Optimal. Leaf size=88 \[ -\frac {8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d} \]
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Rubi [A]
time = 0.05, 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))^{11/2}}{11 a^5 d}+\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac {8 i (a+i a \tan (c+d x))^{7/2}}{7 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) \sqrt {a+i a \tan (c+d x)} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^2 (a+x)^{5/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {Subst}\left (\int \left (4 a^2 (a+x)^{5/2}-4 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 77, normalized size = 0.88 \begin {gather*} \frac {2 \sec ^5(c+d x) (44+107 \cos (2 (c+d x))-91 i \sin (2 (c+d x))) (-i \cos (3 (c+d x))+\sin (3 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{693 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.97, size = 114, normalized size = 1.30
method | result | size |
default | \(\frac {2 \left (-128 i \left (\cos ^{5}\left (d x +c \right )\right )+128 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-16 i \left (\cos ^{3}\left (d x +c \right )\right )+80 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-7 i \cos \left (d x +c \right )+63 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{693 d \cos \left (d x +c \right )^{5}}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 58, normalized size = 0.66 \begin {gather*} -\frac {2 i \, {\left (63 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 308 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 396 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}\right )}}{693 \, a^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 119, normalized size = 1.35 \begin {gather*} -\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (8 i \, e^{\left (11 i \, d x + 11 i \, c\right )} + 44 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 99 i \, e^{\left (7 i \, d x + 7 i \, c\right )}\right )}}{693 \, {\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \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 = 7.01, size = 352, normalized size = 4.00 \begin {gather*} -\frac {\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}}{693\,d}-\frac {\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}}\,256{}\mathrm {i}}{693\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\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}}{231\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\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}}\,7232{}\mathrm {i}}{693\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\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}}\,1472{}\mathrm {i}}{99\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\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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