Optimal. Leaf size=153 \[ -\frac {2 a^{3/2} (c-d) (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{5/2} f \sqrt {c+d}}+\frac {2 a^2 (-3 A d+3 B c-4 B d) \cos (e+f x)}{3 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2976, 2981, 2773, 208} \[ \frac {2 a^2 (-3 A d+3 B c-4 B d) \cos (e+f x)}{3 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^{3/2} (c-d) (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{5/2} f \sqrt {c+d}}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 2773
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx &=-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d f}+\frac {2 \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (B c+3 A d)-\frac {1}{2} a (3 B c-3 A d-4 B d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{3 d}\\ &=\frac {2 a^2 (3 B c-3 A d-4 B d) \cos (e+f x)}{3 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d f}+\frac {(a (c-d) (B c-A d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{d^2}\\ &=\frac {2 a^2 (3 B c-3 A d-4 B d) \cos (e+f x)}{3 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d f}-\frac {\left (2 a^2 (c-d) (B c-A d)\right ) \operatorname {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d^2 f}\\ &=-\frac {2 a^{3/2} (c-d) (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{5/2} \sqrt {c+d} f}+\frac {2 a^2 (3 B c-3 A d-4 B d) \cos (e+f x)}{3 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 3.54, size = 356, normalized size = 2.33 \[ \frac {(a (\sin (e+f x)+1))^{3/2} \left (6 \sqrt {d} (2 A d-2 B c+3 B d) \sin \left (\frac {1}{2} (e+f x)\right )-6 \sqrt {d} (2 A d-2 B c+3 B d) \cos \left (\frac {1}{2} (e+f x)\right )+\frac {3 (c-d) (B c-A d) \left (2 \log \left (\sqrt {d} \sqrt {c+d} \left (\tan ^2\left (\frac {1}{4} (e+f x)\right )+2 \tan \left (\frac {1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{\sqrt {c+d}}-\frac {3 (c-d) (B c-A d) \left (2 \log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )+c+d\right )\right )-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+e+f x\right )}{\sqrt {c+d}}-2 B d^{3/2} \sin \left (\frac {3}{2} (e+f x)\right )-2 B d^{3/2} \cos \left (\frac {3}{2} (e+f x)\right )\right )}{6 d^{5/2} f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.47, size = 880, normalized size = 5.75 \[ \left [-\frac {3 \, {\left (B a c^{2} - {\left (A + B\right )} a c d + A a d^{2} + {\left (B a c^{2} - {\left (A + B\right )} a c d + A a d^{2}\right )} \cos \left (f x + e\right ) + {\left (B a c^{2} - {\left (A + B\right )} a c d + A a d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{c d + d^{2}}} \log \left (\frac {a d^{2} \cos \left (f x + e\right )^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left (6 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {\frac {a}{c d + d^{2}}} - {\left (a c^{2} + 8 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (a d^{2} \cos \left (f x + e\right )^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left (3 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{3} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - c^{2} - 2 \, c d - d^{2} - {\left (c^{2} + d^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \cos \left (f x + e\right ) - c^{2} - 2 \, c d - d^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (B a d \cos \left (f x + e\right )^{2} - 3 \, B a c + {\left (3 \, A + 4 \, B\right )} a d - {\left (3 \, B a c - {\left (3 \, A + 5 \, B\right )} a d\right )} \cos \left (f x + e\right ) + {\left (B a d \cos \left (f x + e\right ) + 3 \, B a c - {\left (3 \, A + 4 \, B\right )} a d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{6 \, {\left (d^{2} f \cos \left (f x + e\right ) + d^{2} f \sin \left (f x + e\right ) + d^{2} f\right )}}, -\frac {3 \, {\left (B a c^{2} - {\left (A + B\right )} a c d + A a d^{2} + {\left (B a c^{2} - {\left (A + B\right )} a c d + A a d^{2}\right )} \cos \left (f x + e\right ) + {\left (B a c^{2} - {\left (A + B\right )} a c d + A a d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{c d + d^{2}}} \arctan \left (\frac {\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) - c - 2 \, d\right )} \sqrt {-\frac {a}{c d + d^{2}}}}{2 \, a \cos \left (f x + e\right )}\right ) + 2 \, {\left (B a d \cos \left (f x + e\right )^{2} - 3 \, B a c + {\left (3 \, A + 4 \, B\right )} a d - {\left (3 \, B a c - {\left (3 \, A + 5 \, B\right )} a d\right )} \cos \left (f x + e\right ) + {\left (B a d \cos \left (f x + e\right ) + 3 \, B a c - {\left (3 \, A + 4 \, B\right )} a d\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3 \, {\left (d^{2} f \cos \left (f x + e\right ) + d^{2} f \sin \left (f x + e\right ) + d^{2} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.73, size = 291, normalized size = 1.90 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (3 A \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a^{2} c d -3 A \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a^{2} d^{2}-3 B \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a^{2} c^{2}+3 B \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a^{2} c d +B \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, d -3 A \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\, a d +3 B \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\, a c -6 B \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\, a d \right )}{3 d^{2} \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{d \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{c+d\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________