Optimal. Leaf size=88 \[ -\frac {2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac {8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]
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Rubi [A] time = 0.08, 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 = {3487, 43} \[ -\frac {2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac {8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^2 (a+x)^{11/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{11/2}-4 a (a+x)^{13/2}+(a+x)^{15/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 97, normalized size = 1.10 \[ \frac {2 a^3 \sec ^8(c+d x) \sqrt {a+i a \tan (c+d x)} (-247 i \sin (2 (c+d x))+263 \cos (2 (c+d x))+68) (\sin (6 c+9 d x)-i \cos (6 c+9 d x))}{3315 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.80, size = 164, normalized size = 1.86 \[ \frac {\sqrt {2} {\left (-4096 i \, a^{3} e^{\left (17 i \, d x + 17 i \, c\right )} - 34816 i \, a^{3} e^{\left (15 i \, d x + 15 i \, c\right )} - 130560 i \, a^{3} e^{\left (13 i \, d x + 13 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{3315 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.77, size = 154, normalized size = 1.75 \[ -\frac {2 \left (1024 i \left (\cos ^{8}\left (d x +c \right )\right )-1024 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )+128 i \left (\cos ^{6}\left (d x +c \right )\right )-640 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+56 i \left (\cos ^{4}\left (d x +c \right )\right )-504 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1072 i \left (\cos ^{2}\left (d x +c \right )\right )+676 \cos \left (d x +c \right ) \sin \left (d x +c \right )+195 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{3}}{3315 d \cos \left (d x +c \right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 58, normalized size = 0.66 \[ -\frac {2 i \, {\left (195 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {17}{2}} - 884 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a + 1020 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a^{2}\right )}}{3315 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.66, size = 562, normalized size = 6.39 \[ -\frac {a^3\,\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}}\,4096{}\mathrm {i}}{3315\,d}-\frac {a^3\,\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}}\,2048{}\mathrm {i}}{3315\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a^3\,\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}}{1105\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {a^3\,\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}}\,56320{}\mathrm {i}}{663\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {a^3\,\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}}\,205312{}\mathrm {i}}{663\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {a^3\,\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}}\,540672{}\mathrm {i}}{1105\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {a^3\,\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}}\,1341952{}\mathrm {i}}{3315\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}+\frac {a^3\,\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}}\,44032{}\mathrm {i}}{255\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^7}-\frac {a^3\,\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}}{17\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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