Optimal. Leaf size=113 \[ -\frac {16 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}+\frac {8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 113, 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^7 d}-\frac {12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac {8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac {i \text {Subst}\left (\int \frac {(a-x)^3}{\sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {Subst}\left (\int \left (\frac {8 a^3}{\sqrt {a+x}}-12 a^2 \sqrt {a+x}+6 a (a+x)^{3/2}-(a+x)^{5/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}+\frac {8 i (a+i a \tan (c+d x))^{3/2}}{a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{5/2}}{5 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{7/2}}{7 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 110, normalized size = 0.97 \begin {gather*} \frac {2 \sec ^7(c+d x) (126 \cos (c+d x)+51 \cos (3 (c+d x))-i (14 \sin (c+d x)+19 \sin (3 (c+d x)))) (\cos (4 (c+d x))+i \sin (4 (c+d x)))}{35 a^3 d (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 90, normalized size = 0.80
method | result | size |
default | \(-\frac {2 \left (204 i \left (\cos ^{3}\left (d x +c \right )\right )+76 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-27 i \cos \left (d x +c \right )-5 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{35 d \cos \left (d x +c \right )^{3} a^{4}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 0.67 \begin {gather*} \frac {2 i \, {\left (5 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 42 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 140 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} - 280 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{3}\right )}}{35 \, a^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 119, normalized size = 1.05 \begin {gather*} -\frac {16 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (16 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 56 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 70 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 35 i \, e^{\left (i \, d x + i \, c\right )}\right )}}{35 \, {\left (a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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 = 6.74, size = 242, normalized size = 2.14 \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}}\,256{}\mathrm {i}}{35\,a^4\,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}}\,128{}\mathrm {i}}{35\,a^4\,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}}\,96{}\mathrm {i}}{35\,a^4\,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\,a^4\,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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