Optimal. Leaf size=248 \[ -\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d (a+b \text {ArcSin}(c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \text {ArcSin}(c+d x))}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \text {ArcSin}(c+d x))}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{8 b^3 d}-\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{8 b^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.37, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4889, 12,
4729, 4807, 4731, 4491, 3384, 3380, 3383, 4719} \begin {gather*} -\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{8 b^3 d}-\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c+d x))}{b}\right )}{8 b^3 d}+\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \text {ArcSin}(c+d x))}-\frac {e^2 (c+d x)}{b^2 d (a+b \text {ArcSin}(c+d x))}-\frac {e^2 \sqrt {1-(c+d x)^2} (c+d x)^2}{2 b d (a+b \text {ArcSin}(c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4719
Rule 4729
Rule 4731
Rule 4807
Rule 4889
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \sin ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {1}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b^3 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 (a+b x)}-\frac {\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac {\left (9 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {9 e^2 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac {9 e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.51, size = 219, normalized size = 0.88 \begin {gather*} \frac {e^2 \left (-\frac {4 b^2 (c+d x)^2 \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^2}+\frac {4 b \left (-2 (c+d x)+3 (c+d x)^3\right )}{a+b \text {ArcSin}(c+d x)}+8 \left (\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+9 \left (-\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )+\cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )-\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )+\sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )\right )}{8 b^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(475\) vs.
\(2(234)=468\).
time = 0.24, size = 476, normalized size = 1.92
method | result | size |
derivativedivides | \(\frac {e^{2} \left (9 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b^{2}+9 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b^{2}+18 \arcsin \left (d x +c \right ) \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a b +18 \arcsin \left (d x +c \right ) \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a b -3 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{2}+\arcsin \left (d x +c \right ) b^{2} \left (d x +c \right )+9 \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a^{2}-\sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{2}-\cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{2}+9 \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a^{2}+\cos \left (3 \arcsin \left (d x +c \right )\right ) b^{2}-3 \sin \left (3 \arcsin \left (d x +c \right )\right ) a b -\sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+a b \left (d x +c \right )\right )}{8 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(476\) |
default | \(\frac {e^{2} \left (9 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b^{2}+9 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b^{2}+18 \arcsin \left (d x +c \right ) \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a b +18 \arcsin \left (d x +c \right ) \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a b -3 \arcsin \left (d x +c \right ) \sin \left (3 \arcsin \left (d x +c \right )\right ) b^{2}+\arcsin \left (d x +c \right ) b^{2} \left (d x +c \right )+9 \cosineIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a^{2}-\sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{2}-\cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{2}+9 \sinIntegral \left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a^{2}+\cos \left (3 \arcsin \left (d x +c \right )\right ) b^{2}-3 \sin \left (3 \arcsin \left (d x +c \right )\right ) a b -\sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+a b \left (d x +c \right )\right )}{8 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(476\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{2} \left (\int \frac {c^{2}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1641 vs.
\(2 (234) = 468\).
time = 0.63, size = 1641, normalized size = 6.62 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________