Optimal. Leaf size=324 \[ \frac{\left (-3 a^2 B+2 a A b+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac{\left (3 a^2 A b-4 a^3 B+6 a b^2 B-3 A b^3\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left (4 a^3 A b+9 a^2 b^2 B-5 a^4 B-6 a A b^3-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac{\left (-9 a^2 A b^3+5 a^4 A b+12 a^3 b^2 B-6 a^5 B-6 a b^4 B+3 A b^5\right ) \sin (c+d x)}{b^7 d}+\frac{\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}+\frac{\left (a^2-b^2\right )^2 \left (-7 a^2 B+6 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac{B \sin ^6(c+d x)}{6 b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.399274, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2837, 772} \[ \frac{\left (-3 a^2 B+2 a A b+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac{\left (3 a^2 A b-4 a^3 B+6 a b^2 B-3 A b^3\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left (4 a^3 A b+9 a^2 b^2 B-5 a^4 B-6 a A b^3-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac{\left (-9 a^2 A b^3+5 a^4 A b+12 a^3 b^2 B-6 a^5 B-6 a b^4 B+3 A b^5\right ) \sin (c+d x)}{b^7 d}+\frac{\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}+\frac{\left (a^2-b^2\right )^2 \left (-7 a^2 B+6 a A b+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac{B \sin ^6(c+d x)}{6 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2837
Rule 772
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (A+\frac{B x}{b}\right ) \left (b^2-x^2\right )^3}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-5 a^4 A b+9 a^2 A b^3-3 A b^5+6 a^5 B-12 a^3 b^2 B+6 a b^4 B}{b}-\frac{\left (-4 a^3 A b+6 a A b^3+5 a^4 B-9 a^2 b^2 B+3 b^4 B\right ) x}{b}+\frac{\left (-3 a^2 A b+3 A b^3+4 a^3 B-6 a b^2 B\right ) x^2}{b}+\frac{\left (2 a A b-3 a^2 B+3 b^2 B\right ) x^3}{b}-\frac{(A b-2 a B) x^4}{b}-\frac{B x^5}{b}+\frac{\left (-a^2+b^2\right )^3 (A b-a B)}{b (a+x)^2}+\frac{\left (-a^2+b^2\right )^2 \left (6 a A b-7 a^2 B+b^2 B\right )}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac{\left (a^2-b^2\right )^2 \left (6 a A b-7 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac{\left (5 a^4 A b-9 a^2 A b^3+3 A b^5-6 a^5 B+12 a^3 b^2 B-6 a b^4 B\right ) \sin (c+d x)}{b^7 d}+\frac{\left (4 a^3 A b-6 a A b^3-5 a^4 B+9 a^2 b^2 B-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac{\left (3 a^2 A b-3 A b^3-4 a^3 B+6 a b^2 B\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left (2 a A b-3 a^2 B+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac{(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac{B \sin ^6(c+d x)}{6 b^2 d}+\frac{\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.66063, size = 396, normalized size = 1.22 \[ \frac{\frac{6 (A b-a B) \left (-4 a^2 b^4 \sin ^4(c+d x)+2 a b^3 \left (5 a^2-7 b^2\right ) \sin ^3(c+d x)-2 b^2 \left (-29 a^2 b^2+15 a^4+8 b^4\right ) \sin ^2(c+d x)+4 \left (a^2-b^2\right )^2 \left (15 a^2 \log (a+b \sin (c+d x))+4 a^2-4 b^2\right )+4 a b \sin (c+d x) \left (15 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))+18 a^2 b^2-11 a^4-4 b^4\right )+b^4 \cos ^4(c+d x) \left (-a^2+3 a b \sin (c+d x)+4 b^2\right )+2 b^6 \cos ^6(c+d x)\right )}{a+b \sin (c+d x)}+B \left (20 a b^3 \left (a^2-3 b^2\right ) \sin ^3(c+d x)-30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+60 a b \left (-3 a^2 b^2+a^4+3 b^4\right ) \sin (c+d x)+15 b^4 \left (b^2-a^2\right ) \cos ^4(c+d x)-60 \left (a^2-b^2\right )^3 \log (a+b \sin (c+d x))+12 a b^5 \sin ^5(c+d x)+10 b^6 \cos ^6(c+d x)\right )}{60 b^8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.126, size = 721, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.98725, size = 509, normalized size = 1.57 \begin{align*} -\frac{\frac{60 \,{\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )}}{b^{9} \sin \left (d x + c\right ) + a b^{8}} + \frac{10 \, B b^{5} \sin \left (d x + c\right )^{6} - 12 \,{\left (2 \, B a b^{4} - A b^{5}\right )} \sin \left (d x + c\right )^{5} + 15 \,{\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4} - 3 \, B b^{5}\right )} \sin \left (d x + c\right )^{4} - 20 \,{\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - 6 \, B a b^{4} + 3 \, A b^{5}\right )} \sin \left (d x + c\right )^{3} + 30 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2} - 9 \, B a^{2} b^{3} + 6 \, A a b^{4} + 3 \, B b^{5}\right )} \sin \left (d x + c\right )^{2} - 60 \,{\left (6 \, B a^{5} - 5 \, A a^{4} b - 12 \, B a^{3} b^{2} + 9 \, A a^{2} b^{3} + 6 \, B a b^{4} - 3 \, A b^{5}\right )} \sin \left (d x + c\right )}{b^{7}} + \frac{60 \,{\left (7 \, B a^{6} - 6 \, A a^{5} b - 15 \, B a^{4} b^{2} + 12 \, A a^{3} b^{3} + 9 \, B a^{2} b^{4} - 6 \, A a b^{5} - B b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.13177, size = 1220, normalized size = 3.77 \begin{align*} -\frac{480 \, B a^{7} - 480 \, A a^{6} b - 3720 \, B a^{5} b^{2} + 3360 \, A a^{4} b^{3} + 5705 \, B a^{3} b^{4} - 4710 \, A a^{2} b^{5} - 2402 \, B a b^{6} + 1536 \, A b^{7} + 16 \,{\left (7 \, B a b^{6} - 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{6} - 8 \,{\left (35 \, B a^{3} b^{4} - 30 \, A a^{2} b^{5} - 33 \, B a b^{6} + 24 \, A b^{7}\right )} \cos \left (d x + c\right )^{4} + 16 \,{\left (105 \, B a^{5} b^{2} - 90 \, A a^{4} b^{3} - 190 \, B a^{3} b^{4} + 150 \, A a^{2} b^{5} + 81 \, B a b^{6} - 48 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 480 \,{\left (7 \, B a^{7} - 6 \, A a^{6} b - 15 \, B a^{5} b^{2} + 12 \, A a^{4} b^{3} + 9 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} - B a b^{6} +{\left (7 \, B a^{6} b - 6 \, A a^{5} b^{2} - 15 \, B a^{4} b^{3} + 12 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 6 \, A a b^{6} - B b^{7}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (80 \, B b^{7} \cos \left (d x + c\right )^{6} + 2880 \, B a^{6} b - 2400 \, A a^{5} b^{2} - 5720 \, B a^{4} b^{3} + 4320 \, A a^{3} b^{4} + 2967 \, B a^{2} b^{5} - 1626 \, A a b^{6} - 190 \, B b^{7} - 24 \,{\left (7 \, B a^{2} b^{5} - 6 \, A a b^{6} - 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 16 \,{\left (35 \, B a^{4} b^{3} - 30 \, A a^{3} b^{4} - 54 \, B a^{2} b^{5} + 42 \, A a b^{6} + 15 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \,{\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}} \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 = 1.27108, size = 770, normalized size = 2.38 \begin{align*} -\frac{\frac{60 \,{\left (7 \, B a^{6} - 6 \, A a^{5} b - 15 \, B a^{4} b^{2} + 12 \, A a^{3} b^{3} + 9 \, B a^{2} b^{4} - 6 \, A a b^{5} - B b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}} - \frac{60 \,{\left (7 \, B a^{6} b \sin \left (d x + c\right ) - 6 \, A a^{5} b^{2} \sin \left (d x + c\right ) - 15 \, B a^{4} b^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} b^{4} \sin \left (d x + c\right ) + 9 \, B a^{2} b^{5} \sin \left (d x + c\right ) - 6 \, A a b^{6} \sin \left (d x + c\right ) - B b^{7} \sin \left (d x + c\right ) + 6 \, B a^{7} - 5 \, A a^{6} b - 12 \, B a^{5} b^{2} + 9 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{8}} + \frac{10 \, B b^{10} \sin \left (d x + c\right )^{6} - 24 \, B a b^{9} \sin \left (d x + c\right )^{5} + 12 \, A b^{10} \sin \left (d x + c\right )^{5} + 45 \, B a^{2} b^{8} \sin \left (d x + c\right )^{4} - 30 \, A a b^{9} \sin \left (d x + c\right )^{4} - 45 \, B b^{10} \sin \left (d x + c\right )^{4} - 80 \, B a^{3} b^{7} \sin \left (d x + c\right )^{3} + 60 \, A a^{2} b^{8} \sin \left (d x + c\right )^{3} + 120 \, B a b^{9} \sin \left (d x + c\right )^{3} - 60 \, A b^{10} \sin \left (d x + c\right )^{3} + 150 \, B a^{4} b^{6} \sin \left (d x + c\right )^{2} - 120 \, A a^{3} b^{7} \sin \left (d x + c\right )^{2} - 270 \, B a^{2} b^{8} \sin \left (d x + c\right )^{2} + 180 \, A a b^{9} \sin \left (d x + c\right )^{2} + 90 \, B b^{10} \sin \left (d x + c\right )^{2} - 360 \, B a^{5} b^{5} \sin \left (d x + c\right ) + 300 \, A a^{4} b^{6} \sin \left (d x + c\right ) + 720 \, B a^{3} b^{7} \sin \left (d x + c\right ) - 540 \, A a^{2} b^{8} \sin \left (d x + c\right ) - 360 \, B a b^{9} \sin \left (d x + c\right ) + 180 \, A b^{10} \sin \left (d x + c\right )}{b^{12}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]