Optimal. Leaf size=88 \[ -\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 88, 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))^{13/2}}{13 a^5 d}+\frac {8 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}-\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^2 (a+x)^{7/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {Subst}\left (\int \left (4 a^2 (a+x)^{7/2}-4 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.67, size = 93, normalized size = 1.06 \begin {gather*} \frac {2 a \sec ^6(c+d x) (\cos (d x)-i \sin (d x)) (52+151 \cos (2 (c+d x))-135 i \sin (2 (c+d x))) (-i \cos (4 c+5 d x)+\sin (4 c+5 d x)) \sqrt {a+i a \tan (c+d x)}}{1287 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 125, normalized size = 1.42
method | result | size |
default | \(-\frac {2 \left (256 i \left (\cos ^{6}\left (d x +c \right )\right )-256 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+32 i \left (\cos ^{4}\left (d x +c \right )\right )-160 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+14 i \left (\cos ^{2}\left (d x +c \right )\right )-126 \sin \left (d x +c \right ) \cos \left (d x +c \right )-99 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{1287 d \cos \left (d x +c \right )^{6}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 58, normalized size = 0.66 \begin {gather*} -\frac {2 i \, {\left (99 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 468 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 572 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2}\right )}}{1287 \, a^{5} 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. 134 vs. \(2 (64) = 128\).
time = 0.40, size = 134, normalized size = 1.52 \begin {gather*} -\frac {128 \, \sqrt {2} {\left (8 i \, a e^{\left (13 i \, d x + 13 i \, c\right )} + 52 i \, a e^{\left (11 i \, d x + 11 i \, c\right )} + 143 i \, a 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}}}{1287 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \sec ^{6}{\left (c + d x \right )}\, dx \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 = 7.31, size = 420, normalized size = 4.77 \begin {gather*} -\frac {a\,\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}}\,1024{}\mathrm {i}}{1287\,d}-\frac {a\,\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}}{1287\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a\,\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}}{429\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {a\,\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}}\,27136{}\mathrm {i}}{1287\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {a\,\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}}\,58624{}\mathrm {i}}{1287\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {a\,\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}}\,5120{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {a\,\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}}{13\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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