Optimal. Leaf size=101 \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {7 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \sin (c+d x)}{d}-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {7 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \sin (c+d x)}{d}-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x^4}{a^4 (a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^4}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (-1+\frac {a^3}{2 (a-x)^3}-\frac {7 a^2}{4 (a-x)^2}+\frac {17 a}{8 (a-x)}+\frac {a}{8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {a^2 \log (1+\sin (c+d x))}{8 d}-\frac {a^2 \sin (c+d x)}{d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {7 a^3}{4 d (a-a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 67, normalized size = 0.66 \[ -\frac {a^2 \left (8 \sin (c+d x)-\frac {14}{\sin (c+d x)-1}-\frac {2}{(\sin (c+d x)-1)^2}+17 \log (1-\sin (c+d x))-\log (\sin (c+d x)+1)\right )}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 154, normalized size = 1.52 \[ \frac {16 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 88, normalized size = 0.87 \[ \frac {2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 34 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 16 \, a^{2} \sin \left (d x + c\right ) + \frac {51 \, a^{2} \sin \left (d x + c\right )^{2} - 74 \, a^{2} \sin \left (d x + c\right ) + 27 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.29, size = 213, normalized size = 2.11 \[ \frac {a^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {3 a^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {3 a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {9 a^{2} \sin \left (d x +c \right )}{4 d}+\frac {9 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 83, normalized size = 0.82 \[ \frac {a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - 8 \, a^{2} \sin \left (d x + c\right ) + \frac {2 \, {\left (7 \, a^{2} \sin \left (d x + c\right ) - 6 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.27, size = 225, normalized size = 2.23 \[ \frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{4\,d}-\frac {17\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{4\,d}-\frac {\frac {9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-14\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+17\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-14\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}+\frac {2\,a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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]
________________________________________________________________________________________