Optimal. Leaf size=132 \[ \frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^3 (3 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^3 (A-B)}{16 d (a \sin (c+d x)+a)}+\frac{a^2 (2 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.156748, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2836, 77, 206} \[ \frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^3 (3 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^3 (A-B)}{16 d (a \sin (c+d x)+a)}+\frac{a^2 (2 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2836
Rule 77
Rule 206
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac{a^7 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{(a-x)^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^7 \operatorname{Subst}\left (\int \left (\frac{A+B}{4 a^2 (a-x)^4}+\frac{A}{4 a^3 (a-x)^3}+\frac{3 A-B}{16 a^4 (a-x)^2}+\frac{A-B}{16 a^4 (a+x)^2}+\frac{2 A-B}{8 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2}+\frac{a^3 (3 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^3 (A-B)}{16 d (a+a \sin (c+d x))}+\frac{\left (a^3 (2 A-B)\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac{a^2 (2 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2}+\frac{a^3 (3 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^3 (A-B)}{16 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.656443, size = 90, normalized size = 0.68 \[ \frac{a^2 \left (\frac{3 B-9 A}{\sin (c+d x)-1}-\frac{3 (A-B)}{\sin (c+d x)+1}-\frac{4 (A+B)}{(\sin (c+d x)-1)^3}+6 (2 A-B) \tanh ^{-1}(\sin (c+d x))+\frac{6 A}{(\sin (c+d x)-1)^2}\right )}{48 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.115, size = 379, normalized size = 2.9 \begin{align*}{\frac{{a}^{2}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}A\sin \left ( dx+c \right ) }{16\,d}}+{\frac{{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{12\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}A}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{8\,d}}-{\frac{B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{{a}^{2}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{5\,{a}^{2}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{5\,{a}^{2}A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{B{a}^{2}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02922, size = 200, normalized size = 1.52 \begin{align*} \frac{3 \,{\left (2 \, A - B\right )} a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (2 \, A - B\right )} a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (3 \,{\left (2 \, A - B\right )} a^{2} \sin \left (d x + c\right )^{3} - 6 \,{\left (2 \, A - B\right )} a^{2} \sin \left (d x + c\right )^{2} +{\left (2 \, A - B\right )} a^{2} \sin \left (d x + c\right ) + 2 \,{\left (4 \, A + B\right )} a^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + 2 \, \sin \left (d x + c\right ) - 1}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.77586, size = 640, normalized size = 4.85 \begin{align*} -\frac{12 \,{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{2} - 8 \,{\left (A - 2 \, B\right )} a^{2} - 3 \,{\left ({\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \,{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left ({\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \,{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (3 \,{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right )^{2} - 4 \,{\left (2 \, A - B\right )} a^{2}\right )} \sin \left (d x + c\right )}{48 \,{\left (d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, 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 [A] time = 1.36955, size = 282, normalized size = 2.14 \begin{align*} \frac{6 \,{\left (2 \, A a^{2} - B a^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \,{\left (2 \, A a^{2} - B a^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{6 \,{\left (2 \, A a^{2} \sin \left (d x + c\right ) - B a^{2} \sin \left (d x + c\right ) + 3 \, A a^{2} - 2 \, B a^{2}\right )}}{\sin \left (d x + c\right ) + 1} + \frac{22 \, A a^{2} \sin \left (d x + c\right )^{3} - 11 \, B a^{2} \sin \left (d x + c\right )^{3} - 84 \, A a^{2} \sin \left (d x + c\right )^{2} + 39 \, B a^{2} \sin \left (d x + c\right )^{2} + 114 \, A a^{2} \sin \left (d x + c\right ) - 45 \, B a^{2} \sin \left (d x + c\right ) - 60 \, A a^{2} + 9 \, B a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]