Optimal. Leaf size=98 \[ \frac{8 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}+\frac{3 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{A \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.164468, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2966, 3770, 2650, 2648} \[ \frac{8 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}+\frac{3 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{A \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2966
Rule 3770
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{\csc (c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx &=\int \left (\frac{A \csc (c+d x)}{a^3}-\frac{2 A}{a^3 (1+\sin (c+d x))^3}-\frac{A}{a^3 (1+\sin (c+d x))^2}-\frac{A}{a^3 (1+\sin (c+d x))}\right ) \, dx\\ &=\frac{A \int \csc (c+d x) \, dx}{a^3}-\frac{A \int \frac{1}{(1+\sin (c+d x))^2} \, dx}{a^3}-\frac{A \int \frac{1}{1+\sin (c+d x)} \, dx}{a^3}-\frac{(2 A) \int \frac{1}{(1+\sin (c+d x))^3} \, dx}{a^3}\\ &=-\frac{A \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac{A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac{A \int \frac{1}{1+\sin (c+d x)} \, dx}{3 a^3}-\frac{(4 A) \int \frac{1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}\\ &=-\frac{A \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{3 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^2}+\frac{4 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}-\frac{(4 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{15 a^3}\\ &=-\frac{A \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{3 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^2}+\frac{8 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [B] time = 1.0052, size = 313, normalized size = 3.19 \[ \frac{(A-A \sin (c+d x)) \left (2 \sin \left (\frac{d x}{2}\right ) (-19 \sin (c+d x)+4 \cos (2 (c+d x))-17)+\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (3 \cos \left (\frac{c}{2}\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2-3 \sin \left (\frac{c}{2}\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2-5 \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4+5 \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4-2 \sin \left (\frac{c}{2}\right )+2 \cos \left (\frac{c}{2}\right )\right )\right )}{5 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.153, size = 130, normalized size = 1.3 \begin{align*}{\frac{16\,A}{5\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-8\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+12\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-10\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+8\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}+{\frac{A}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.01314, size = 585, normalized size = 5.97 \begin{align*} \frac{A{\left (\frac{2 \,{\left (\frac{115 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{185 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{135 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 32\right )}}{a^{3} + \frac{5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac{15 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + \frac{2 \, A{\left (\frac{20 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{30 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac{5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.06006, size = 817, normalized size = 8.34 \begin{align*} \frac{16 \, A \cos \left (d x + c\right )^{3} - 22 \, A \cos \left (d x + c\right )^{2} - 42 \, A \cos \left (d x + c\right ) - 5 \,{\left (A \cos \left (d x + c\right )^{3} + 3 \, A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) +{\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) - 4 \, A\right )} \sin \left (d x + c\right ) - 4 \, A\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 5 \,{\left (A \cos \left (d x + c\right )^{3} + 3 \, A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) +{\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) - 4 \, A\right )} \sin \left (d x + c\right ) - 4 \, A\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \,{\left (8 \, A \cos \left (d x + c\right )^{2} + 19 \, A \cos \left (d x + c\right ) - 2 \, A\right )} \sin \left (d x + c\right ) - 4 \, A}{10 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{A \left (\int - \frac{\csc{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin{\left (c + d x \right )} + 1}\, dx + \int \frac{\sin{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin{\left (c + d x \right )} + 1}\, dx\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17805, size = 134, normalized size = 1.37 \begin{align*} \frac{\frac{5 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{2 \,{\left (20 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 55 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 75 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 45 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 13 \, A\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]