Optimal. Leaf size=156 \[ -\frac {d \left (6 a b c d-2 a^2 d^2-b^2 \left (6 c^2+d^2\right )\right ) x}{2 b^3}+\frac {2 (b c-a d)^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} f}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f} \]
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Rubi [A]
time = 0.26, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2872, 3102,
2814, 2739, 632, 210} \begin {gather*} \frac {2 (b c-a d)^3 \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 f \sqrt {a^2-b^2}}-\frac {d x \left (-2 a^2 d^2+6 a b c d-\left (b^2 \left (6 c^2+d^2\right )\right )\right )}{2 b^3}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2872
Rule 3102
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^3}{a+b \sin (e+f x)} \, dx &=-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}+\frac {\int \frac {2 b c^3+a d^3-d \left (a c d-b \left (6 c^2+d^2\right )\right ) \sin (e+f x)+d^2 (5 b c-2 a d) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{2 b}\\ &=-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}+\frac {\int \frac {b \left (2 b c^3+a d^3\right )-d \left (6 a b c d-2 a^2 d^2-b^2 \left (6 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^2}\\ &=-\frac {d \left (6 a b c d-2 a^2 d^2-b^2 \left (6 c^2+d^2\right )\right ) x}{2 b^3}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}+\frac {(b c-a d)^3 \int \frac {1}{a+b \sin (e+f x)} \, dx}{b^3}\\ &=-\frac {d \left (6 a b c d-2 a^2 d^2-b^2 \left (6 c^2+d^2\right )\right ) x}{2 b^3}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}+\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^3 f}\\ &=-\frac {d \left (6 a b c d-2 a^2 d^2-b^2 \left (6 c^2+d^2\right )\right ) x}{2 b^3}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}-\frac {\left (4 (b c-a d)^3\right ) \text {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} (e+f x)\right )\right )}{b^3 f}\\ &=-\frac {d \left (6 a b c d-2 a^2 d^2-b^2 \left (6 c^2+d^2\right )\right ) x}{2 b^3}+\frac {2 (b c-a d)^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} f}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 138, normalized size = 0.88 \begin {gather*} \frac {2 d \left (-6 a b c d+2 a^2 d^2+b^2 \left (6 c^2+d^2\right )\right ) (e+f x)+\frac {8 (b c-a d)^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-4 b d^2 (3 b c-a d) \cos (e+f x)-b^2 d^3 \sin (2 (e+f x))}{4 b^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 229, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {\frac {2 d \left (\frac {\frac {d^{2} b^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (a b \,d^{2}-3 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+a b \,d^{2}-3 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{3}}+\frac {2 \left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{3} \sqrt {a^{2}-b^{2}}}}{f}\) | \(229\) |
default | \(\frac {\frac {2 d \left (\frac {\frac {d^{2} b^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (a b \,d^{2}-3 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+a b \,d^{2}-3 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{3}}+\frac {2 \left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{3} \sqrt {a^{2}-b^{2}}}}{f}\) | \(229\) |
risch | \(\frac {d^{3} x \,a^{2}}{b^{3}}-\frac {3 d^{2} x a c}{b^{2}}+\frac {3 d x \,c^{2}}{b}+\frac {d^{3} x}{2 b}+\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )} a}{2 b^{2} f}-\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} c}{2 b f}+\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )} a}{2 b^{2} f}-\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} c}{2 b f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{3} d^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2} c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a \,c^{2} d}{\sqrt {-a^{2}+b^{2}}\, f b}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c^{3}}{\sqrt {-a^{2}+b^{2}}\, f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{3} d^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2} c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a \,c^{2} d}{\sqrt {-a^{2}+b^{2}}\, f b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c^{3}}{\sqrt {-a^{2}+b^{2}}\, f}-\frac {d^{3} \sin \left (2 f x +2 e \right )}{4 b f}\) | \(733\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 581, normalized size = 3.72 \begin {gather*} \left [-\frac {{\left (a^{2} b^{2} - b^{4}\right )} d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} f x - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (3 \, {\left (a^{2} b^{2} - b^{4}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} f}, -\frac {{\left (a^{2} b^{2} - b^{4}\right )} d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} f x + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) + 2 \, {\left (3 \, {\left (a^{2} b^{2} - b^{4}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 252, normalized size = 1.62 \begin {gather*} \frac {\frac {{\left (6 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 2 \, a^{2} d^{3} + b^{2} d^{3}\right )} {\left (f x + e\right )}}{b^{3}} + \frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{3}} + \frac {2 \, {\left (b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, b c d^{2} + 2 \, a d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.77, size = 2500, normalized size = 16.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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