Optimal. Leaf size=237 \[ \frac {2 a b \left (3 a^2-10 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac {6 b \left (a^2-2 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}-\frac {6 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^2}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {6 \left (a^4-4 a^2 b^2-4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.47, antiderivative size = 237, 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 = {2691, 2861, 2862, 2669, 2640, 2639} \[ \frac {2 a b \left (3 a^2-10 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac {6 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^2}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {6 b \left (a^2-2 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}-\frac {6 \left (-4 a^2 b^2+a^4-4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2640
Rule 2669
Rule 2691
Rule 2861
Rule 2862
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(a+b \sin (c+d x))^2 \left (-\frac {3 a^2}{2}+3 b^2+\frac {3}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 (a+b \sin (c+d x))^2 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {4 \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (-\frac {3}{4} a \left (a^2-6 b^2\right )-\frac {15}{4} b \left (a^2-2 b^2\right ) \sin (c+d x)\right ) \, dx}{5 e^4}\\ &=\frac {6 b \left (a^2-2 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 (a+b \sin (c+d x))^2 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {8 \int \sqrt {e \cos (c+d x)} \left (-\frac {15}{8} \left (a^4-4 a^2 b^2-4 b^4\right )-\frac {15}{8} a b \left (3 a^2-10 b^2\right ) \sin (c+d x)\right ) \, dx}{25 e^4}\\ &=\frac {2 a b \left (3 a^2-10 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac {6 b \left (a^2-2 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 (a+b \sin (c+d x))^2 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (3 \left (a^4-4 a^2 b^2-4 b^4\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^4}\\ &=\frac {2 a b \left (3 a^2-10 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac {6 b \left (a^2-2 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 (a+b \sin (c+d x))^2 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (3 \left (a^4-4 a^2 b^2-4 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^4 \sqrt {\cos (c+d x)}}\\ &=\frac {2 a b \left (3 a^2-10 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac {6 \left (a^4-4 a^2 b^2-4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {6 b \left (a^2-2 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 (a+b \sin (c+d x))^2 \left (a b-\left (a^2-2 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.63, size = 152, normalized size = 0.64 \[ \frac {2 \left (3 a^4 \sin (c+d x)-12 a^2 b^2 \sin (c+d x)+4 a b \left (a^2+b^2\right ) \sec ^2(c+d x)-3 \left (a^4-4 a^2 b^2-4 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (a^4+6 a^2 b^2+b^4\right ) \tan (c+d x) \sec (c+d x)-20 a b^3-7 b^4 \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{4} \cos \left (d x + c\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 5.96, size = 874, normalized size = 3.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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]
________________________________________________________________________________________