Optimal. Leaf size=55 \[ \frac {i (a+i a \tan (c+d x))^8}{8 a^3 d}-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac {i (a+i a \tan (c+d x))^8}{8 a^3 d}-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x) (a+x)^6 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (2 a (a+x)^6-(a+x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d}+\frac {i (a+i a \tan (c+d x))^8}{8 a^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 2.09, size = 143, normalized size = 2.60 \[ \frac {a^5 \sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+14 \sin (3 c+4 d x)-14 \sin (5 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)+28 i \cos (c+2 d x)+28 i \cos (3 c+2 d x)+14 i \cos (3 c+4 d x)+14 i \cos (5 c+4 d x)-35 \sin (c)+35 i \cos (c))}{56 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.47, size = 191, normalized size = 3.47 \[ \frac {896 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} + 1792 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 2240 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 1792 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 896 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a^{5}}{7 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.45, size = 108, normalized size = 1.96 \[ -\frac {-7 i \, a^{5} \tan \left (d x + c\right )^{8} - 40 \, a^{5} \tan \left (d x + c\right )^{7} + 84 i \, a^{5} \tan \left (d x + c\right )^{6} + 56 \, a^{5} \tan \left (d x + c\right )^{5} + 70 i \, a^{5} \tan \left (d x + c\right )^{4} + 168 \, a^{5} \tan \left (d x + c\right )^{3} - 140 i \, a^{5} \tan \left (d x + c\right )^{2} - 56 \, a^{5} \tan \left (d x + c\right )}{56 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.45, size = 213, normalized size = 3.87 \[ \frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{4 \cos \left (d x +c \right )^{4}}-a^{5} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 108, normalized size = 1.96 \[ \frac {21 i \, a^{5} \tan \left (d x + c\right )^{8} + 120 \, a^{5} \tan \left (d x + c\right )^{7} - 252 i \, a^{5} \tan \left (d x + c\right )^{6} - 168 \, a^{5} \tan \left (d x + c\right )^{5} - 210 i \, a^{5} \tan \left (d x + c\right )^{4} - 504 \, a^{5} \tan \left (d x + c\right )^{3} + 420 i \, a^{5} \tan \left (d x + c\right )^{2} + 168 \, a^{5} \tan \left (d x + c\right )}{168 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.30, size = 151, normalized size = 2.75 \[ \frac {a^5\,\sin \left (c+d\,x\right )\,\left (56\,{\cos \left (c+d\,x\right )}^7+{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )\,140{}\mathrm {i}-168\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3\,70{}\mathrm {i}-56\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^5\,84{}\mathrm {i}+40\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^6+{\sin \left (c+d\,x\right )}^7\,7{}\mathrm {i}\right )}{56\,d\,{\cos \left (c+d\,x\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ i a^{5} \left (\int \left (- i \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________