Optimal. Leaf size=80 \[ -\frac {4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac {i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{3 d (a-i a \tan (c+d x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 80, 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 {4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac {i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{3 d (a-i a \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \cos ^{10}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^{11}\right ) \text {Subst}\left (\int \frac {(a+x)^2}{(a-x)^6} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^{11}\right ) \text {Subst}\left (\int \left (\frac {4 a^2}{(a-x)^6}-\frac {4 a}{(a-x)^5}+\frac {1}{(a-x)^4}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {4 i a^{13}}{5 d (a-i a \tan (c+d x))^5}+\frac {i a^{12}}{d (a-i a \tan (c+d x))^4}-\frac {i a^{11}}{3 d (a-i a \tan (c+d x))^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.61, size = 55, normalized size = 0.69 \begin {gather*} \frac {a^8 (15+16 \cos (2 (c+d x))-4 i \sin (2 (c+d x))) (-i \cos (8 (c+d x))+\sin (8 (c+d x)))}{240 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. 587 vs. \(2 (70 ) = 140\).
time = 0.25, size = 588, normalized size = 7.35 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 152 vs. \(2 (64) = 128\).
time = 0.50, size = 152, normalized size = 1.90 \begin {gather*} -\frac {5 \, a^{8} \tan \left (d x + c\right )^{7} - 30 i \, a^{8} \tan \left (d x + c\right )^{6} - 77 \, a^{8} \tan \left (d x + c\right )^{5} + 110 i \, a^{8} \tan \left (d x + c\right )^{4} + 95 \, a^{8} \tan \left (d x + c\right )^{3} - 50 i \, a^{8} \tan \left (d x + c\right )^{2} - 15 \, a^{8} \tan \left (d x + c\right ) + 2 i \, a^{8}}{15 \, {\left (\tan \left (d x + c\right )^{10} + 5 \, \tan \left (d x + c\right )^{8} + 10 \, \tan \left (d x + c\right )^{6} + 10 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 48, normalized size = 0.60 \begin {gather*} \frac {-6 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.58, size = 121, normalized size = 1.51 \begin {gather*} \begin {cases} \frac {- 384 i a^{8} d^{2} e^{10 i c} e^{10 i d x} - 960 i a^{8} d^{2} e^{8 i c} e^{8 i d x} - 640 i a^{8} d^{2} e^{6 i c} e^{6 i d x}}{15360 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (\frac {a^{8} e^{10 i c}}{4} + \frac {a^{8} e^{8 i c}}{2} + \frac {a^{8} e^{6 i c}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 409 vs. \(2 (64) = 128\).
time = 1.24, size = 409, normalized size = 5.11 \begin {gather*} -\frac {6 i \, a^{8} e^{\left (38 i \, d x + 24 i \, c\right )} + 99 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} + 766 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} + 3689 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} + 12376 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} + 30667 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} + 58058 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} + 85657 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} + 99528 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} + 91377 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} + 66066 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} + 37219 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} + 5089 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} + 1126 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} + 155 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} + 10 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} + 16016 i \, a^{8} e^{\left (14 i \, d x\right )}}{240 \, {\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.48, size = 82, normalized size = 1.02 \begin {gather*} \frac {a^8\,\left (-5\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,5{}\mathrm {i}+2\right )}{15\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,5{}\mathrm {i}-10\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,10{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________