Optimal. Leaf size=177 \[ -\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {(c+d \tan (e+f x))^{3/2}}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2}}+\frac {i \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}} \]
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Rubi [A] time = 0.32, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3547, 3546, 3544, 208} \[ -\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {(c+d \tan (e+f x))^{3/2}}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2}}+\frac {i \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3544
Rule 3546
Rule 3547
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{3/2}} \, dx &=-\frac {(c+d \tan (e+f x))^{3/2}}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2}}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx}{2 a}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}-\frac {(c+d \tan (e+f x))^{3/2}}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2}}+\frac {(c-i d) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}-\frac {(c+d \tan (e+f x))^{3/2}}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2}}-\frac {(i c+d) \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 )}{2 f}\\ &=-\frac {i \sqrt {c-i d} \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 )}{2 \sqrt {2} a^{3/2} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}-\frac {(c+d \tan (e+f x))^{3/2}}{3 (i c-d) f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 4.42, size = 249, normalized size = 1.41 \[ \frac {\sec ^{\frac {3}{2}}(e+f x) \left (-i \sqrt {2} \sqrt {c-i d} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{3/2} \left (1+e^{2 i (e+f x)}\right )^{3/2} \log \left (2 \left (\sqrt {c-i d} e^{i (e+f x)}+\sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )\right )-\frac {2 ((3 c+i d) \tan (e+f x)-5 i c+3 d) \sqrt {c+d \tan (e+f x)}}{3 (c+i d) \sec ^{\frac {3}{2}}(e+f x)}\right )}{4 f (a+i a \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 419, normalized size = 2.37 \[ \frac {{\left (3 \, \sqrt {\frac {1}{2}} {\left (i \, a^{2} c - a^{2} d\right )} f \sqrt {-\frac {c - i \, d}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (2 i \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {c - i \, d}{a^{3} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (-i \, a^{2} c + a^{2} d\right )} f \sqrt {-\frac {c - i \, d}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (-2 i \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {c - i \, d}{a^{3} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - \sqrt {2} {\left (2 \, {\left (2 \, c + i \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (5 \, c + 3 i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{12 \, {\left (i \, a^{2} c - a^{2} d\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 1180, normalized size = 6.67 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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