Optimal. Leaf size=77 \[ \frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {a^4}{d (a-a \sin (c+d x))}-\frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 77, 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, 44} \[ \frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {a^4}{d (a-a \sin (c+d x))}-\frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2836
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {a}{(a-x)^3 x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \left (\frac {1}{a (a-x)^3}+\frac {1}{a^2 (a-x)^2}+\frac {1}{a^3 (a-x)}+\frac {1}{a^3 x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {a^3 \log (\sin (c+d x))}{d}+\frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {a^4}{d (a-a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 54, normalized size = 0.70 \[ \frac {a^3 \left (\frac {3-2 \sin (c+d x)}{(\sin (c+d x)-1)^2}-2 \log (1-\sin (c+d x))+2 \log (\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 126, normalized size = 1.64 \[ \frac {2 \, a^{3} \sin \left (d x + c\right ) - 3 \, a^{3} + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 123, normalized size = 1.60 \[ -\frac {12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - 6 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 76 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 114 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 76 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.61, size = 172, normalized size = 2.23 \[ \frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {a^{3} \sin \left (d x +c \right )}{8 d}+\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{3}}{d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {9 a^{3} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {a^{3}}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 70, normalized size = 0.91 \[ -\frac {2 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, a^{3} \sin \left (d x + c\right ) - 3 \, a^{3}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.12, size = 61, normalized size = 0.79 \[ \frac {2\,a^3\,\mathrm {atanh}\left (2\,\sin \left (c+d\,x\right )-1\right )}{d}-\frac {a^3\,\sin \left (c+d\,x\right )-\frac {3\,a^3}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^2-2\,\sin \left (c+d\,x\right )+1\right )} \]
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]
________________________________________________________________________________________