Optimal. Leaf size=101 \[ -\frac {a \cos (c+d x)}{d}+\frac {5 a \tan ^3(c+d x)}{6 d}-\frac {5 a \tan (c+d x)}{2 d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {2 a \sec (c+d x)}{d}-\frac {a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {5 a x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2838, 2590, 270, 2591, 288, 302, 203} \[ -\frac {a \cos (c+d x)}{d}+\frac {5 a \tan ^3(c+d x)}{6 d}-\frac {5 a \tan (c+d x)}{2 d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {2 a \sec (c+d x)}{d}-\frac {a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {5 a x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 270
Rule 288
Rule 302
Rule 2590
Rule 2591
Rule 2838
Rubi steps
\begin {align*} \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx &=a \int \sin (c+d x) \tan ^4(c+d x) \, dx+a \int \sin ^2(c+d x) \tan ^4(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}-\frac {a \operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {a \cos (c+d x)}{d}-\frac {2 a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {(5 a) \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {a \cos (c+d x)}{d}-\frac {2 a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {5 a \tan (c+d x)}{2 d}+\frac {5 a \tan ^3(c+d x)}{6 d}-\frac {a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {5 a x}{2}-\frac {a \cos (c+d x)}{d}-\frac {2 a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {5 a \tan (c+d x)}{2 d}+\frac {5 a \tan ^3(c+d x)}{6 d}-\frac {a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 76, normalized size = 0.75 \[ \frac {a \left (-3 \sin (2 (c+d x))-12 \cos (c+d x)-28 \tan (c+d x)+4 \sec ^3(c+d x)-24 \sec (c+d x)+4 \tan (c+d x) \sec ^2(c+d x)+30 c+30 d x\right )}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 98, normalized size = 0.97 \[ \frac {3 \, a \cos \left (d x + c\right )^{4} - 15 \, a d x \cos \left (d x + c\right ) + 17 \, a \cos \left (d x + c\right )^{2} + {\left (15 \, a d x \cos \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right ) - 4 \, a}{6 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 134, normalized size = 1.33 \[ \frac {15 \, {\left (d x + c\right )} a + \frac {3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} + \frac {6 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 23 \, a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.46, size = 154, normalized size = 1.52 \[ \frac {a \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+a \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 87, normalized size = 0.86 \[ \frac {{\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a - 2 \, a {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 14.62, size = 243, normalized size = 2.41 \[ \frac {5\,a\,x}{2}-\frac {\left (\frac {a\,\left (30\,d\,x-30\right )}{6}-5\,a\,d\,x\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (5\,a\,d\,x-\frac {a\,\left (30\,d\,x-60\right )}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {a\,\left (30\,d\,x-50\right )}{6}-5\,a\,d\,x\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\left (5\,a\,d\,x-\frac {a\,\left (30\,d\,x-14\right )}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {a\,\left (30\,d\,x-4\right )}{6}-5\,a\,d\,x\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (5\,a\,d\,x-\frac {a\,\left (30\,d\,x-34\right )}{6}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a\,\left (15\,d\,x-32\right )}{6}-\frac {5\,a\,d\,x}{2}}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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]
________________________________________________________________________________________