Optimal. Leaf size=218 \[ -\frac {3 \sqrt [4]{-1} \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d} \]
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Rubi [A]
time = 0.45, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3639, 3676,
3678, 3682, 3625, 211, 3680, 65, 223, 209} \begin {gather*} -\frac {3 \sqrt [4]{-1} \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 211
Rule 223
Rule 3625
Rule 3639
Rule 3676
Rule 3678
Rule 3680
Rule 3682
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}-\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\frac {5 a}{2}+4 i a \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)} \left (-\frac {39 i a^2}{4}-\frac {21}{2} a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {\int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {21 a^3}{4}-\frac {9}{2} i a^3 \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{3 a^5}\\ &=-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {3 \int \frac {(a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx}{2 a^3}+\frac {\int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx}{4 a^2}\\ &=-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}-\frac {i \text {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{2 d}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d}\\ &=\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}+\frac {3 \text {Subst}\left (\int \frac {1}{1-i a x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a d}\\ &=-\frac {3 \sqrt [4]{-1} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {\tan ^{\frac {5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {13 i \tan ^{\frac {3}{2}}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {7 \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}{2 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 4.28, size = 240, normalized size = 1.10 \begin {gather*} \frac {-\frac {12 e^{3 i (c+d x)} \sqrt {-1+e^{2 i (c+d x)}} \left (\tanh ^{-1}\left (\frac {e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )+6 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} e^{i (c+d x)}}{\sqrt {-1+e^{2 i (c+d x)}}}\right )\right )}{\sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (1+e^{2 i (c+d x)}\right )^2}+i \sec ^2(c+d x) (15+27 \cos (2 (c+d x))+29 i \sin (2 (c+d x))) \sqrt {\tan (c+d x)}}{12 a d (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \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. 814 vs. \(2 (170 ) = 340\).
time = 0.18, size = 815, normalized size = 3.74 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 \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. 612 vs. \(2 (160) = 320\).
time = 0.62, size = 612, normalized size = 2.81 \begin {gather*} \frac {{\left (3 \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {1}{2} \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) - 3 \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-\frac {1}{2} \, a^{2} d \sqrt {-\frac {i}{2 \, a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) + 3 \, a^{2} d \sqrt {-\frac {9 i}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {104 \, {\left (6 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )} + {\left (3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {-\frac {9 i}{a^{3} d^{2}}}\right )}}{1815 \, {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}}\right ) - 3 \, a^{2} d \sqrt {-\frac {9 i}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (\frac {104 \, {\left (6 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )} - {\left (3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {-\frac {9 i}{a^{3} d^{2}}}\right )}}{1815 \, {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (28 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\mathrm {tan}\left (c+d\,x\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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