Optimal. Leaf size=69 \[ -\frac{2 a^5 B \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 B x+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.154012, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {2855, 2670, 2680, 8} \[ -\frac{2 a^5 B \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 B x+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2855
Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-(a B) \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\left (a^5 B\right ) \int \frac{\cos ^2(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac{2 a^5 B \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+\left (a^3 B\right ) \int 1 \, dx\\ &=a^3 B x+\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac{2 a^5 B \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.05802, size = 121, normalized size = 1.75 \[ -\frac{a^3 \left (-3 \cos \left (\frac{1}{2} (c+d x)\right ) (2 A+3 B (c+d x+2))+\cos \left (\frac{3}{2} (c+d x)\right ) (2 A+B (3 c+3 d x+14))+6 B \sin \left (\frac{1}{2} (c+d x)\right ) (2 (c+d x+2)+(c+d x) \cos (c+d x))\right )}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.126, size = 248, normalized size = 3.6 \begin{align*}{\frac{1}{d} \left ({a}^{3}A \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,\cos \left ( dx+c \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{3}} \right ) +B{a}^{3} \left ({\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-\tan \left ( dx+c \right ) +dx+c \right ) +{\frac{{a}^{3}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+3\,B{a}^{3} \left ( 1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}-1/3\, \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +{\frac{{a}^{3}A}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{B{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{a}^{3}A \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) +{\frac{B{a}^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.65021, size = 221, normalized size = 3.2 \begin{align*} \frac{3 \, A a^{3} \tan \left (d x + c\right )^{3} + 3 \, B a^{3} \tan \left (d x + c\right )^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} B a^{3} - \frac{{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} A a^{3}}{\cos \left (d x + c\right )^{3}} - \frac{3 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} B a^{3}}{\cos \left (d x + c\right )^{3}} + \frac{3 \, A a^{3}}{\cos \left (d x + c\right )^{3}} + \frac{B a^{3}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.60412, size = 402, normalized size = 5.83 \begin{align*} -\frac{6 \, B a^{3} d x + 2 \,{\left (A + B\right )} a^{3} -{\left (3 \, B a^{3} d x +{\left (A + 7 \, B\right )} a^{3}\right )} \cos \left (d x + c\right )^{2} +{\left (3 \, B a^{3} d x +{\left (A - 5 \, B\right )} a^{3}\right )} \cos \left (d x + c\right ) -{\left (6 \, B a^{3} d x - 2 \,{\left (A + B\right )} a^{3} +{\left (3 \, B a^{3} d x -{\left (A + 7 \, B\right )} a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\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 = 1.3302, size = 126, normalized size = 1.83 \begin{align*} \frac{3 \,{\left (d x + c\right )} B a^{3} - \frac{2 \,{\left (3 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a^{3} - 5 \, B a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]