Optimal. Leaf size=471 \[ -\frac{a \left (-170 a^2 b^2+105 a^4+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^8 d}-\frac{\left (-20 a^2 b^2+14 a^4+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left (-61 a^2 b^2+42 a^4+16 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac{\left (-52 a^2 b^2+35 a^4+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}+\frac{\left (-86 a^2 b^2+56 a^4+27 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac{x \left (-200 a^4 b^2+90 a^2 b^4+112 a^6-5 b^6\right )}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.54977, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2896, 3047, 3049, 3023, 2735, 2660, 618, 204} \[ -\frac{a \left (-170 a^2 b^2+105 a^4+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^8 d}-\frac{\left (-20 a^2 b^2+14 a^4+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left (-61 a^2 b^2+42 a^4+16 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac{\left (-52 a^2 b^2+35 a^4+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}+\frac{\left (-86 a^2 b^2+56 a^4+27 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac{x \left (-200 a^4 b^2+90 a^2 b^4+112 a^6-5 b^6\right )}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2896
Rule 3047
Rule 3049
Rule 3023
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\int \frac{\sin ^4(c+d x) \left (60 \left (7 a^4-10 a^2 b^2+3 b^4\right )-12 a b \left (2 a^2-5 b^2\right ) \sin (c+d x)-12 \left (42 a^4-65 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{360 a^2 b^2}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^3(c+d x) \left (-144 \left (14 a^6-34 a^4 b^2+25 a^2 b^4-5 b^6\right )+12 a b \left (7 a^4-17 a^2 b^2+10 b^4\right ) \sin (c+d x)+60 \left (42 a^6-103 a^4 b^2+77 a^2 b^4-16 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 a^2 b^3 \left (a^2-b^2\right )}\\ &=\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^2(c+d x) \left (180 a \left (42 a^6-103 a^4 b^2+77 a^2 b^4-16 b^6\right )-36 a^2 b \left (14 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x)-288 a \left (35 a^6-87 a^4 b^2+67 a^2 b^4-15 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^2 b^4 \left (a^2-b^2\right )}\\ &=-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac{\int \frac{\sin (c+d x) \left (-576 a^2 \left (35 a^6-87 a^4 b^2+67 a^2 b^4-15 b^6\right )+36 a^3 b \left (70 a^4-153 a^2 b^2+83 b^4\right ) \sin (c+d x)+540 a^2 \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^2 b^5 \left (a^2-b^2\right )}\\ &=\frac{\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac{\int \frac{540 a^3 \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right )-36 a^2 b \left (280 a^6-654 a^4 b^2+449 a^2 b^4-75 b^6\right ) \sin (c+d x)-576 a^3 \left (105 a^6-275 a^4 b^2+231 a^2 b^4-61 b^6\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{8640 a^2 b^6 \left (a^2-b^2\right )}\\ &=-\frac{a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac{\int \frac{540 a^3 b \left (56 a^6-142 a^4 b^2+113 a^2 b^4-27 b^6\right )+540 a^2 \left (112 a^8-312 a^6 b^2+290 a^4 b^4-95 a^2 b^6+5 b^8\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8640 a^2 b^7 \left (a^2-b^2\right )}\\ &=-\frac{\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac{a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\left (a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^8}\\ &=-\frac{\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac{a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\left (2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^2\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^8 d}\\ &=-\frac{\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac{a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}-\frac{\left (4 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^2\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^8 d}\\ &=-\frac{\left (112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac{2 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^8 d}-\frac{a \left (105 a^4-170 a^2 b^2+61 b^4\right ) \cos (c+d x)}{15 b^7 d}+\frac{\left (56 a^4-86 a^2 b^2+27 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}-\frac{\left (35 a^4-52 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{15 a b^5 d}+\frac{\left (42 a^4-61 a^2 b^2+16 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^4 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{\left (14 a^4-20 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) \sin ^6(c+d x)}{6 b d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 8.43162, size = 462, normalized size = 0.98 \[ \frac{3840 a \left (7 a^2-2 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )-\frac{3360 a^5 b^2 \sin (2 (c+d x))-24000 a^4 b^3 c \sin (c+d x)-24000 a^4 b^3 d x \sin (c+d x)-5440 a^3 b^4 \sin (2 (c+d x))-140 a^3 b^4 \sin (4 (c+d x))+10800 a^2 b^5 c \sin (c+d x)+10800 a^2 b^5 d x \sin (c+d x)-42 a^2 b^5 \cos (5 (c+d x))+15 b \left (-1488 a^4 b^2+576 a^2 b^4+896 a^6-15 b^6\right ) \cos (c+d x)+10 \left (-79 a^2 b^5+56 a^4 b^3+18 b^7\right ) \cos (3 (c+d x))-24000 a^5 b^2 c+10800 a^3 b^4 c-24000 a^5 b^2 d x+10800 a^3 b^4 d x+13440 a^6 b c \sin (c+d x)+13440 a^6 b d x \sin (c+d x)+13440 a^7 c+13440 a^7 d x+1910 a b^6 \sin (2 (c+d x))+166 a b^6 \sin (4 (c+d x))+14 a b^6 \sin (6 (c+d x))-600 a b^6 c-600 a b^6 d x-600 b^7 c \sin (c+d x)-600 b^7 d x \sin (c+d x)+40 b^7 \cos (5 (c+d x))+5 b^7 \cos (7 (c+d x))}{a+b \sin (c+d x)}}{1920 b^8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.151, size = 1817, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.47191, size = 1898, normalized size = 4.03 \begin{align*} \text{result too large to display} \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 = 1.27245, size = 1127, normalized size = 2.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]