Optimal. Leaf size=282 \[ \frac {a \left (5 a^2-6 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac {a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^5 d}-\frac {2 a^3 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d}+\frac {x \left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right )}{16 b^7}+\frac {a \sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 b d} \]
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Rubi [A] time = 1.00, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2895, 3049, 3023, 2735, 2660, 618, 204} \[ \frac {a \left (-20 a^2 b^2+15 a^4+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {2 a^3 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d}-\frac {\left (6 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac {a \left (5 a^2-6 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{15 b^4 d}-\frac {\left (-10 a^2 b^2+8 a^4+b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^5 d}+\frac {x \left (-24 a^4 b^2+6 a^2 b^4+16 a^6+b^6\right )}{16 b^7}+\frac {a \sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}-\frac {\sin ^5(c+d x) \cos (c+d x)}{6 b d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2735
Rule 2895
Rule 3023
Rule 3049
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\int \frac {\sin ^3(c+d x) \left (6 \left (4 a^2-5 b^2\right )-a b \sin (c+d x)-5 \left (6 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{30 b^2}\\ &=-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\int \frac {\sin ^2(c+d x) \left (-15 a \left (6 a^2-7 b^2\right )+3 b \left (2 a^2-5 b^2\right ) \sin (c+d x)+24 a \left (5 a^2-6 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 b^3}\\ &=\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\int \frac {\sin (c+d x) \left (48 a^2 \left (5 a^2-6 b^2\right )-3 a b \left (10 a^2-9 b^2\right ) \sin (c+d x)-45 \left (8 a^4-10 a^2 b^2+b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 b^4}\\ &=-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\int \frac {-45 a \left (8 a^4-10 a^2 b^2+b^4\right )+3 b \left (40 a^4-42 a^2 b^2-15 b^4\right ) \sin (c+d x)+48 a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{720 b^5}\\ &=\frac {a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\int \frac {-45 a b \left (8 a^4-10 a^2 b^2+b^4\right )-45 \left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{720 b^6}\\ &=\frac {\left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}+\frac {a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\left (a^3 \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^7}\\ &=\frac {\left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}+\frac {a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac {\left (2 a^3 \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^7 d}\\ &=\frac {\left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}+\frac {a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}+\frac {\left (4 a^3 \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^7 d}\\ &=\frac {\left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}-\frac {2 a^3 \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^7 d}+\frac {a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac {\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac {a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac {\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac {a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 b d}\\ \end {align*}
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Mathematica [A] time = 2.21, size = 274, normalized size = 0.97 \[ \frac {960 a^6 c+960 a^6 d x-240 a^4 b^2 \sin (2 (c+d x))-1440 a^4 b^2 c-1440 a^4 b^2 d x+\left (60 a b^5-80 a^3 b^3\right ) \cos (3 (c+d x))+240 a^2 b^4 \sin (2 (c+d x))+30 a^2 b^4 \sin (4 (c+d x))+360 a^2 b^4 c+360 a^2 b^4 d x+120 a b \left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x)-1920 a^3 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )+12 a b^5 \cos (5 (c+d x))+15 b^6 \sin (2 (c+d x))-15 b^6 \sin (4 (c+d x))-5 b^6 \sin (6 (c+d x))+60 b^6 c+60 b^6 d x}{960 b^7 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 526, normalized size = 1.87 \[ \left [\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, a^{3} b^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 24 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} d x - 120 \, {\left (a^{5} - a^{3} b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 240 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 6 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}, \frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, a^{3} b^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 24 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} d x + 240 \, {\left (a^{5} - a^{3} b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 240 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 6 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 726, normalized size = 2.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 1501, normalized size = 5.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 14.64, size = 600, normalized size = 2.13 \[ \frac {\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,b\,d}+\frac {\sin \left (2\,c+2\,d\,x\right )}{64\,b\,d}-\frac {\sin \left (4\,c+4\,d\,x\right )}{64\,b\,d}-\frac {\sin \left (6\,c+6\,d\,x\right )}{192\,b\,d}+\frac {a\,\cos \left (3\,c+3\,d\,x\right )}{16\,b^2\,d}+\frac {a\,\cos \left (5\,c+5\,d\,x\right )}{80\,b^2\,d}-\frac {5\,a^3\,\cos \left (c+d\,x\right )}{4\,b^4\,d}+\frac {a^5\,\cos \left (c+d\,x\right )}{b^6\,d}+\frac {3\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,b^3\,d}-\frac {3\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^5\,d}+\frac {2\,a^6\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^7\,d}-\frac {a^3\,\cos \left (3\,c+3\,d\,x\right )}{12\,b^4\,d}+\frac {a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,b^3\,d}+\frac {a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,b^3\,d}-\frac {a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,b^5\,d}+\frac {a\,\cos \left (c+d\,x\right )}{8\,b^2\,d}-\frac {2\,a^3\,\mathrm {atanh}\left (\frac {2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}-a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{b^7\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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