Optimal. Leaf size=117 \[ \frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^7 d}-\frac {4 i (a+i a \tan (c+d x))^{9/2}}{3 a^6 d}+\frac {24 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}-\frac {16 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 117, 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} \[ \frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^7 d}-\frac {4 i (a+i a \tan (c+d x))^{9/2}}{3 a^6 d}+\frac {24 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}-\frac {16 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^3 (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{3/2}-12 a^2 (a+x)^{5/2}+6 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {16 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}-\frac {4 i (a+i a \tan (c+d x))^{9/2}}{3 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.81, size = 110, normalized size = 0.94 \[ \frac {2 i \sec ^6(c+d x) (\cos (4 (c+d x))+i \sin (4 (c+d x))) (494 i \cos (2 (c+d x))+110 \tan (c+d x)+215 \sin (3 (c+d x)) \sec (c+d x)+39 i)}{1155 a d (\tan (c+d x)-i) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 149, normalized size = 1.27 \[ \frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-1024 i \, e^{\left (11 i \, d x + 11 i \, c\right )} - 5632 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 12672 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 14784 i \, e^{\left (5 i \, d x + 5 i \, c\right )}\right )}}{1155 \, {\left (a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{8}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.37, size = 117, normalized size = 1.00 \[ -\frac {2 \left (256 i \left (\cos ^{5}\left (d x +c \right )\right )-256 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+32 i \left (\cos ^{3}\left (d x +c \right )\right )-160 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+245 i \cos \left (d x +c \right )+105 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{1155 d \cos \left (d x +c \right )^{5} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 76, normalized size = 0.65 \[ \frac {2 i \, {\left (105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 770 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 1980 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 1848 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3}\right )}}{1155 \, a^{7} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.64, size = 370, normalized size = 3.16 \[ -\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,1024{}\mathrm {i}}{1155\,a^2\,d}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{1155\,a^2\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{385\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{231\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,256{}\mathrm {i}}{33\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{11\,a^2\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{8}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________