Optimal. Leaf size=88 \[ -\frac{2 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{5 a^3 \log (1-\cos (c+d x))}{d}-\frac{5 a^3 \log (\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.156347, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2836, 12, 77} \[ -\frac{2 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{5 a^3 \log (1-\cos (c+d x))}{d}-\frac{5 a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2836
Rule 12
Rule 77
Rubi steps
\begin{align*} \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^3(c+d x) \sec ^3(c+d x) \, dx\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{a^3 (-a+x)}{(-a-x)^2 x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \frac{-a+x}{(-a-x)^2 x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{a x^3}+\frac{3}{a^2 x^2}-\frac{5}{a^3 x}+\frac{2}{a^2 (a+x)^2}+\frac{5}{a^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac{2 a^4}{d (a-a \cos (c+d x))}+\frac{5 a^3 \log (1-\cos (c+d x))}{d}-\frac{5 a^3 \log (\cos (c+d x))}{d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.861069, size = 88, normalized size = 1. \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (2 \csc ^2\left (\frac{1}{2} (c+d x)\right )-\sec ^2(c+d x)-6 \sec (c+d x)+10 \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.074, size = 67, normalized size = 0.8 \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}-2\,{\frac{{a}^{3}}{d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+5\,{\frac{{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00001, size = 113, normalized size = 1.28 \begin{align*} \frac{10 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 10 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac{10 \, a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74009, size = 319, normalized size = 3.62 \begin{align*} \frac{10 \, a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) - a^{3} - 10 \,{\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 10 \,{\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.37479, size = 255, normalized size = 2.9 \begin{align*} \frac{10 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 10 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{2 \,{\left (a^{3} - \frac{5 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} + \frac{27 \, a^{3} + \frac{38 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{15 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]