Optimal. Leaf size=117 \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {11 a \log (1-\sin (c+d x))}{16 d}+\frac {a \log (\sin (c+d x))}{d}-\frac {5 a \log (\sin (c+d x)+1)}{16 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2836, 12, 88} \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {11 a \log (1-\sin (c+d x))}{16 d}+\frac {a \log (\sin (c+d x))}{d}-\frac {5 a \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {a}{(a-x)^3 x (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 x (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \left (\frac {1}{4 a^3 (a-x)^3}+\frac {1}{2 a^4 (a-x)^2}+\frac {11}{16 a^5 (a-x)}+\frac {1}{a^5 x}-\frac {1}{8 a^4 (a+x)^2}-\frac {5}{16 a^5 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {11 a \log (1-\sin (c+d x))}{16 d}+\frac {a \log (\sin (c+d x))}{d}-\frac {5 a \log (1+\sin (c+d x))}{16 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a^2}{8 d (a+a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 99, normalized size = 0.85 \[ \frac {a \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {a \left (-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}+\frac {3 a \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 175, normalized size = 1.50 \[ -\frac {6 \, a \cos \left (d x + c\right )^{2} - 16 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 5 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 11 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, a \sin \left (d x + c\right ) + 6 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 104, normalized size = 0.89 \[ -\frac {10 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 22 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 32 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {2 \, {\left (5 \, a \sin \left (d x + c\right ) + 7 \, a\right )}}{\sin \left (d x + c\right ) + 1} - \frac {33 \, a \sin \left (d x + c\right )^{2} - 82 \, a \sin \left (d x + c\right ) + 53 \, a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 100, normalized size = 0.85 \[ \frac {a \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a}{4 d \cos \left (d x +c \right )^{4}}+\frac {a}{2 d \cos \left (d x +c \right )^{2}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 95, normalized size = 0.81 \[ -\frac {5 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - 16 \, a \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (3 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) - 6 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 99, normalized size = 0.85 \[ \frac {a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{8}+\frac {a\,\sin \left (c+d\,x\right )}{8}-\frac {3\,a}{4}}{d\,\left ({\cos \left (c+d\,x\right )}^2+{\sin \left (c+d\,x\right )}^3-\sin \left (c+d\,x\right )\right )}-\frac {11\,a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{16\,d}-\frac {5\,a\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{16\,d} \]
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]
________________________________________________________________________________________