Optimal. Leaf size=62 \[ \frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {i a \cos ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3486, 2633} \[ \frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {i a \cos ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2633
Rule 3486
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {i a \cos ^5(c+d x)}{5 d}+a \int \cos ^5(c+d x) \, dx\\ &=-\frac {i a \cos ^5(c+d x)}{5 d}-\frac {a \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac {i a \cos ^5(c+d x)}{5 d}+\frac {a \sin (c+d x)}{d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 62, normalized size = 1.00 \[ \frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {i a \cos ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 66, normalized size = 1.06 \[ \frac {{\left (-3 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 20 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 90 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 60 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i \, a\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.04, size = 220, normalized size = 3.55 \[ -\frac {{\left (135 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 90 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 135 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 90 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 45 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 45 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 12 i \, a e^{\left (8 i \, d x + 6 i \, c\right )} + 80 i \, a e^{\left (6 i \, d x + 4 i \, c\right )} + 360 i \, a e^{\left (4 i \, d x + 2 i \, c\right )} - 240 i \, a e^{\left (2 i \, d x\right )} - 20 i \, a e^{\left (-2 i \, c\right )}\right )} e^{\left (-3 i \, d x - i \, c\right )}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 47, normalized size = 0.76 \[ \frac {-\frac {i a \left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 49, normalized size = 0.79 \[ -\frac {3 i \, a \cos \left (d x + c\right )^{5} - {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.93, size = 70, normalized size = 1.13 \[ -\frac {2\,a\,\left (-\frac {75\,\sin \left (c+d\,x\right )}{16}-\frac {25\,\sin \left (3\,c+3\,d\,x\right )}{32}-\frac {3\,\sin \left (5\,c+5\,d\,x\right )}{32}+\frac {\cos \left (c+d\,x\right )\,15{}\mathrm {i}}{16}+\frac {\cos \left (3\,c+3\,d\,x\right )\,15{}\mathrm {i}}{32}+\frac {\cos \left (5\,c+5\,d\,x\right )\,3{}\mathrm {i}}{32}\right )}{15\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.44, size = 187, normalized size = 3.02 \[ \begin {cases} - \frac {\left (18432 i a d^{4} e^{9 i c} e^{5 i d x} + 122880 i a d^{4} e^{7 i c} e^{3 i d x} + 552960 i a d^{4} e^{5 i c} e^{i d x} - 368640 i a d^{4} e^{3 i c} e^{- i d x} - 30720 i a d^{4} e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{1474560 d^{5}} & \text {for}\: 1474560 d^{5} e^{4 i c} \neq 0 \\\frac {x \left (a e^{8 i c} + 4 a e^{6 i c} + 6 a e^{4 i c} + 4 a e^{2 i c} + a\right ) e^{- 3 i c}}{16} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________