Optimal. Leaf size=173 \[ -\frac {3 a \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \sqrt {a^2-b^2}}+\frac {3 x \left (4 a^2-b^2\right )}{2 b^5}+\frac {3 \cos (c+d x) \left (4 a^2+2 a b \sin (c+d x)-b^2\right )}{2 b^4 d (a+b \sin (c+d x))}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2863, 2735, 2660, 618, 204} \[ -\frac {3 a \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \sqrt {a^2-b^2}}+\frac {3 \cos (c+d x) \left (4 a^2+2 a b \sin (c+d x)-b^2\right )}{2 b^4 d (a+b \sin (c+d x))}+\frac {3 x \left (4 a^2-b^2\right )}{2 b^5}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 2660
Rule 2735
Rule 2863
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}-\frac {3 \int \frac {\cos ^2(c+d x) (-2 b-4 a \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx}{4 b^2}\\ &=\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \cos (c+d x) \left (4 a^2-b^2+2 a b \sin (c+d x)\right )}{2 b^4 d (a+b \sin (c+d x))}+\frac {3 \int \frac {4 a b+2 \left (4 a^2-b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{4 b^4}\\ &=\frac {3 \left (4 a^2-b^2\right ) x}{2 b^5}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \cos (c+d x) \left (4 a^2-b^2+2 a b \sin (c+d x)\right )}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (3 \left (-4 a b^2+2 a \left (4 a^2-b^2\right )\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{4 b^5}\\ &=\frac {3 \left (4 a^2-b^2\right ) x}{2 b^5}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \cos (c+d x) \left (4 a^2-b^2+2 a b \sin (c+d x)\right )}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (3 a \left (4 a^2-3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d}\\ &=\frac {3 \left (4 a^2-b^2\right ) x}{2 b^5}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \cos (c+d x) \left (4 a^2-b^2+2 a b \sin (c+d x)\right )}{2 b^4 d (a+b \sin (c+d x))}+\frac {\left (6 a \left (4 a^2-3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d}\\ &=\frac {3 \left (4 a^2-b^2\right ) x}{2 b^5}-\frac {3 a \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \sqrt {a^2-b^2} d}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \cos (c+d x) \left (4 a^2-b^2+2 a b \sin (c+d x)\right )}{2 b^4 d (a+b \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.46, size = 274, normalized size = 1.58 \[ \frac {\frac {96 a^4 c+96 a^4 d x+192 a^3 b c \sin (c+d x)+192 a^3 b d x \sin (c+d x)+96 a^3 b \cos (c+d x)+72 a^2 b^2 \sin (2 (c+d x))+12 b^2 \left (b^2-4 a^2\right ) (c+d x) \cos (2 (c+d x))+24 a^2 b^2 c+24 a^2 b^2 d x-48 a b^3 c \sin (c+d x)-48 a b^3 d x \sin (c+d x)-8 a b^3 \cos (3 (c+d x))-10 b^4 \sin (2 (c+d x))+b^4 \sin (4 (c+d x))-12 b^4 c-12 b^4 d x}{(a+b \sin (c+d x))^2}-\frac {48 a \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}}{16 b^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.80, size = 837, normalized size = 4.84 \[ \left [\frac {6 \, {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 8 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (4 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + b^{6}\right )} d x - 3 \, {\left (4 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4} - {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 6 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x + 3 \, {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{3} b^{6} - a b^{8}\right )} d \sin \left (d x + c\right ) - {\left (a^{4} b^{5} - b^{9}\right )} d\right )}}, \frac {3 \, {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + b^{6}\right )} d x - 3 \, {\left (4 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4} - {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 3 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x + 3 \, {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{3} b^{6} - a b^{8}\right )} d \sin \left (d x + c\right ) - {\left (a^{4} b^{5} - b^{9}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 429, normalized size = 2.48 \[ \frac {\frac {3 \, {\left (4 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {2 \, {\left (6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 15 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 54 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 45 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 29 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a^{4} - a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a b^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.54, size = 639, normalized size = 3.69 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6 a}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {12 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{5}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{3}}+\frac {5 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{3} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}+\frac {6 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{4} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}+\frac {11 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,b^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2} a}+\frac {19 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{3} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}+\frac {6 a^{3}}{d \,b^{4} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}-\frac {a}{d \,b^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )^{2}}-\frac {12 a^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{5} \sqrt {a^{2}-b^{2}}}+\frac {9 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{3} \sqrt {a^{2}-b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.41, size = 1743, normalized size = 10.08 \[ \text {result too large to display} \]
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]
________________________________________________________________________________________