Optimal. Leaf size=116 \[ -\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}-\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.103942, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4397, 2721, 894} \[ -\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}-\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4397
Rule 2721
Rule 894
Rubi steps
\begin{align*} \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx &=\int (b+a \cos (c+d x))^3 \tan ^3(c+d x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(b+x)^3 \left (a^2-x^2\right )}{x^3} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1-\frac{3 b^2}{a^2}\right )+\frac{a^2 b^3}{x^3}+\frac{3 a^2 b^2}{x^2}+\frac{3 a^2 b-b^3}{x}-3 b x-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.310056, size = 102, normalized size = 0.88 \[ \frac{-9 a \left (a^2-4 b^2\right ) \cos (c+d x)+9 a^2 b \cos (2 (c+d x))-36 a^2 b \log (\cos (c+d x))+a^3 \cos (3 (c+d x))+36 a b^2 \sec (c+d x)+6 b^3 \sec ^2(c+d x)+12 b^3 \log (\cos (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 164, normalized size = 1.4 \begin{align*} -{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{3\,d}}-{\frac{2\,{a}^{3}\cos \left ( dx+c \right ) }{3\,d}}-{\frac{3\,{a}^{2}b \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{{a}^{2}b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}+3\,{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}a{b}^{2}}{d}}+6\,{\frac{a{b}^{2}\cos \left ( dx+c \right ) }{d}}+{\frac{{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04589, size = 153, normalized size = 1.32 \begin{align*} \frac{{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{3}}{3 \, d} - \frac{3 \,{\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} a^{2} b}{2 \, d} - \frac{b^{3}{\left (\frac{1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{2 \, d} + \frac{3 \, a b^{2}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.536226, size = 300, normalized size = 2.59 \begin{align*} \frac{4 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{2} b \cos \left (d x + c\right )^{4} - 9 \, a^{2} b \cos \left (d x + c\right )^{2} + 36 \, a b^{2} \cos \left (d x + c\right ) - 12 \,{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 12 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 6 \, b^{3}}{12 \, d \cos \left (d x + c\right )^{2}} \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*} \int \left (a \sin{\left (c + d x \right )} + b \tan{\left (c + d x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]