Optimal. Leaf size=105 \[ -\frac {\tan ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec (c+d x)}{a d}+\frac {x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2839, 2606, 194, 3473, 8} \[ -\frac {\tan ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec (c+d x)}{a d}+\frac {x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 194
Rule 2606
Rule 2839
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec (c+d x) \tan ^5(c+d x) \, dx}{a}-\frac {\int \tan ^6(c+d x) \, dx}{a}\\ &=-\frac {\tan ^5(c+d x)}{5 a d}+\frac {\int \tan ^4(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d}-\frac {\int \tan ^2(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {\sec (c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d}+\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}+\frac {\sec (c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.61, size = 191, normalized size = 1.82 \[ -\frac {\sec ^3(c+d x) \left (8 \sin (c+d x)-30 c \sin (2 (c+d x))-30 d x \sin (2 (c+d x))+\frac {89}{4} \sin (2 (c+d x))+16 \sin (3 (c+d x))-15 c \sin (4 (c+d x))-15 d x \sin (4 (c+d x))+\frac {89}{8} \sin (4 (c+d x))+\left (-90 c-90 d x+\frac {267}{4}\right ) \cos (c+d x)-16 \cos (2 (c+d x))-30 c \cos (3 (c+d x))-30 d x \cos (3 (c+d x))+\frac {89}{4} \cos (3 (c+d x))-23 \cos (4 (c+d x))-25\right )}{120 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 98, normalized size = 0.93 \[ \frac {15 \, d x \cos \left (d x + c\right )^{3} + 23 \, \cos \left (d x + c\right )^{4} - 19 \, \cos \left (d x + c\right )^{2} + {\left (15 \, d x \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 4}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 130, normalized size = 1.24 \[ \frac {\frac {120 \, {\left (d x + c\right )}}{a} + \frac {5 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {3 \, {\left (55 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 71\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.43, size = 166, normalized size = 1.58 \[ -\frac {1}{6 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {2}{5 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {11}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 318, normalized size = 3.03 \[ \frac {2 \, {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {46 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {13 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {100 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {30 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 8}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 16.14, size = 131, normalized size = 1.25 \[ \frac {x}{a}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}-\frac {92\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {16}{15}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________