Optimal. Leaf size=98 \[ \frac {2 \sin (c+d x)}{3 a d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}-\frac {b \cos (c+d x)-a \sin (c+d x)}{3 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3076, 3075} \[ \frac {2 \sin (c+d x)}{3 a d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}-\frac {b \cos (c+d x)-a \sin (c+d x)}{3 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3075
Rule 3076
Rubi steps
\begin {align*} \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx &=-\frac {b \cos (c+d x)-a \sin (c+d x)}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {2 \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx}{3 \left (a^2+b^2\right )}\\ &=-\frac {b \cos (c+d x)-a \sin (c+d x)}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {2 \sin (c+d x)}{3 a \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 85, normalized size = 0.87 \[ \frac {\sin (c+d x) \left (\left (a^2-b^2\right ) \cos (2 (c+d x))+2 a^2+b^2\right )-a b \cos (3 (c+d x))}{3 a d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.67, size = 217, normalized size = 2.21 \[ -\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right ) - {\left (a^{3} + 3 \, a b^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, {\left ({\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 50, normalized size = 0.51 \[ -\frac {3 \, b^{2} \tan \left (d x + c\right )^{2} + 3 \, a b \tan \left (d x + c\right ) + a^{2} + b^{2}}{3 \, {\left (b \tan \left (d x + c\right ) + a\right )}^{3} b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.56, size = 64, normalized size = 0.65 \[ \frac {\frac {a}{b^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{2}+b^{2}}{3 b^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {1}{b^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 85, normalized size = 0.87 \[ -\frac {3 \, b^{2} \tan \left (d x + c\right )^{2} + 3 \, a b \tan \left (d x + c\right ) + a^{2} + b^{2}}{3 \, {\left (b^{6} \tan \left (d x + c\right )^{3} + 3 \, a b^{5} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b^{4} \tan \left (d x + c\right ) + a^{3} b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.12, size = 222, normalized size = 2.27 \[ \frac {\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^2-2\,b^2\right )}{3\,a^3}+\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^2}-\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{a^2}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (12\,a\,b^2-3\,a^3\right )-a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (12\,a\,b^2-3\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^2\,b-8\,b^3\right )+a^3+6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________