Optimal. Leaf size=235 \[ \frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^4 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2} f}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f} \]
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Rubi [A]
time = 0.48, antiderivative size = 235, 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 = {2872, 3112,
3102, 2814, 2739, 632, 210} \begin {gather*} \frac {2 (b c-a d)^4 \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 f \sqrt {a^2-b^2}}+\frac {d^2 \left (-3 a^2 d^2+12 a b c d-\left (b^2 \left (17 c^2+2 d^2\right )\right )\right ) \cos (e+f x)}{3 b^3 f}+\frac {d x \left (-2 a^3 d^3+8 a^2 b c d^2-a b^2 d \left (12 c^2+d^2\right )+4 b^3 c \left (2 c^2+d^2\right )\right )}{2 b^4}-\frac {d^3 (8 b c-3 a d) \sin (e+f x) \cos (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2872
Rule 3102
Rule 3112
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^4}{a+b \sin (e+f x)} \, dx &=-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\int \frac {(c+d \sin (e+f x)) \left (3 b c^3+2 a d^3+d \left (9 b c^2-a c d+2 b d^2\right ) \sin (e+f x)+d^2 (8 b c-3 a d) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{3 b}\\ &=-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\int \frac {3 \left (2 b^2 c^4+4 a b c d^3-a^2 d^4\right )-b d \left (a d \left (2 c^2-d^2\right )-12 b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)-2 d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{6 b^2}\\ &=\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\int \frac {3 b \left (2 b^2 c^4+4 a b c d^3-a^2 d^4\right )+3 d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{6 b^3}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {(b c-a d)^4 \int \frac {1}{a+b \sin (e+f x)} \, dx}{b^4}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\left (2 (b c-a d)^4\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^4 f}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}-\frac {\left (4 (b c-a d)^4\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^4 f}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^4 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2} f}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 203, normalized size = 0.86 \begin {gather*} \frac {-6 d \left (-8 a^2 b c d^2+2 a^3 d^3-4 b^3 c \left (2 c^2+d^2\right )+a b^2 d \left (12 c^2+d^2\right )\right ) (e+f x)+\frac {24 (b c-a d)^4 \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}}-3 b d^2 \left (-16 a b c d+4 a^2 d^2+3 b^2 \left (8 c^2+d^2\right )\right ) \cos (e+f x)+b^3 d^4 \cos (3 (e+f x))-3 b^2 d^3 (4 b c-a d) \sin (2 (e+f x))}{12 b^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 375, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{4} \sqrt {a^{2}-b^{2}}}-\frac {2 d \left (\frac {\left (\frac {1}{2} a \,b^{2} d^{3}-2 b^{3} c \,d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (a^{2} b \,d^{3}-4 a \,b^{2} c \,d^{2}+6 b^{3} c^{2} d \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 a^{2} b \,d^{3}-8 a \,b^{2} c \,d^{2}+12 b^{3} c^{2} d +2 b^{3} d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {1}{2} a \,b^{2} d^{3}+2 b^{3} c \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a^{2} b \,d^{3}-4 a \,b^{2} c \,d^{2}+6 b^{3} c^{2} d +\frac {2 b^{3} d^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {\left (2 a^{3} d^{3}-8 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +a \,b^{2} d^{3}-8 b^{3} c^{3}-4 b^{3} c \,d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}}{f}\) | \(375\) |
default | \(\frac {\frac {2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{4} \sqrt {a^{2}-b^{2}}}-\frac {2 d \left (\frac {\left (\frac {1}{2} a \,b^{2} d^{3}-2 b^{3} c \,d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (a^{2} b \,d^{3}-4 a \,b^{2} c \,d^{2}+6 b^{3} c^{2} d \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 a^{2} b \,d^{3}-8 a \,b^{2} c \,d^{2}+12 b^{3} c^{2} d +2 b^{3} d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {1}{2} a \,b^{2} d^{3}+2 b^{3} c \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a^{2} b \,d^{3}-4 a \,b^{2} c \,d^{2}+6 b^{3} c^{2} d +\frac {2 b^{3} d^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {\left (2 a^{3} d^{3}-8 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +a \,b^{2} d^{3}-8 b^{3} c^{3}-4 b^{3} c \,d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}}{f}\) | \(375\) |
risch | \(\frac {4 d^{3} x \,a^{2} c}{b^{3}}-\frac {6 d^{2} x a \,c^{2}}{b^{2}}-\frac {d^{4} {\mathrm e}^{i \left (f x +e \right )} a^{2}}{2 b^{3} f}-\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} c^{2}}{b f}-\frac {d^{4} {\mathrm e}^{-i \left (f x +e \right )} a^{2}}{2 b^{3} f}-\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} c^{2}}{b f}-\frac {3 d^{4} {\mathrm e}^{i \left (f x +e \right )}}{8 b f}-\frac {3 d^{4} {\mathrm e}^{-i \left (f x +e \right )}}{8 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 ) c^{4}}{\sqrt {-a^{2}+b^{2}}\, f}+\frac {d^{4} \sin \left (2 f x +2 e \right ) a}{4 b^{2} f}-\frac {d^{3} \sin \left (2 f x +2 e \right ) c}{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 ) c^{4}}{\sqrt {-a^{2}+b^{2}}\, f}+\frac {d^{4} \cos \left (3 f x +3 e \right )}{12 b f}-\frac {d^{4} x \,a^{3}}{b^{4}}-\frac {d^{4} x a}{2 b^{2}}+\frac {4 d x \,c^{3}}{b}+\frac {2 d^{3} x c}{b}+\frac {2 d^{3} {\mathrm e}^{i \left (f x +e \right )} a c}{b^{2} f}+\frac {2 d^{3} {\mathrm e}^{-i \left (f x +e \right )} a c}{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^{4} d^{4}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{4}}+\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^{4} d^{4}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{4}}+\frac {4 \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} c \,d^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{3}}-\frac {6 \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 ) d^{2} a^{2} c^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{2}}+\frac {4 \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 ) d a \,c^{3}}{\sqrt {-a^{2}+b^{2}}\, f b}-\frac {4 \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} c \,d^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{3}}+\frac {6 \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 ) d^{2} a^{2} c^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{2}}-\frac {4 \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 ) d a \,c^{3}}{\sqrt {-a^{2}+b^{2}}\, f b}\) | \(1050\) |
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 = 0.41, size = 812, normalized size = 3.46 \begin {gather*} \left [\frac {2 \, {\left (a^{2} b^{3} - b^{5}\right )} d^{4} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{3} d - 12 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{4} b - a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} f x - 3 \, {\left (4 \, {\left (a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 ) - 6 \, {\left (6 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{2} d^{2} - 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d^{3} + {\left (a^{4} b - b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )} f}, \frac {2 \, {\left (a^{2} b^{3} - b^{5}\right )} d^{4} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{3} d - 12 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{4} b - a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} f x - 3 \, {\left (4 \, {\left (a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 ) - 6 \, {\left (6 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{2} d^{2} - 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d^{3} + {\left (a^{4} b - b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 465 vs.
\(2 (228) = 456\).
time = 0.48, size = 465, normalized size = 1.98 \begin {gather*} \frac {\frac {3 \, {\left (8 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 8 \, a^{2} b c d^{3} + 4 \, b^{3} c d^{3} - 2 \, a^{3} d^{4} - a b^{2} d^{4}\right )} {\left (f x + e\right )}}{b^{4}} + \frac {12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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^{4}} + \frac {2 \, {\left (12 \, b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 36 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 24 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 72 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 48 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, b^{2} c^{2} d^{2} + 24 \, a b c d^{3} - 6 \, a^{2} d^{4} - 4 \, b^{2} d^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} b^{3}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.49, size = 2500, normalized size = 10.64 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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