Optimal. Leaf size=200 \[ -\frac {\sqrt [4]{-1} a^{5/2} (c+5 i d) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {4 i \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {c-i d} f}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f} \]
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Rubi [A]
time = 0.44, antiderivative size = 200, 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 = {3637, 3682,
3625, 214, 3680, 65, 223, 212} \begin {gather*} -\frac {\sqrt [4]{-1} a^{5/2} (c+5 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 )}{d^{3/2} f}-\frac {4 i \sqrt {2} a^{5/2} \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 \sqrt {c-i d}}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 3625
Rule 3637
Rule 3680
Rule 3682
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{5/2}}{\sqrt {c+d \tan (e+f x)}} \, dx &=-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\frac {a \int \frac {\sqrt {a+i a \tan (e+f x)} \left (\frac {1}{2} a (i c+3 d)+\frac {1}{2} a (c+5 i d) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{d}\\ &=-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\left (4 a^2\right ) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {(a (i c-5 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 d}\\ &=-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}-\frac {\left (8 i a^4\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 {\left (a^3 (i c-5 d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 d f}\\ &=-\frac {4 i \sqrt {2} a^{5/2} \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 )}{\sqrt {c-i d} f}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\frac {\left (a^2 (c+5 i d)\right ) \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 )}{d f}\\ &=-\frac {4 i \sqrt {2} a^{5/2} \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 )}{\sqrt {c-i d} f}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\frac {\left (a^2 (c+5 i d)\right ) \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 )}{d f}\\ &=-\frac {\sqrt [4]{-1} a^{5/2} (c+5 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 )}{d^{3/2} f}-\frac {4 i \sqrt {2} a^{5/2} \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 )}{\sqrt {c-i d} f}-\frac {a^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d 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. \(602\) vs. \(2(200)=400\).
time = 7.59, size = 602, normalized size = 3.01 \begin {gather*} \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \cos ^2(e+f x) (a+i a \tan (e+f x))^{5/2} \left (\frac {\cos (e+f x) \left (\sqrt {c-i d} (c+5 i d) \log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (-i d+d e^{i (e+f x)}+i 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 )}{\sqrt {d} (-i c+5 d) \left (i+e^{i (e+f x)}\right )}\right )-\sqrt {c-i d} (c+5 i 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 )}{\sqrt {d} (-i c+5 d) \left (-i+e^{i (e+f x)}\right )}\right )+(8+8 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 ) (-\cos (2 e)+i \sin (2 e))}{\sqrt {c-i d} \sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))}}-(1-i) \sqrt {d} \cos (2 e) \sqrt {c+d \tan (e+f x)}+(1+i) \sqrt {d} \sin (2 e) \sqrt {c+d \tan (e+f x)}\right )}{d^{3/2} f (\cos (f x)+i \sin (f x))^2} \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. 1294 vs. \(2 (157 ) = 314\).
time = 0.66, size = 1295, normalized size = 6.48
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1295\) |
default | \(\text {Expression too large to display}\) | \(1295\) |
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: ValueError} \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. 809 vs. \(2 (156) = 312\).
time = 1.55, size = 809, normalized size = 4.04 \begin {gather*} -\frac {2 \, \sqrt {2} a^{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}} e^{\left (i \, f x + i \, e\right )} + d f \sqrt {\frac {i \, a^{5} c^{2} - 10 \, a^{5} c d - 25 i \, a^{5} d^{2}}{d^{3} f^{2}}} \log \left (\frac {{\left (2 \, d^{2} f \sqrt {\frac {i \, a^{5} c^{2} - 10 \, a^{5} c d - 25 i \, a^{5} d^{2}}{d^{3} f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (-i \, a^{2} c + 5 \, a^{2} d + {\left (-i \, a^{2} c + 5 \, a^{2} d\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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{-i \, a^{2} c + 5 \, a^{2} d}\right ) - d f \sqrt {\frac {i \, a^{5} c^{2} - 10 \, a^{5} c d - 25 i \, a^{5} d^{2}}{d^{3} f^{2}}} \log \left (-\frac {{\left (2 \, d^{2} f \sqrt {\frac {i \, a^{5} c^{2} - 10 \, a^{5} c d - 25 i \, a^{5} d^{2}}{d^{3} f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (-i \, a^{2} c + 5 \, a^{2} d + {\left (-i \, a^{2} c + 5 \, a^{2} d\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}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{-i \, a^{2} c + 5 \, a^{2} d}\right ) - \sqrt {-\frac {32 i \, a^{5}}{{\left (i \, c + d\right )} f^{2}}} d f \log \left (\frac {{\left (\sqrt {-\frac {32 i \, a^{5}}{{\left (i \, c + d\right )} f^{2}}} {\left (i \, c + d\right )} f e^{\left (i \, f x + i \, e\right )} + 4 \, \sqrt {2} {\left (a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}\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 )}}{4 \, a^{2}}\right ) + \sqrt {-\frac {32 i \, a^{5}}{{\left (i \, c + d\right )} f^{2}}} d f \log \left (\frac {{\left (\sqrt {-\frac {32 i \, a^{5}}{{\left (i \, c + d\right )} f^{2}}} {\left (-i \, c - d\right )} f e^{\left (i \, f x + i \, e\right )} + 4 \, \sqrt {2} {\left (a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2}\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 )}}{4 \, a^{2}}\right )}{2 \, d 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 \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}}}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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