Optimal. Leaf size=59 \[ -\frac {4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45}
\begin {gather*} \frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}-\frac {4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac {i \text {Subst}\left (\int (a-x) (a+x)^{7/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (2 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {4 i (a+i a \tan (c+d x))^{9/2}}{9 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 85, normalized size = 1.44 \begin {gather*} -\frac {2 a^2 \sec ^4(c+d x) (\cos (4 c+6 d x)+i \sin (4 c+6 d x)) (13 i+9 \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}{99 d (\cos (d x)+i \sin (d 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. 116 vs. \(2 (47 ) = 94\).
time = 0.76, size = 117, normalized size = 1.98
method | result | size |
default | \(-\frac {2 \left (32 i \left (\cos ^{5}\left (d x +c \right )\right )-32 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+4 i \left (\cos ^{3}\left (d x +c \right )\right )-20 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-23 i \cos \left (d x +c \right )+9 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{99 d \cos \left (d x +c \right )^{5}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 40, normalized size = 0.68 \begin {gather*} \frac {2 i \, {\left (9 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 22 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a\right )}}{99 \, a^{3} d} \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. 114 vs. \(2 (43) = 86\).
time = 0.38, size = 114, normalized size = 1.93 \begin {gather*} -\frac {64 \, \sqrt {2} {\left (2 i \, a^{2} e^{\left (11 i \, d x + 11 i \, c\right )} + 11 i \, a^{2} e^{\left (9 i \, d x + 9 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{99 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.42, size = 370, normalized size = 6.27 \begin {gather*} -\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{99\,d}-\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{99\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}+\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{33\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,2944{}\mathrm {i}}{99\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,2176{}\mathrm {i}}{99\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {a^2\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{11\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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