Optimal. Leaf size=196 \[ -\frac {\sqrt [4]{-1} \sqrt {a} (3 c-i d) \sqrt {d} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {i \sqrt {2} \sqrt {a} (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {d \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f} \]
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Rubi [A]
time = 0.47, 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 = {3641, 3682,
3625, 214, 3680, 65, 223, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 3625
Rule 3641
Rule 3680
Rule 3682
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 ) \text {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)) \text {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) \text {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) \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(507\) vs. \(2(196)=392\).
time = 7.85, size = 507, normalized size = 2.59 \begin {gather*} \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a+i a \tan (e+f x)} \left (-\frac {\cos (e+f x) \left ((3 c-i d) \sqrt {d} \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (d+i d e^{i (e+f x)}-c \left (i+e^{i (e+f x)}\right )+(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)}}}\right )}{d^{3/2} (3 i c+d) \left (i+e^{i (e+f x)}\right )}\right )+i \sqrt {d} (3 i c+d) \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (c+i d+i c e^{i (e+f x)}+d e^{i (e+f x)}+(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)}}}\right )}{(3 c-i d) d^{3/2} \left (-i+e^{i (e+f x)}\right )}\right )+(2+2 i) (c-i d)^{3/2} \log \left (2 \left (\sqrt {c-i d} \cos (e+f x)+i \sqrt {c-i d} \sin (e+f x)+\sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))} \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{\sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))}}+(1-i) d \sqrt {c+d \tan (e+f x)}\right )}{f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1701 vs. \(2 (153 ) = 306\).
time = 0.60, size = 1702, normalized size = 8.68
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1702\) |
default | \(\text {Expression too large to display}\) | \(1702\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 796 vs. \(2 (152) = 304\).
time = 0.89, size = 796, normalized size = 4.06 \begin {gather*} \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 )} - \sqrt {2} f \sqrt {-\frac {a c^{3} - 3 i \, a c^{2} d - 3 \, a c d^{2} + i \, a d^{3}}{f^{2}}} \log \left (\frac {{\left (\sqrt {2} f \sqrt {-\frac {a c^{3} - 3 i \, a c^{2} d - 3 \, a c d^{2} + i \, a d^{3}}{f^{2}}} e^{\left (i \, f x + i \, e\right )} + \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}}\right )} e^{\left (-i \, f x - i \, e\right )}}{i \, c + d}\right ) + \sqrt {2} f \sqrt {-\frac {a c^{3} - 3 i \, a c^{2} d - 3 \, a c d^{2} + i \, a d^{3}}{f^{2}}} \log \left (-\frac {{\left (\sqrt {2} f \sqrt {-\frac {a c^{3} - 3 i \, a c^{2} d - 3 \, a c d^{2} + i \, a d^{3}}{f^{2}}} e^{\left (i \, f x + i \, e\right )} - \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}}\right )} e^{\left (-i \, f x - i \, e\right )}}{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 {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 )}{2 \, f} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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