Optimal. Leaf size=196 \[ \frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {i \sqrt {2} \sqrt {a} (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}-\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} (3 c-i d) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f} \]
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Rubi [A] time = 0.74, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3560, 3601, 3544, 208, 3599, 63, 217, 206} \[ \frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {i \sqrt {2} \sqrt {a} (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}-\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} (3 c-i d) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 3544
Rule 3560
Rule 3599
Rule 3601
Rubi steps
\begin {align*} \int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx &=\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (-\frac {1}{2} a \left (2 c^2-i c d-d^2\right )-\frac {1}{2} a (3 c-i d) d \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{a}\\ &=\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+(c-i d)^2 \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {(d (3 i c+d)) \int \frac {(a-i a \tan (e+f x)) \sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 a}\\ &=\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\left (2 i a^2 (c-i d)^2\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 {(a d (3 i c+d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=-\frac {i \sqrt {2} \sqrt {a} (c-i d)^{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}+\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {((3 c-i d) d) \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 {i \sqrt {2} \sqrt {a} (c-i d)^{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}+\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {((3 c-i d) d) \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 {\sqrt [4]{-1} \sqrt {a} (3 c-i d) \sqrt {d} \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 )}{f}-\frac {i \sqrt {2} \sqrt {a} (c-i d)^{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}+\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\\ \end {align*}
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Mathematica [B] time = 7.28, size = 507, normalized size = 2.59 \[ \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a+i a \tan (e+f x)} \left ((1-i) d \sqrt {c+d \tan (e+f x)}-\frac {\cos (e+f x) \left (\sqrt {d} (3 c-i d) \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left ((1-i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}-c \left (e^{i (e+f x)}+i\right )+i d e^{i (e+f x)}+d\right )}{d^{3/2} (d+3 i c) \left (e^{i (e+f x)}+i\right )}\right )+i \sqrt {d} (d+3 i c) \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left ((1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right )}{d^{3/2} (3 c-i d) \left (e^{i (e+f x)}-i\right )}\right )+(2+2 i) (c-i d)^{3/2} \log \left (2 \left (i \sqrt {c-i d} \sin (e+f x)+\sqrt {c-i d} \cos (e+f x)+\sqrt {i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{\sqrt {i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right )}{f} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.73, size = 759, normalized size = 3.87 \[ \frac {2 \, \sqrt {2} d \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}} e^{\left (i \, f x + i \, e\right )} + f \sqrt {\frac {9 i \, a c^{2} d + 6 \, a c d^{2} - i \, a d^{3}}{f^{2}}} \log \left (\frac {{\left (\sqrt {2} {\left ({\left (3 i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c + d\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}} + 2 \, f \sqrt {\frac {9 i \, a c^{2} d + 6 \, a c d^{2} - i \, a d^{3}}{f^{2}}} e^{\left (i \, f x + i \, e\right )}\right )} e^{\left (-i \, f x - i \, e\right )}}{3 i \, c + d}\right ) - f \sqrt {\frac {9 i \, a c^{2} d + 6 \, a c d^{2} - i \, a d^{3}}{f^{2}}} \log \left (\frac {{\left (\sqrt {2} {\left ({\left (3 i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c + d\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}} - 2 \, f \sqrt {\frac {9 i \, a c^{2} d + 6 \, a c d^{2} - i \, a d^{3}}{f^{2}}} e^{\left (i \, f x + i \, e\right )}\right )} e^{\left (-i \, f x - i \, e\right )}}{3 i \, c + d}\right ) - f \sqrt {-\frac {2 \, a c^{3} - 6 i \, a c^{2} d - 6 \, a c d^{2} + 2 i \, a d^{3}}{f^{2}}} \log \left (\frac {{\left (\sqrt {2} {\left ({\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c + d\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}} + f \sqrt {-\frac {2 \, a c^{3} - 6 i \, a c^{2} d - 6 \, a c d^{2} + 2 i \, a d^{3}}{f^{2}}} e^{\left (i \, f x + i \, e\right )}\right )} e^{\left (-i \, f x - i \, e\right )}}{i \, c + d}\right ) + f \sqrt {-\frac {2 \, a c^{3} - 6 i \, a c^{2} d - 6 \, a c d^{2} + 2 i \, a d^{3}}{f^{2}}} \log \left (\frac {{\left (\sqrt {2} {\left ({\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c + d\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}} - f \sqrt {-\frac {2 \, a c^{3} - 6 i \, a c^{2} d - 6 \, a c d^{2} + 2 i \, a d^{3}}{f^{2}}} e^{\left (i \, f x + i \, e\right )}\right )} e^{\left (-i \, f x - i \, e\right )}}{i \, c + d}\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 1701, normalized size = 8.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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