Optimal. Leaf size=113 \[ \frac{\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac{\left (-3 a^2 B+2 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.168508, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2837, 772} \[ \frac{\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac{\left (-3 a^2 B+2 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2837
Rule 772
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (A+\frac{B x}{b}\right ) \left (b^2-x^2\right )}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-A b+2 a B}{b}-\frac{B x}{b}+\frac{\left (-a^2+b^2\right ) (A b-a B)}{b (a+x)^2}+\frac{2 a A b-3 a^2 B+b^2 B}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\left (2 a A b-3 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d}+\frac{\left (a^2-b^2\right ) (A b-a B)}{b^4 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.504945, size = 111, normalized size = 0.98 \[ \frac{\frac{B \left (b^2-a^2\right ) \log (a+b \sin (c+d x))}{b}+\left (A-\frac{a B}{b}\right ) \left (\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)\right )+a B \sin (c+d x)-\frac{1}{2} b B \sin ^2(c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.121, size = 202, normalized size = 1.8 \begin{align*} -{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{b}^{2}d}}-{\frac{A\sin \left ( dx+c \right ) }{{b}^{2}d}}+2\,{\frac{aB\sin \left ( dx+c \right ) }{d{b}^{3}}}+2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) Aa}{d{b}^{3}}}-3\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B{a}^{2}}{d{b}^{4}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B}{{b}^{2}d}}+{\frac{{a}^{2}A}{d{b}^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{A}{bd \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{B{a}^{3}}{d{b}^{4} \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{aB}{{b}^{2}d \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.976019, size = 159, normalized size = 1.41 \begin{align*} -\frac{\frac{2 \,{\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )}}{b^{5} \sin \left (d x + c\right ) + a b^{4}} + \frac{B b \sin \left (d x + c\right )^{2} - 2 \,{\left (2 \, B a - A b\right )} \sin \left (d x + c\right )}{b^{3}} + \frac{2 \,{\left (3 \, B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55436, size = 408, normalized size = 3.61 \begin{align*} -\frac{4 \, B a^{3} - 4 \, A a^{2} b - 11 \, B a b^{2} + 8 \, A b^{3} + 2 \,{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b - B a b^{2} +{\left (3 \, B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (2 \, B b^{3} \cos \left (d x + c\right )^{2} + 8 \, B a^{2} b - 4 \, A a b^{2} - B b^{3}\right )} \sin \left (d x + c\right )}{4 \,{\left (b^{5} d \sin \left (d x + c\right ) + a b^{4} 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 = 1.18877, size = 254, normalized size = 2.25 \begin{align*} -\frac{\frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}{\left (B - \frac{2 \,{\left (3 \, B a b - A b^{2}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b}\right )}}{b^{4}} - \frac{2 \,{\left (3 \, B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\frac{{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{2 \,{\left (\frac{B a^{3} b^{2}}{b \sin \left (d x + c\right ) + a} - \frac{A a^{2} b^{3}}{b \sin \left (d x + c\right ) + a} - \frac{B a b^{4}}{b \sin \left (d x + c\right ) + a} + \frac{A b^{5}}{b \sin \left (d x + c\right ) + a}\right )}}{b^{6}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]