Optimal. Leaf size=591 \[ \frac {7 a e^{9/2} \left (5 a^2-6 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{9/2} d \left (b^2-a^2\right )^{5/4}}-\frac {7 a e^{9/2} \left (5 a^2-6 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{9/2} d \left (b^2-a^2\right )^{5/4}}-\frac {7 a^2 e^5 \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 d \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {7 a^2 e^5 \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 d \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}+\frac {7 e^4 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{8 b^4 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}+\frac {7 e^3 \left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.43, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2693, 2863, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac {7 e^3 \left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {7 a e^{9/2} \left (5 a^2-6 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{9/2} d \left (b^2-a^2\right )^{5/4}}-\frac {7 a e^{9/2} \left (5 a^2-6 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{9/2} d \left (b^2-a^2\right )^{5/4}}+\frac {7 e^4 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{8 b^4 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}-\frac {7 a^2 e^5 \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 d \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {7 a^2 e^5 \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 d \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 208
Rule 298
Rule 329
Rule 2639
Rule 2640
Rule 2693
Rule 2701
Rule 2805
Rule 2807
Rule 2863
Rule 2864
Rule 2867
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx &=-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {\left (7 e^2\right ) \int \frac {(e \cos (c+d x))^{5/2} \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{6 b}\\ &=-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (7 e^4\right ) \int \frac {\sqrt {e \cos (c+d x)} \left (-2 b-\frac {5}{2} a \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{4 b^3}\\ &=-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (7 e^4\right ) \int \frac {\sqrt {e \cos (c+d x)} \left (-\frac {a b}{2}-\frac {1}{4} \left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )}\\ &=-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^4\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{16 b^4 \left (a^2-b^2\right )}+\frac {\left (7 \left (5 a^2-4 b^2\right ) e^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{16 b^4 \left (a^2-b^2\right )}\\ &=-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{32 b^5 \left (a^2-b^2\right )}-\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{32 b^5 \left (a^2-b^2\right )}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{16 b^3 \left (a^2-b^2\right ) d}+\frac {\left (7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{16 b^4 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}\\ &=\frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 b^3 \left (a^2-b^2\right ) d}+\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{32 b^5 \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{32 b^5 \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}\\ &=\frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{16 b^4 \left (a^2-b^2\right ) d}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{16 b^4 \left (a^2-b^2\right ) d}\\ &=\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}-\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}+\frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 26.83, size = 900, normalized size = 1.52 \[ \frac {\sec ^4(c+d x) \left (-\frac {5 a \cos (c+d x)}{4 b^3 (a+b \sin (c+d x))^2}+\frac {12 b^2 \cos (c+d x)-19 a^2 \cos (c+d x)}{8 b^3 \left (b^2-a^2\right ) (a+b \sin (c+d x))}+\frac {a^2 \cos (c+d x)-b^2 \cos (c+d x)}{3 b^3 (a+b \sin (c+d x))^3}\right ) (e \cos (c+d x))^{9/2}}{d}+\frac {7 \left (-\frac {\left (5 a^2-4 b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d x) b^{5/2}+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac {4 a b \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (i b \cos (c+d x)-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )+\log \left (i b \cos (c+d x)+(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )\right )}{\sqrt {b} \sqrt [4]{b^2-a^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right ) (e \cos (c+d x))^{9/2}}{16 (a-b) b^3 (a+b) d \cos ^{\frac {9}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
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 [C] time = 100.68, size = 237416, normalized size = 401.72 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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]
________________________________________________________________________________________