Optimal. Leaf size=151 \[ -\frac {2 (-1)^{3/4} a^{3/2} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d}} \]
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Rubi [A] time = 0.43, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {3555, 3544, 208, 3599, 63, 217, 206} \[ -\frac {2 (-1)^{3/4} a^{3/2} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f \sqrt {c-i d}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 3544
Rule 3555
Rule 3599
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{3/2}}{\sqrt {c+d \tan (e+f x)}} \, dx &=i \int \frac {\sqrt {a+i a \tan (e+f x)} (i a+a \tan (e+f x))}{\sqrt {c+d \tan (e+f x)}} \, dx+(2 a) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{f}-\frac {\left (4 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{f}\\ &=-\frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {c-i d} f}+\frac {(2 i a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+i d-\frac {i d x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{f}\\ &=-\frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {c-i d} f}+\frac {(2 i a) \operatorname {Subst}\left (\int \frac {1}{1+\frac {i d x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}\\ &=-\frac {2 (-1)^{3/4} a^{3/2} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {c-i d} f}\\ \end {align*}
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Mathematica [B] time = 5.39, size = 505, normalized size = 3.34 \[ \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \sqrt {i \sin (2 (e+f x))+\cos (2 (e+f x))+1} (\cos (2 e+f x)-i \sin (2 e+f x)) \left (\sqrt {c-i d} \log \left (\frac {(2-2 i) e^{\frac {3 i e}{2}} \left (-(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}+i c \left (e^{i (e+f x)}+i\right )+d e^{i (e+f x)}-i d\right )}{\sqrt {d} \left (e^{i (e+f x)}+i\right )}\right )-\sqrt {c-i d} \log \left (\frac {(2+2 i) e^{\frac {3 i e}{2}} \left ((1-i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}+c \left (e^{i (e+f x)}-i\right )-i d e^{i (e+f x)}+d\right )}{\sqrt {d} \left (e^{i (e+f x)}-i\right )}\right )+(2-2 i) \sqrt {d} \log \left (2 \left (i \sqrt {c-i d} \sin (e+f x)+\sqrt {c-i d} \cos (e+f x)+\sqrt {i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{\sqrt {d} f \sqrt {c-i d}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.49, size = 533, normalized size = 3.53 \[ \frac {1}{2} \, \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} \log \left (\frac {{\left ({\left (i \, c + d\right )} f \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + 2 \, \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a}\right ) - \frac {1}{2} \, \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} \log \left (\frac {{\left ({\left (-i \, c - d\right )} f \sqrt {-\frac {8 i \, a^{3}}{{\left (i \, c + d\right )} f^{2}}} e^{\left (i \, f x + i \, e\right )} + 2 \, \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} \log \left (\frac {{\left (d f \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} \log \left (-\frac {{\left (d f \sqrt {-\frac {4 i \, a^{3}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 183, normalized size = 1.21 \[ \frac {\sqrt {2 \, a d^{2} + 2 \, \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} + d^{2}} a d} a {\left (\frac {i \, {\left (d \tan \left (f x + e\right ) + c\right )} a d - i \, a c d}{a d^{2} + \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} d^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} c d^{2} + a^{2} c^{2} d^{2} + a^{2} d^{4}}} + 1\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{2 \, {\left ({\left (-i \, d \tan \left (f x + e\right ) - i \, c\right )} d + i \, c d + d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 986, normalized size = 6.53 \[ \frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {c +d \tan \left (f x +e \right )}\, a^{2} \left (i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a \tan \left (f x +e \right ) d +i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i d a}+d a}{2 \sqrt {i d a}}\right ) c -i \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a \tan \left (f x +e \right ) d +i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i d a}+d a}{2 \sqrt {i d a}}\right ) d -i \ln \left (\frac {3 c a +i a \tan \left (f x +e \right ) c -i d a +3 a \tan \left (f x +e \right ) d +2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {i d a}\, c +i \ln \left (\frac {3 c a +i a \tan \left (f x +e \right ) c -i d a +3 a \tan \left (f x +e \right ) d +2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {i d a}\, d -\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a \tan \left (f x +e \right ) d +i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i d a}+d a}{2 \sqrt {i d a}}\right ) c^{2}-\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a \tan \left (f x +e \right ) d +i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i d a}+d a}{2 \sqrt {i d a}}\right ) d^{2}+\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a \tan \left (f x +e \right ) d +i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i d a}+d a}{2 \sqrt {i d a}}\right ) c +\sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \ln \left (\frac {2 i a \tan \left (f x +e \right ) d +i a c +2 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {i d a}+d a}{2 \sqrt {i d a}}\right ) d -\ln \left (\frac {3 c a +i a \tan \left (f x +e \right ) c -i d a +3 a \tan \left (f x +e \right ) d +2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {i d a}\, c -\ln \left (\frac {3 c a +i a \tan \left (f x +e \right ) c -i d a +3 a \tan \left (f x +e \right ) d +2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{\tan \left (f x +e \right )+i}\right ) \sqrt {i d a}\, d \right ) \sqrt {2}}{2 f \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \left (c^{2}+d^{2}\right ) \sqrt {i d a}\, \sqrt {-a \left (i d -c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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