Optimal. Leaf size=131 \[ -\frac {4 i a^2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \]
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Rubi [A]
time = 0.22, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3624, 3609,
3618, 65, 214} \begin {gather*} -\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a^2 (d+i c) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 i a^2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3609
Rule 3618
Rule 3624
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx &=-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \left (2 a^2+2 i a^2 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2} \, dx\\ &=\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \sqrt {c+d \tan (e+f x)} \left (2 a^2 (c-i d)+2 a^2 (i c+d) \tan (e+f x)\right ) \, dx\\ &=\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \frac {2 a^2 (c-i d)^2+2 i a^2 (c-i d)^2 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {\left (4 i a^4 (c-i d)^4\right ) \text {Subst}\left (\int \frac {1}{\left (-4 a^4 (c-i d)^4+2 a^2 (c-i d)^2 x\right ) \sqrt {c-\frac {i d x}{2 a^2 (c-i d)^2}}} \, dx,x,2 i a^2 (c-i d)^2 \tan (e+f x)\right )}{f}\\ &=\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}-\frac {\left (16 a^6 (c-i d)^6\right ) \text {Subst}\left (\int \frac {1}{-4 a^4 (c-i d)^4-\frac {4 i a^4 c (c-i d)^4}{d}+\frac {4 i a^4 (c-i d)^4 x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac {4 i a^2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {4 a^2 (i c+d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}\\ \end {align*}
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Mathematica [A]
time = 3.98, size = 221, normalized size = 1.69 \begin {gather*} \frac {a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left (-4 i (c-i d)^{3/2} e^{-2 i e} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}}{\sqrt {c-i d}}\right )-\frac {\sec ^2(e+f x) (\cos (2 e)-i \sin (2 e)) \left (3 c^2-40 i c d-27 d^2+\left (3 c^2-40 i c d-33 d^2\right ) \cos (2 (e+f x))+2 (3 c-5 i d) d \sin (2 (e+f x))\right ) \sqrt {c+d \tan (e+f x)}}{15 d}\right )}{f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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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. 871 vs. \(2 (110 ) = 220\).
time = 0.27, size = 872, normalized size = 6.66
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i d c \sqrt {c +d \tan \left (f x +e \right )}+2 d^{2} \sqrt {c +d \tan \left (f x +e \right )}-2 d \left (\frac {\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}-\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )\right )}{f d}\) | \(872\) |
default | \(\frac {2 a^{2} \left (-\frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 i d c \sqrt {c +d \tan \left (f x +e \right )}+2 d^{2} \sqrt {c +d \tan \left (f x +e \right )}-2 d \left (\frac {\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c -\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}+\frac {\left (-i c^{2} \sqrt {c^{2}+d^{2}}+i d^{2} \sqrt {c^{2}+d^{2}}-i c^{3}-i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-c^{2} d -d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}+\frac {\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{3}+i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c^{2} d +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, d^{3}-\frac {\left (i c^{2} \sqrt {c^{2}+d^{2}}-i d^{2} \sqrt {c^{2}+d^{2}}+i c^{3}+i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+c^{2} d +d^{3}\right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}}\right )\right )}{f d}\) | \(872\) |
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 \begin {gather*} \text {Failed to integrate} \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. 666 vs. \(2 (109) = 218\).
time = 1.46, size = 666, normalized size = 5.08 \begin {gather*} \frac {15 \, {\left (d f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \sqrt {-\frac {a^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} \log \left (\frac {2 \, {\left (-i \, a^{2} c^{2} - a^{2} c d + {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\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^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} + {\left (-i \, a^{2} c^{2} - 2 \, a^{2} c d + i \, a^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{-i \, a^{2} c - a^{2} d}\right ) - 15 \, {\left (d f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )} \sqrt {-\frac {a^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} \log \left (\frac {2 \, {\left (-i \, a^{2} c^{2} - a^{2} c d - {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\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^{4} c^{3} - 3 i \, a^{4} c^{2} d - 3 \, a^{4} c d^{2} + i \, a^{4} d^{3}}{f^{2}}} + {\left (-i \, a^{2} c^{2} - 2 \, a^{2} c d + i \, a^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{-i \, a^{2} c - a^{2} d}\right ) - 2 \, {\left (3 \, a^{2} c^{2} - 34 i \, a^{2} c d - 23 \, a^{2} d^{2} + {\left (3 \, a^{2} c^{2} - 46 i \, a^{2} c d - 43 \, a^{2} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (3 \, a^{2} c^{2} - 40 i \, a^{2} c d - 27 \, a^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\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}}}{15 \, {\left (d f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f\right )}} \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*} - a^{2} \left (\int \left (- c \sqrt {c + d \tan {\left (e + f x \right )}}\right )\, dx + \int c \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx + \int d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 2 i c \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 i d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 288 vs. \(2 (109) = 218\).
time = 0.81, size = 288, normalized size = 2.20 \begin {gather*} -\frac {8 \, {\left (-i \, a^{2} c^{2} - 2 \, a^{2} c d + i \, a^{2} d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{\sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 \, {\left (3 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} a^{2} d^{4} f^{4} - 10 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{2} d^{5} f^{4} - 30 i \, \sqrt {d \tan \left (f x + e\right ) + c} a^{2} c d^{5} f^{4} - 30 \, \sqrt {d \tan \left (f x + e\right ) + c} a^{2} d^{6} f^{4}\right )}}{15 \, d^{5} f^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.32, size = 196, normalized size = 1.50 \begin {gather*} -\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{3\,d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{3\,d\,f}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}-\left (c-d\,1{}\mathrm {i}\right )\,\left (\frac {2\,a^2\,\left (c-d\,1{}\mathrm {i}\right )}{d\,f}-\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{d\,f}\right )\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}-\frac {2\,a^2\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}+\frac {\sqrt {4{}\mathrm {i}}\,a^2\,\mathrm {atan}\left (\frac {\sqrt {4{}\mathrm {i}}\,{\left (-d-c\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{2\,\left (c^2\,1{}\mathrm {i}+2\,c\,d-d^2\,1{}\mathrm {i}\right )}\right )\,{\left (-d-c\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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