Optimal. Leaf size=127 \[ -\frac {4 i a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {2 a^2 (i c-d)}{3 d (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {4 i a^2}{(c-i d)^2 f \sqrt {c+d \tan (e+f x)}} \]
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Rubi [A]
time = 0.24, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3623, 3610,
3618, 65, 214} \begin {gather*} \frac {4 i a^2}{f (c-i d)^2 \sqrt {c+d \tan (e+f x)}}+\frac {2 a^2 (-d+i c)}{3 d f (d+i c) (c+d \tan (e+f x))^{3/2}}-\frac {4 i a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3610
Rule 3618
Rule 3623
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^2}{(c+d \tan (e+f x))^{5/2}} \, dx &=\frac {2 a^2 (i c-d)}{3 d (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {\int \frac {2 a^2 (c+i d)+2 a^2 (i c-d) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{c^2+d^2}\\ &=\frac {2 a^2 (i c-d)}{3 d (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {4 i a^2}{(c-i d)^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\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}{\left (c^2+d^2\right )^2}\\ &=\frac {2 a^2 (i c-d)}{3 d (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {4 i a^2}{(c-i d)^2 f \sqrt {c+d \tan (e+f x)}}+\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 )}{\left (c^2+d^2\right )^2 f}\\ &=\frac {2 a^2 (i c-d)}{3 d (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {4 i a^2}{(c-i d)^2 f \sqrt {c+d \tan (e+f x)}}-\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 \left (c^2+d^2\right )^2 f}\\ &=-\frac {4 i a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {2 a^2 (i c-d)}{3 d (i c+d) f (c+d \tan (e+f x))^{3/2}}+\frac {4 i a^2}{(c-i d)^2 f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 5.14, size = 218, normalized size = 1.72 \begin {gather*} \frac {a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left (-\frac {4 i 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 )}{(c-i d)^{5/2}}+\frac {2 \cos (e+f x) (\cos (2 e)-i \sin (2 e)) \left (\left (c^2+6 i c d+d^2\right ) \cos (e+f x)+6 i d^2 \sin (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{3 (c-i d)^2 d (c \cos (e+f x)+d \sin (e+f x))^2}\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. 898 vs. \(2 (108 ) = 216\).
time = 0.29, size = 899, normalized size = 7.08
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (-\frac {-2 i c d -c^{2}+d^{2}}{3 \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 d \left (i c^{2}-i d^{2}-2 c d \right )}{\left (c^{2}+d^{2}\right )^{2} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {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}-3 i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-3 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}-3 i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-3 \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}-3 i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-3 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}+3 i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+3 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}-3 i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-3 \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}+3 i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+3 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 )}{\left (c^{2}+d^{2}\right )^{2}}\right )}{f d}\) | \(899\) |
default | \(\frac {2 a^{2} \left (-\frac {-2 i c d -c^{2}+d^{2}}{3 \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 d \left (i c^{2}-i d^{2}-2 c d \right )}{\left (c^{2}+d^{2}\right )^{2} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {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}-3 i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-3 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}-3 i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-3 \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}-3 i c \,d^{2}-2 c d \sqrt {c^{2}+d^{2}}-3 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}+3 i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+3 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}-3 i \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, c \,d^{2}-3 \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}+3 i c \,d^{2}+2 c d \sqrt {c^{2}+d^{2}}+3 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 )}{\left (c^{2}+d^{2}\right )^{2}}\right )}{f d}\) | \(899\) |
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. 929 vs. \(2 (104) = 208\).
time = 1.04, size = 929, normalized size = 7.31 \begin {gather*} \frac {3 \, {\left ({\left (c^{4} d - 4 i \, c^{3} d^{2} - 6 \, c^{2} d^{3} + 4 i \, c d^{4} + d^{5}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{4} d - 2 i \, c^{3} d^{2} - 2 i \, c d^{4} - d^{5}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{4} d + 2 \, c^{2} d^{3} + d^{5}\right )} f\right )} \sqrt {-\frac {16 i \, a^{4}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{2}}} \log \left (\frac {{\left (4 \, a^{2} c + {\left ({\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}\right )} f\right )} \sqrt {-\frac {16 i \, a^{4}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{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}} + 4 \, {\left (a^{2} c - i \, a^{2} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2}}\right ) - 3 \, {\left ({\left (c^{4} d - 4 i \, c^{3} d^{2} - 6 \, c^{2} d^{3} + 4 i \, c d^{4} + d^{5}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{4} d - 2 i \, c^{3} d^{2} - 2 i \, c d^{4} - d^{5}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{4} d + 2 \, c^{2} d^{3} + d^{5}\right )} f\right )} \sqrt {-\frac {16 i \, a^{4}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{2}}} \log \left (\frac {{\left (4 \, a^{2} c + {\left ({\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} f\right )} \sqrt {-\frac {16 i \, a^{4}}{{\left (i \, c^{5} + 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 5 i \, c d^{4} + d^{5}\right )} f^{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}} + 4 \, {\left (a^{2} c - i \, a^{2} d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2}}\right ) + 8 \, {\left (a^{2} c^{2} + 6 i \, a^{2} c d - 5 \, a^{2} d^{2} + {\left (a^{2} c^{2} + 6 i \, a^{2} c d + 7 \, a^{2} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (a^{2} c^{2} + 6 i \, a^{2} c d + 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}}}{12 \, {\left ({\left (c^{4} d - 4 i \, c^{3} d^{2} - 6 \, c^{2} d^{3} + 4 i \, c d^{4} + d^{5}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (c^{4} d - 2 i \, c^{3} d^{2} - 2 i \, c d^{4} - d^{5}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (c^{4} d + 2 \, c^{2} d^{3} + d^{5}\right )} 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 \frac {\tan ^{2}{\left (e + f x \right )}}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\, dx + \int \left (- \frac {2 i \tan {\left (e + f x \right )}}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}\right )\, dx + \int \left (- \frac {1}{c^{2} \sqrt {c + d \tan {\left (e + f x \right )}} + 2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} + d^{2} \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. 250 vs. \(2 (104) = 208\).
time = 0.90, size = 250, normalized size = 1.97 \begin {gather*} \frac {8 \, a^{2} \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 )}{{\left (-i \, c^{2} f - 2 \, c d f + i \, d^{2} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 \, {\left (-i \, a^{2} c^{2} + 6 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} d - i \, a^{2} d^{2}\right )}}{-3 \, {\left (-i \, c^{2} d f - 2 \, c d^{2} f + i \, d^{3} f\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.90, size = 221, normalized size = 1.74 \begin {gather*} \frac {\frac {a^2\,\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,4{}\mathrm {i}}{f\,{\left (c-d\,1{}\mathrm {i}\right )}^2}+\frac {2\,a^2\,\left (c+d\,1{}\mathrm {i}\right )}{3\,d\,f\,\left (c-d\,1{}\mathrm {i}\right )}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {a^2\,\mathrm {atan}\left (\frac {\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,\left (2\,c^8\,f^2+8\,c^6\,d^2\,f^2+12\,c^4\,d^4\,f^2+8\,c^2\,d^6\,f^2+2\,d^8\,f^2\right )}{2\,f\,{\left (-c+d\,1{}\mathrm {i}\right )}^{5/2}\,\left (f\,c^6+2{}\mathrm {i}\,f\,c^5\,d+f\,c^4\,d^2+4{}\mathrm {i}\,f\,c^3\,d^3-f\,c^2\,d^4+2{}\mathrm {i}\,f\,c\,d^5-f\,d^6\right )}\right )\,4{}\mathrm {i}}{f\,{\left (-c+d\,1{}\mathrm {i}\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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