Optimal. Leaf size=30 \[ \frac {\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3088, 37} \[ \frac {\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 3088
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^2}{x^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {(b+a \cot (c+d x))^3 \tan ^3(c+d x)}{3 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 46, normalized size = 1.53 \[ \frac {a^2 \tan (c+d x)}{d}+\frac {a b \tan ^2(c+d x)}{d}+\frac {b^2 \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 55, normalized size = 1.83 \[ \frac {3 \, a b \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 41, normalized size = 1.37 \[ \frac {b^{2} \tan \left (d x + c\right )^{3} + 3 \, a b \tan \left (d x + c\right )^{2} + 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 10.89, size = 48, normalized size = 1.60 \[ \frac {a^{2} \tan \left (d x +c \right )+\frac {a b}{\cos \left (d x +c \right )^{2}}+\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 45, normalized size = 1.50 \[ \frac {b^{2} \tan \left (d x + c\right )^{3} + 3 \, a^{2} \tan \left (d x + c\right ) - \frac {3 \, a b}{\sin \left (d x + c\right )^{2} - 1}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.49, size = 68, normalized size = 2.27 \[ \frac {\frac {b^2\,\sin \left (c+d\,x\right )}{3}+\frac {{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,\left (3\,a^2-b^2\right )}{3}+a\,b\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^2}{d\,{\cos \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{2} \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________