Optimal. Leaf size=59 \[ -\frac {4 i (a+i a \tan (c+d x))^{5/2}}{5 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d} \]
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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))^{7/2}}{7 a^3 d}-\frac {4 i (a+i a \tan (c+d x))^{5/2}}{5 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) \sqrt {a+i a \tan (c+d x)} \, dx &=-\frac {i \text {Subst}\left (\int (a-x) (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (2 a (a+x)^{3/2}-(a+x)^{5/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {4 i (a+i a \tan (c+d x))^{5/2}}{5 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 58, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {a+i a \tan (c+d x)} \left (8 (-i+\tan (c+d x))+\sec ^2(c+d x) (-i+5 \tan (c+d x))\right )}{35 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 87, normalized size = 1.47
method | result | size |
default | \(-\frac {2 \left (8 i \left (\cos ^{3}\left (d x +c \right )\right )-8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+i \cos \left (d x +c \right )-5 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{35 d \cos \left (d x +c \right )^{3}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 40, normalized size = 0.68 \begin {gather*} \frac {2 i \, {\left (5 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 14 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a\right )}}{35 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 84, normalized size = 1.42 \begin {gather*} -\frac {16 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (2 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 7 i \, e^{\left (5 i \, d x + 5 i \, c\right )}\right )}}{35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 ^{4}{\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 = 6.25, size = 230, normalized size = 3.90 \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}}\,32{}\mathrm {i}}{35\,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}}\,16{}\mathrm {i}}{35\,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}}\,128{}\mathrm {i}}{35\,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}}\,16{}\mathrm {i}}{7\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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