Optimal. Leaf size=109 \[ -\frac {8 i (a+i a \tan (c+d x))^7}{7 a^4 d}+\frac {3 i (a+i a \tan (c+d x))^8}{2 a^5 d}-\frac {2 i (a+i a \tan (c+d x))^9}{3 a^6 d}+\frac {i (a+i a \tan (c+d x))^{10}}{10 a^7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 109, 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 = {3568, 45}
\begin {gather*} \frac {i (a+i a \tan (c+d x))^{10}}{10 a^7 d}-\frac {2 i (a+i a \tan (c+d x))^9}{3 a^6 d}+\frac {3 i (a+i a \tan (c+d x))^8}{2 a^5 d}-\frac {8 i (a+i a \tan (c+d x))^7}{7 a^4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 (a+x)^6 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {Subst}\left (\int \left (8 a^3 (a+x)^6-12 a^2 (a+x)^7+6 a (a+x)^8-(a+x)^9\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {8 i (a+i a \tan (c+d x))^7}{7 a^4 d}+\frac {3 i (a+i a \tan (c+d x))^8}{2 a^5 d}-\frac {2 i (a+i a \tan (c+d x))^9}{3 a^6 d}+\frac {i (a+i a \tan (c+d x))^{10}}{10 a^7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.17, size = 117, normalized size = 1.07 \begin {gather*} \frac {a^3 \sec (c) \sec ^{10}(c+d x) (126 i \cos (c)+105 i \cos (c+2 d x)+105 i \cos (3 c+2 d x)-126 \sin (c)+105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+120 \sin (3 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x))}{840 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 219 vs. \(2 (93 ) = 186\).
time = 0.27, size = 220, normalized size = 2.02
method | result | size |
risch | \(\frac {128 i a^{3} \left (210 \,{\mathrm e}^{12 i \left (d x +c \right )}+252 \,{\mathrm e}^{10 i \left (d x +c \right )}+210 \,{\mathrm e}^{8 i \left (d x +c \right )}+120 \,{\mathrm e}^{6 i \left (d x +c \right )}+45 \,{\mathrm e}^{4 i \left (d x +c \right )}+10 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(91\) |
derivativedivides | \(\frac {-i a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{4}}\right )-3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+\frac {3 i a^{3}}{8 \cos \left (d x +c \right )^{8}}-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(220\) |
default | \(\frac {-i a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{4}}\right )-3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+\frac {3 i a^{3}}{8 \cos \left (d x +c \right )^{8}}-a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(220\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 108, normalized size = 0.99 \begin {gather*} -\frac {21 i \, a^{3} \tan \left (d x + c\right )^{10} + 70 \, a^{3} \tan \left (d x + c\right )^{9} + 240 \, a^{3} \tan \left (d x + c\right )^{7} - 210 i \, a^{3} \tan \left (d x + c\right )^{6} + 252 \, a^{3} \tan \left (d x + c\right )^{5} - 420 i \, a^{3} \tan \left (d x + c\right )^{4} - 315 i \, a^{3} \tan \left (d x + c\right )^{2} - 210 \, a^{3} \tan \left (d x + c\right )}{210 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 215 vs. \(2 (85) = 170\).
time = 0.37, size = 215, normalized size = 1.97 \begin {gather*} -\frac {128 \, {\left (-210 i \, a^{3} e^{\left (12 i \, d x + 12 i \, c\right )} - 252 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} - 210 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 120 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 45 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 10 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3}\right )}}{105 \, {\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i a^{3} \left (\int i \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.65, size = 108, normalized size = 0.99 \begin {gather*} -\frac {21 i \, a^{3} \tan \left (d x + c\right )^{10} + 70 \, a^{3} \tan \left (d x + c\right )^{9} + 240 \, a^{3} \tan \left (d x + c\right )^{7} - 210 i \, a^{3} \tan \left (d x + c\right )^{6} + 252 \, a^{3} \tan \left (d x + c\right )^{5} - 420 i \, a^{3} \tan \left (d x + c\right )^{4} - 315 i \, a^{3} \tan \left (d x + c\right )^{2} - 210 \, a^{3} \tan \left (d x + c\right )}{210 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.31, size = 151, normalized size = 1.39 \begin {gather*} -\frac {a^3\,\sin \left (c+d\,x\right )\,\left (-210\,{\cos \left (c+d\,x\right )}^9-{\cos \left (c+d\,x\right )}^8\,\sin \left (c+d\,x\right )\,315{}\mathrm {i}-{\cos \left (c+d\,x\right )}^6\,{\sin \left (c+d\,x\right )}^3\,420{}\mathrm {i}+252\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^5\,210{}\mathrm {i}+240\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^6+70\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^8+{\sin \left (c+d\,x\right )}^9\,21{}\mathrm {i}\right )}{210\,d\,{\cos \left (c+d\,x\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________