Optimal. Leaf size=120 \[ \frac{b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac{a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}+\frac{b^3 \tan ^6(c+d x)}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0979102, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 894} \[ \frac{b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac{a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}+\frac{b^3 \tan ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3088
Rule 894
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^3 \left (1+x^2\right )}{x^7} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{b^3}{x^7}+\frac{3 a b^2}{x^6}+\frac{3 a^2 b+b^3}{x^5}+\frac{a^3+3 a b^2}{x^4}+\frac{3 a^2 b}{x^3}+\frac{a^3}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a^3 \tan (c+d x)}{d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a \left (a^2+3 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{b \left (3 a^2+b^2\right ) \tan ^4(c+d x)}{4 d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}+\frac{b^3 \tan ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.372622, size = 54, normalized size = 0.45 \[ \frac{(a+b \tan (c+d x))^4 \left (a^2-4 a b \tan (c+d x)+10 b^2 \tan ^2(c+d x)+15 b^2\right )}{60 b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.13, size = 127, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( -{a}^{3} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) +{\frac{3\,{a}^{2}b}{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+3\,a{b}^{2} \left ( 1/5\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+2/15\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{b}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{12\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.18704, size = 165, normalized size = 1.38 \begin{align*} \frac{20 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + 12 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac{5 \,{\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} b^{3}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + \frac{45 \, a^{2} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.510117, size = 246, normalized size = 2.05 \begin{align*} \frac{10 \, b^{3} + 15 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (2 \,{\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 9 \, a b^{2} \cos \left (d x + c\right ) +{\left (5 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{6}} \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 = 1.2001, size = 151, normalized size = 1.26 \begin{align*} \frac{10 \, b^{3} \tan \left (d x + c\right )^{6} + 36 \, a b^{2} \tan \left (d x + c\right )^{5} + 45 \, a^{2} b \tan \left (d x + c\right )^{4} + 15 \, b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} \tan \left (d x + c\right )^{3} + 60 \, a b^{2} \tan \left (d x + c\right )^{3} + 90 \, a^{2} b \tan \left (d x + c\right )^{2} + 60 \, a^{3} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]