )}{2}\right )\right )}{d \sin \left (d x +c \right ) \sqrt {e \cos \left (d x +c \right )}\, \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )-1\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.88, size = 1400, normalized size = 4.18 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}{\sqrt {e \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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