Optimal. Leaf size=143 \[ \frac {a^3 \left (2 c^2-6 c d+7 d^2\right ) x}{2 d^3}-\frac {2 a^3 (c-d)^3 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \sqrt {c^2-d^2} f}+\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f} \]
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Rubi [A]
time = 0.27, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2842, 3047,
3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {2 a^3 (c-d)^3 \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \sqrt {c^2-d^2}}+\frac {a^3 x \left (2 c^2-6 c d+7 d^2\right )}{2 d^3}+\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2842
Rule 3047
Rule 3102
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{c+d \sin (e+f x)} \, dx &=-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}+\frac {\int \frac {(a+a \sin (e+f x)) \left (a^2 (c+2 d)-a^2 (2 c-5 d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{2 d}\\ &=-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}+\frac {\int \frac {a^3 (c+2 d)+\left (-a^3 (2 c-5 d)+a^3 (c+2 d)\right ) \sin (e+f x)-a^3 (2 c-5 d) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{2 d}\\ &=\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}+\frac {\int \frac {a^3 d (c+2 d)+a^3 \left (2 c^2-6 c d+7 d^2\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2}\\ &=\frac {a^3 \left (2 c^2-6 c d+7 d^2\right ) x}{2 d^3}+\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}-\frac {\left (a^3 (c-d)^3\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^3}\\ &=\frac {a^3 \left (2 c^2-6 c d+7 d^2\right ) x}{2 d^3}+\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}-\frac {\left (2 a^3 (c-d)^3\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 f}\\ &=\frac {a^3 \left (2 c^2-6 c d+7 d^2\right ) x}{2 d^3}+\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}+\frac {\left (4 a^3 (c-d)^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 f}\\ &=\frac {a^3 \left (2 c^2-6 c d+7 d^2\right ) x}{2 d^3}-\frac {2 a^3 (c-d)^3 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \sqrt {c^2-d^2} f}+\frac {a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac {\cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d f}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 162, normalized size = 1.13 \begin {gather*} \frac {a^3 (1+\sin (e+f x))^3 \left (-8 (c-d)^3 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )+\sqrt {c^2-d^2} \left (2 \left (2 c^2-6 c d+7 d^2\right ) (e+f x)+4 (c-3 d) d \cos (e+f x)-d^2 \sin (2 (e+f x))\right )\right )}{4 d^3 \sqrt {c^2-d^2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 184, normalized size = 1.29
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (\frac {\frac {\frac {d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (c d -3 d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+c d -3 d^{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (2 c^{2}-6 c d +7 d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}}{d^{3}}+\frac {\left (-c^{3}+3 c^{2} d -3 c \,d^{2}+d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{d^{3} \sqrt {c^{2}-d^{2}}}\right )}{f}\) | \(184\) |
default | \(\frac {2 a^{3} \left (\frac {\frac {\frac {d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (c d -3 d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+c d -3 d^{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (2 c^{2}-6 c d +7 d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}}{d^{3}}+\frac {\left (-c^{3}+3 c^{2} d -3 c \,d^{2}+d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{d^{3} \sqrt {c^{2}-d^{2}}}\right )}{f}\) | \(184\) |
risch | \(\frac {a^{3} x \,c^{2}}{d^{3}}-\frac {3 a^{3} x c}{d^{2}}+\frac {7 a^{3} x}{2 d}+\frac {a^{3} {\mathrm e}^{i \left (f x +e \right )} c}{2 d^{2} f}-\frac {3 a^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 d f}+\frac {a^{3} {\mathrm e}^{-i \left (f x +e \right )} c}{2 d^{2} f}-\frac {3 a^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 d f}+\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right ) f \,d^{3}}-\frac {2 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right ) f \,d^{2}}+\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{\left (c +d \right ) f d}-\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right ) f \,d^{3}}+\frac {2 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right ) f \,d^{2}}-\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{\left (c +d \right ) f d}-\frac {a^{3} \sin \left (2 f x +2 e \right )}{4 d f}\) | \(503\) |
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 = 0.39, size = 419, normalized size = 2.93 \begin {gather*} \left [-\frac {a^{3} d^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} f x - {\left (a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right )} \sqrt {-\frac {c - d}{c + d}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 2 \, {\left (a^{3} c d - 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )}{2 \, d^{3} f}, -\frac {a^{3} d^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} f x - 2 \, {\left (a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right )} \sqrt {\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (c \sin \left (f x + e\right ) + d\right )} \sqrt {\frac {c - d}{c + d}}}{{\left (c - d\right )} \cos \left (f x + e\right )}\right ) - 2 \, {\left (a^{3} c d - 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )}{2 \, d^{3} 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 = 0.49, size = 239, normalized size = 1.67 \begin {gather*} \frac {\frac {{\left (2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} {\left (f x + e\right )}}{d^{3}} - \frac {4 \, {\left (a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} 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 (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{\sqrt {c^{2} - d^{2}} d^{3}} + \frac {2 \, {\left (a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c - 6 \, a^{3} d\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} d^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.22, size = 2500, normalized size = 17.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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